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In this paper, a conservative numerical method for the Vlasov–Darwin system is proposed. The Darwin model was assumed to be valid only in the Coulomb gauge, but recently this model has also been extended to the Lorenz gauge [1]. The Darwin model, based on the Lorenz gauge, exhibits a good symmetry between scalar and vector potentials, making the proofs of physical constraints relatively easy. In addition, the total energy was believed to be one of the conservative quantities of the Vlasov–Darwin system. However, the improved theory suggests that the Hamiltonian is the conservative quantity rather than the total energy. The structure-preserving scheme proposed in this paper exactly maintains the Lorenz gauge and the conservation laws of charge, canonical momentum, and Hamiltonian in discrete form. • The first Vlasov-Darwin scheme which exactly conserves the charge, canonical momentum, and Hamiltonian. • The proposed scheme is based on a modified Darwin model that was derived in the previous study. • The Lorenz gauge revealed that the Hamiltonian is the conservative quantity of this system rather than the total energy. [ABSTRACT FROM AUTHOR]

High-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method has become an attractive method for simulating multi-scale flows because of its high accuracy and low dissipation. However, simulating flows with strong shocks is still challenging for FR/CPR. Recently, a priori subcell limiting based on compact nonlinear nonuniform weighted (CNNW) schemes was proposed for the CPR method to solve the Euler equations, which robustly simulates strong shocks and has a good balance between high resolution and shock-capturing robustness. The CPR method with subcell CNNW limiting is a hybrid scheme also called the hybrid CPR-CNNW scheme (abbr. HCCS). This paper extends HCCS to solve compressible Navier-Stokes equations on curvilinear grids. To improve the capability of HCCS in simulating hypersonic flows, a viscous-term subcell limiting (VSL) approach is proposed, which emphasizes the importance of introducing nonlinear mechanisms to the viscous term for capturing shocks. The main idea of the VSL is to replace the unified high-order polynomial distribution of physical variables or fluxes by a limited piecewise polynomial distribution for the viscous-term discretization in troubled cells. In this paper, the VSL is implemented by decomposing the troubled cells into subcells and limiting the distribution of physical variables for each subcell. To make the approach efficient, the VSL reuses the limited results used in the inviscid-term discretization. Moreover, HCCS with the VSL is extended to curvilinear grids. Discrete grid metrics and discrete Jacobian are designed to make HCCS with the VSL satisfy discrete conservation laws and discrete geometric conservation laws and fulfill free-stream preservation, which are validated by two test cases on curvilinear grids. It is shown that HCCS without the VSL blows up in simulating some viscous hypersonic flows. Numerical results on several typical cases show that HCCS with the VSL has obvious superiority in high resolution, shock-capturing robustness,and accurate prediction of surface pressure, skin friction, and heat transfer. • A viscous-term subcell limiting (VSL) approach is proposed for high-order FR/CPR method. • It is shown that the VSL improves ability of the FR/CPR method in simulating hypersonic flows. • Discrete conservation laws, discrete GCL and fulfilling free-stream preservation are satisfied. • Numerical results shows accurate prediction of pressure, skin friction and heat transfer. [ABSTRACT FROM AUTHOR]

Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. To capture the discontinuities in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy and versatility. • This paper presents a NN approach for solving parametric elliptic interface equations. • IONet is capable of approximating operators with discontinuities in both input functions and output solutions. • IONet is an accessible mesh-free surrogate model. • IONet solves parametric interface problems with multi inputs, high-contrast coefficients and irregular geometries. [ABSTRACT FROM AUTHOR]

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art e N prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis. • General very high-order finite difference method which can be tuned to high-order accuracy with sparsity. • Adaptable numerical method to both global and local eigenvalue problem for discrete modes in linear stability theory. • Unified approach capable of computing discrete and continuous normal modes essential for bi-orthogonal decomposition. • Direct discretization on the non-uniform grid to achieve high-order accuracy while maintaining stencil stability. [ABSTRACT FROM AUTHOR]

For multi-scale problems, the conventional physics-informed neural networks (PINNs) face some challenges in obtaining available predictions. In this paper, based on PINNs, we propose a practical deep learning framework for multi-scale problems by reconstructing the loss function and associating it with specialized neural network architectures. New PINN methods derived from the improved PINN framework differ from the conventional PINN method mainly in two aspects. First, the new methods use a novel loss function by modifying the standard loss function through a (grouping) regularization strategy. The regularization strategy implements a different power operation on each loss term so that all loss terms composing the loss function are of approximately the same order of magnitude, which makes all loss terms be optimized synchronously during the optimization process. Second, for the multi-frequency or high-frequency problems, in addition to using the modified loss function, new methods upgrade the neural network architecture from the common fully-connected neural network to specialized network architectures such as the Fourier feature architecture given in Ref. [1] and the integrated architecture developed by us. The combination of the above two techniques leads to a significant improvement in the computational accuracy of multi-scale problems. Several challenging numerical examples demonstrate the effectiveness of the proposed methods. The proposed methods not only significantly outperform the conventional PINN method in terms of computational efficiency and computational accuracy, but also compare favorably with the state-of-the-art methods in the recent literature. The improved PINN framework facilitates better application of PINNs to multi-scale problems. The data and code accompanying this paper are available at https://github.com/wangyong1301108/MMPINN. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic variables at each macroscale point are defined and the resulting multicontinuum equations are formulated. In this paper, we propose a general formulation and associated ingredients that allow performing multicontinuum homogenization. Our derivation consists of several main parts. In the first part, we propose a general expansion, where the solution is expressed via the product of multiple macro variables and associated cell problems. The second part consists of formulating the cell problems. The cell problems are formulated as saddle point problems with constraints for each continua. Defining the continua via test functions, we set the constraints as an integral representation. Finally, substituting the expansion to the original system, we obtain multicontinuum systems. We present an application to the mixed formulation of elliptic equations. This is a challenging system as the system does not have symmetry. We discuss the local problems and various macroscale representations for the solution and its gradient. Using various order approximations, one can obtain different systems of equations. We discuss the applicability of multicontinuum homogenization and relate this to high contrast in the cell problem. Numerical results are presented. • We derive multicontinuum methods for general systems. • We discuss the application of the framework. • We discuss cell solutions. • We consider applications to several equations, including systems and non-symmetric problems. • We present numerical implementation for a mixed formulation. [ABSTRACT FROM AUTHOR]

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions. • Three/four-stage SDIRK methods with controllable numerical dissipation. • Algebraically stable for time-dependent nonlinear simulations. • Embedded formulation with accurate error estimation. • Applications to both first- and second-order time-dependent problems. [ABSTRACT FROM AUTHOR]

A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this paper. The discretization scheme is based on the filtered spherical harmonics (F P N) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. In order to keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo (IMC) linearization method [7] in the construction of the UGKS numerical boundary fluxes. This is an essential point in the construction. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the F P N angular discretization, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the P N reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme, called the P P F P N -based UGKS, is obtained. Furthermore, we can show that in the regime ϵ ≪ 1 and the regime ϵ = O (1) , the second-order fluxes can be simplified. And, a simplified spatial second-order scheme, called the P P F P N -based SUGKS, is thus presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preserving. By employing the F P N method for the angular variable, the proposed schemes are almost free of ray effects. Moreover, the above-mentioned way of imposing the positivity would not destroy both AP and second-order accuracy properties. To our best knowledge, this is the first time that spatial second-order, positive, asymptotic preserving and almost free of ray effects schemes are constructed for the nonlinear radiative transfer equations without operator splitting. Therefore, this paper improves our previous work on the first-order scheme [42] which could not be directly extended to high order, while keeping the solution positive. Various numerical experiments are included to validate the properties of the proposed schemes. • A spatial second-order FPN scheme with both AP and PP properties is developed for nonlinear radiative transfer equations. • The scheme is almost free of ray effects, and meanwhile can reduce the Gibbs phenomena in the PN approximation. • The IMC linearization method is used in the construction of the UGKS numerical fluxes to make the solution positive. • A simplified scheme with all properties of the non-simplified one is proposed in regimes ϵ ≪ 1 and ϵ = O (1) to reduce the computational costs. • Numerical experiments have validated the spatial second-order accuracy, AP, PP and almost ray effects free properties. [ABSTRACT FROM AUTHOR]

• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a hybrid LBM-FVM solver for two-phase fluid flow simulations in which interface dynamics is modeled by a conservative phase-field equation. Integrating fluid equations over time is achieved through a velocity-based lattice Boltzmann solver which is improved by a central-moment multiple-relaxation-time collision model to reach higher accuracy. For interface evolution, we propose a finite-volume-based numerical treatment for the integration of the phase-field equation: we show that the second-order isotropic centered stencils for diffusive and separation fluxes combined with the WENO-5 stencils for advective fluxes achieve similar and sometimes even higher accuracy than the state-of-the-art double-distribution-function LBM methods as well as the DUGKS-based method, while requiring less computations and a smaller amount of memory. Benchmark tests (such as the 2D diagonal translation of a circular interface), along with quantitative evaluations on more complex tests (such as the rising bubble and Rayleigh-Taylor instability simulations) allowing comparisons with prior numerical methods and/or experimental data, are presented to validate the advantage of our hybrid solver. Moreover, 3D simulations (including a dam break simulation) are also compared to the time-lapse photography of physical experiments in order to allow for more qualitative evaluations. • This paper proposes a new hybrid LBM-FVM solver to simulate two-phase flows which reduces memory consumption and improves computational accuracy and efficiency. • The momentum equation is solved by a set of lattice Boltzmann equations with a velocity-based high-order CM-MRT model, while the phase-field equation is solved by a WENO-based finite-volume approach. • Our solver is validated through benchmark tests, comparisons, and validation examples, both quantitatively and qualitatively. • Our massively-parallel implementation on GPU offers efficient simulation of two-phase flows for a low memory footprint. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the fully conservative, high-order numerical method previously developed by Johnson and Kercher (2020) [14] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. (2020) [16] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients, including a simple linear-scaling limiter, required for the discrete solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee discrete satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, advection of a low-density Gaussian wave, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that the formulation can achieve optimal high-order convergence in smooth flows. Furthermore, we find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case. Finally, mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

In this paper we present a scalable algorithm for fault-tolerant quantum computers for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities, where the object walls are aligned with the Cartesian grid, the relative difference of velocities in each dimension is bounded by 1 and the total simulated time is dependent on the discrete velocities chosen. We provide detailed descriptions and complexity analyses of all steps of our quantum transport method (QTM) and present numerical results for 2D flows generated in Qiskit as a proof of concept. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight of this paper we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution consists of a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct particle flow behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe quantum algorithm for the transport equation that can be used for a multitude of flow configurations and leads to physically correct behavior. We finally show that our approach only requires O (n w n g 2 + d n t v n v max 2) CNOT gates, which is quadratic in the amount of qubits necessary to encode the grid and the amount of qubits necessary to encode the discrete velocities in a single spatial dimension. This complexity result makes our approach superior to state-of-the-art approaches known in the literature. • Quantum algorithm for the collisionless Boltzmann equation. • Efficient quantum primitive for streaming and reflection. • Fail-safe specular reflection operation. • Detailed complexity analysis in terms of natively implementable two-qubit gates. • Quantum computational fluid method focused on near-term implementability. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the acceleration of numerical simulations. We focus on a hypersonic planetary reentry problem whose simulation involves coupling fluid dynamics and chemical reactions. Simulating chemical reactions takes most of the computational time but, on the other hand, cannot be avoided to obtain accurate predictions. We face a trade-off between cost-efficiency and accuracy: the numerical scheme has to be sufficiently efficient to be used in an operational context but accurate enough to predict the phenomenon faithfully. To tackle this trade-off, we design a hybrid numerical scheme coupling a traditional fluid dynamic solver with a neural network approximating the chemical reactions. We rely on their power in terms of accuracy and dimension reduction when applied in a big data context and on their efficiency stemming from their matrix-vector structure to achieve important acceleration factors (×10 to ×18.6). This paper aims to explain how we design such cost-effective hybrid numerical schemes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these schemes as references. • Deep Learning-based hybridization speeds up numerical schemes of atmospheric reentry while maintaining high accuracy. • Initializing a scheme with a hybrid code's prediction reduces the convergence time and keeps the exact same guarantees. • Uncertainty analysis provides statistical guarantees concerning approximation errors when using hybridization code. • Neural network approximation error is statistically lower than many other sources of error inherent to numerical simulations. [ABSTRACT FROM AUTHOR]

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n , the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at t n + 1 / 2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature. • This paper proposes a novel and efficient numerical scheme for the Kohn-Sham gradient flow based model. • The scheme is an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme. • The scheme does not modify the original energy, allows large time step sizes, and solves small systems at each time step. • The rigorous proof is presented in the paper. • Several numerical examples are illustrated to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations. • Exponential Parareal notably reduces time-to-solution for non-diffusive equations. • Linear analysis accurately predicts Parareal performance on nonlinear problems. • Repartitioning is essential for stabilizing exponential integrators within Parareal. [ABSTRACT FROM AUTHOR]

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z -transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source. • A new method to track the propagation of transient high-frequency wave energy. • The method can handle a wide range of geometries, including multi-domain problems. • A modified convolution quadrature approach achieves superconvergence. • The method shows potential for extension to vectorial wave problems and from 2D to 3D. [ABSTRACT FROM AUTHOR]

In the realm of steady-state solutions of Euler equations, the challenge of achieving convergence of residue close to machine zero has long plagued high-order shock-capturing schemes, particularly in the presence of strong shock waves. In response to this issue, this paper presents a hybrid limiter designed specifically for the discontinuous Galerkin (DG) method which incorporates a pseudo-time marching method. This hybrid limiter, initially introduced in DG methods for unsteady problems, preserves the local data structure while enhancing resolution through effective and precise shock capturing. Notably, for the steady problem, the hybridization of the DG solution with the cell average is employed, deviating from the low-order limited DG solution typically used for unsteady problems. This approach seamlessly integrates the previously distinct troubled cell indicator and limiter components resulting in a more cohesive and efficient limiter. It obviates the need for characteristic decomposition and intercell communication, leading to substantial reductions in computational costs and enhancement in parallel efficiency. Numerical experiments are presented to demonstrate the robust performance of the hybrid limiter for steady-state Euler equations both on the structured and unstructured meshes. [ABSTRACT FROM AUTHOR]

Incorporating unstructured data into physical models is a challenging problem that is emerging in data assimilation. Traditional approaches focus on well-defined observation operators whose functional forms are typically assumed to be known. This prevents these methods from achieving a consistent model-data synthesis in configurations where the mapping from data-space to model-space is unknown. To address these shortcomings, in this paper we develop a physics-informed dynamical variational autoencoder (Φ-DVAE) to embed diverse data streams into time-evolving physical systems described by differential equations. Our approach combines a standard, possibly nonlinear, filter for the latent state-space model and a VAE, to assimilate the unstructured data into the latent dynamical system. Unstructured data, in our example systems, comes in the form of video data and velocity field measurements, however the methodology is suitably generic to allow for arbitrary unknown observation operators. A variational Bayesian framework is used for the joint estimation of the encoding, latent states, and unknown system parameters. To demonstrate the method, we provide case studies with the Lorenz-63 ordinary differential equation, and the advection and Korteweg-de Vries partial differential equations. Our results, with synthetic data, show that Φ-DVAE provides a data efficient dynamics encoding methodology which is competitive with standard approaches. Unknown parameters are recovered with uncertainty quantification, and unseen data are accurately predicted. • Bayesian inference methodology for unstructured data assimilation. • Variational autoencoder embeds data to observations of latent differential equation. • Statistical FEM construction and parameter estimation account for misspecification. • Demonstrated on Lorenz-63, advection and Korteweg-de Vries differential equations. • Embedding physical prior knowledge produces data-efficient learning. [ABSTRACT FROM AUTHOR]

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time. [ABSTRACT FROM AUTHOR]

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms. The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection. • The introduction of two limiter regions sufficient for flux form incompressible advection to retain a maximum principle. • Some necessary conditions for flux form incompressible advection to remain positive are shown to be incompatible with Spekreijse's limiter region. • Sufficient conditions for maintaining a discrete maximum principle for flux form incompressible advection are shown to be more general than the Sweby region. [ABSTRACT FROM AUTHOR]

• A novel physics-informed convolutional neural network f-PICNN for PDEs without any labelled data. • Nonlinear convolutional units (NCUs). • Memory mechanism (numerical results show it can considerably speed up the convergence). • Finite discretization schemes (e.g., Euler, Crank Nicolson and fourth-order Runge-Kutta). • Auto-regressive framework. The physics and interdisciplinary problems in science and engineering are mainly described as partial differential equations (PDEs). Recently, a novel method using physics-informed neural networks (PINNs) to solve PDEs by employing deep neural networks with physical constraints as data-driven models has been pioneered for surrogate modelling and inverse problems. However, the original PINNs based on fully connected neural networks pose intrinsic limitations and poor performance for the PDEs with nonlinearity, drastic gradients, multiscale characteristics or high dimensionality in which the complex features are hard to capture. This leads to difficulties in convergence to correct solutions and high computational costs. To address the above problems, in this paper, a novel physics-informed convolutional neural network framework based on finite discretization schemes with a stack of a series of nonlinear convolutional units (NCUs) for solving PDEs in the space-time domain without any labelled data (f-PICNN) is proposed, in which the memory mechanism can considerably speed up the convergence. Specifically, the initial conditions (ICs) are hard-encoded into the network as the first time-step solution and used to extrapolate the next time-step solution. The Dirichlet boundary conditions (BCs) are constrained by soft BC enforcement while the Neumann BCs are hard enforced. Furthermore, the loss function is designed as a set of discretized PDE residuals and optimized to conform to physics laws. Finally, the proposed auto-regressive model has been proven to be effective in a wide range of 1D and 2D nonlinear PDEs in both space and time under different finite discretization schemes (e.g., Euler, Crank Nicolson and fourth-order Runge-Kutta). The numerical results demonstrate that the proposed framework not only shows the ability to learn the PDEs efficiently but also provides an opportunity for greater conceptual simplicity, and potential for extrapolation from learning the PDEs using a limited dataset. [ABSTRACT FROM AUTHOR]

The inverse wave scattering problem seeks to estimate a heterogeneous, inaccessible medium, modeled by unknown variable coefficients in wave equations, from transient recordings of waves generated by probing signals. It is a widely studied inverse problem with important applications, that is usually formulated as a nonlinear least squares data fit optimization. For typical measurement setups and band-limited probing signals, the least squares objective function has spurious local minima far and near the true solution, so Newton-type optimization methods fail to obtain satisfactory answers. We introduce a different approach, for electromagnetic inverse wave scattering in lossless, anisotropic media. It is an extension of recently developed data driven reduced order modeling methods for the acoustic wave equation in isotropic media. Our reduced order model (ROM) is an algebraic, discrete time dynamical system derived from Maxwell's equations. It has four important properties: (1) It can be computed in a data driven way, without knowledge of the medium. (2) The data to ROM mapping is nonlinear and yet the ROM can be obtained in a non-iterative fashion, using numerical linear algebra methods. (3) The ROM has a special algebraic structure that captures the causal propagation of the wave field in the unknown medium. (4) It is an interpolation ROM i.e., it fits the data on a uniform time grid. We show how to obtain from the ROM an estimate of the wave field at inaccessible points inside the unknown medium. The use of this wave is twofold: First, it defines a computationally inexpensive imaging function designed to estimate the support of reflective structures in the medium, modeled by jump discontinuities of the matrix valued dielectric permittivity. Second, it gives an objective function for quantitative estimation of the dielectric permittivity, that has better behavior than the least squares data fitting objective function. The methodology introduced in this paper applies to Maxwell's equations in three dimensions. To avoid high computational costs, we limit the study to a cylindrical domain filled with an orthotropic medium, so the problem becomes two dimensional. • Quantitative method for electromagnetic inverse wave scattering in lossless anisotropic media outperforming FWI. • Qualitative imaging method for electromagnetic inverse wave scattering in lossless anisotropic media outperforming migration. • Data-driven, interpolating model reduction capable of estimating internal wavefields and facilitating inversion and imaging. [ABSTRACT FROM AUTHOR]

We propose, in this paper, a Variable Spiking Wavelet Neural Operator (VS-WNO), which aims to bridge the gap between theoretical and practical implementation of Artificial Intelligence (AI) algorithms for mechanics applications. With recent developments like the introduction of neural operators, AI's potential for being used in mechanics applications has increased significantly. However, AI's immense energy and resource requirements are a hurdle in its practical field use case. The proposed VS-WNO is based on the principles of spiking neural networks, which have shown promise in reducing the energy requirements of the neural networks. This makes possible the use of such algorithms in edge computing. The proposed VS-WNO utilizes variable spiking neurons, which promote sparse communication, thus conserving energy, and its use is further supported by its ability to tackle regression tasks, often faced in the field of mechanics. Various examples dealing with partial differential equations, like Burger's equation, Allen Cahn's equation, and Darcy's equation, have been shown. Comparisons have been shown against wavelet neural operator utilizing leaky integrate and fire neurons (direct and encoded inputs) and vanilla wavelet neural operator utilizing artificial neurons. The results produced illustrate the ability of the proposed VS-WNO to converge to ground truth while promoting sparse communication. • Neuroscience inspired operator learning is proposed for scientific computing. • Proposed VS-WNO promotes sparse communications and is energy efficient. • We introduce a tailored spiking loss function to limit spiking activity. • Numerical examples solved illustrate accuracy and energy efficiency of VS-WNO. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models. • We propose a linear relaxation method with regularized energy reformulation for phase field models. • Our approach avoids time derivatives for auxiliary variables, aligning closer with the original PDEs and energy laws. • The proposed schemes for the MBE and PFC models are linear and unconditional energy stable. • The proposed approach is general and applicable to many phase field models and general gradient flow models. [ABSTRACT FROM AUTHOR]

In this paper, we develop a simple and fast algorithm for the time-domain acoustic scattering by a sound-soft or an impedance obstacle in three dimensions. We express the solution to the scattering problem by layer potentials, and then a time-domain boundary integral equation is derived. To numerically solve the resulting boundary integral equation, we propose a full discretization scheme by combining the convolution splines with a Galerkin method. In time, we approximate the density in a backward manner in terms of the convolution splines. In space, we project the density at each time onto the space of spherical harmonics, and then use the spatial discretization of a Nyström type on the surface of an obstacle which is homeomorphic to a sphere. A gallery of numerical examples are presented to show the efficiency of our algorithm. The stability, convergence and accuracy of the algorithm are discussed. • We develop an efficient algorithm for the time-domain acoustic obstacle scattering in three dimensions. • We propose a full discretization scheme by combining the convolution splines with a Galerkin method. • The stability, convergence and accuracy of the algorithm are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified. • Introduces high order finite difference boundary treatment for solving hyperbolic conservation laws. • Use a Cartesian mesh on domain with complex geometries and allow the boundary intersecting the grids in an arbitrary fashion. • Maintain stability with the same Courant-Friedrichs-Lewy number as the periodic boundary cases for any boundary locations. • Employ less terms using the relatively complicated ILW procedure and improve computational efficiency. [ABSTRACT FROM AUTHOR]

As the finite element method (FEM) and the finite volume method (FVM), both their traditional and high-order variants, continue their proliferation into various applied engineering disciplines, adaptive mesh refinement and optimization strategies have increased in their importance when solving real-world computational fluid mechanics applications. The post-processing and visualization of the resulting flow fields present two significant analysis and visualization challenges. The first challenge is the handling of elemental continuity, which is often only C 0 continuous (in continuous Galerkin methods) or piecewise discontinuous (in discontinuous Galerkin methods). The second challenge is that, depending on the flow regime and the geometric configurations for which adaptive meshing strategies are used, the meshes generated are often highly anisotropic. The (uniform knot) line-SIAC (L-SIAC) filter has been proposed as a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous. In this paper, we demonstrate that the state-of-the-art adaptive L-SIAC filter, designed for mildly anisotropic meshes, suffers degradation in the quality of the post-processed solution when applied to the types of highly anisotropic meshes produced through adaptive mesh refinement and optimization. Hence, a new Non-Uniform Knot (NUK) L-SIAC filter is proposed that automatically conforms to the underlying mesh anisotropy. We demonstrate that the new filter behaves similarly to the adaptive L-SIAC filter when applied to uniform and mildly anisotropic meshes, and furthermore we show the superiority of the NUK L-SIAC filter when applied to highly anisotropic meshes. The newly formulated filter is applied to 2D canonical scalar fields and used to visualize 2D and 3D fluid flow simulation results. • This study analyzes and visualizes principle and derived fields from FEM and FVM methods used over highly anisotropic meshes with element length ratios of up to 1000:1. • A new Non-Uniform Knot (NUK) L-SIAC filter is proposed to address issues with the adaptive L-SIAC filter on highly anisotropic meshes. • The NUK L-SIAC filter performs similarly to the adaptive L-SIAC filter on uniform and mildly anisotropic meshes but outperforms it on highly anisotropic meshes. [ABSTRACT FROM AUTHOR]

In this paper, a new type of increasingly high-order hybrid multi-resolution weighted essentially non-oscillatory (HMR-WENO) schemes is presented in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. Based on the reconstruction polynomials defined on the one-point, three-point, five-point, seven-point, and nine-point spatial stencils, we reconstruct one zeroth degree reconstruction polynomial, one quadratic reconstruction polynomial, one quartic reconstruction polynomial, one sextic reconstruction polynomial, one octave reconstruction polynomial, and their derivative polynomials together with a new hierarchical bisection method to design a series of new troubled cell indicators which can precisely find all extreme points of associated unequal degree reconstruction polynomials located inside the smallest interval in one dimension. The new troubled cell indicators do not introduce any manual parameters related to different problems. The new hybrid methodology is divided into two parts: if all extreme points of the reconstruction polynomials are nonexistent or outside the smallest interval, the target cell is not a troubled cell and the simple linear upwind schemes are utilized to obtain high-order approximations. Otherwise, the target cell is a troubled cell and the MR-WENO spatial reconstruction procedures with excellent shock-capture ability are adopted. Then a series of HMR-WENO schemes are proposed by using these new troubled cell indicators, which can be easily expanded to arbitrarily high-order accuracies in multi-dimensions. The main benefits of these HMR-WENO schemes are their efficiency, since they could save about 22%-75% CPU time than that of the same order MR-WENO schemes when simulating some benchmark examples in multi-dimensions. • It is the first time to design new troubled cell indicators to find all extreme points of polynomials. • A new hierarchical bisection method is designed to accurately compute all extreme points. • New hybrid methods can be adopted for designing arbitrarily high-order HMR-WENO schemes. • A series of examples demonstrate that the HMR-WENO schemes can save 20%-75% CPU time than before. [ABSTRACT FROM AUTHOR]

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models. • Propose an augmented subspace approximation based error indicator for the POD approximation of the time dependent PDEs. • Obtain a general framework of augmented subspace based adaptive POD method for solving the time dependent PDEs. • Provide two strategies to obtain the augmented subspaces, the residual type and the coarse-grid approximation type. • Numerical experiments show that the new method is more efficient than the existing adaptive POD methods. [ABSTRACT FROM AUTHOR]

This paper presents p -adaptive hybridized flux reconstruction schemes to reduce the computational cost of implicit discretizations. We first introduce spatial and temporal discretization and discuss the adaptation algorithm via a nondimensional vorticity indicator for hybridized methods with globally continuous and globally discontinuous numerical traces. At each adaptation level, projection operations are applied to determine the new space based on the element-wise projected solution and transmission conditions. We validate our implementation and analyze performance via numerical examples. Specifically, we show via an isentropic vortex that p -adaptive hybridization of both HFR and EFR methods results in comparable numerical error to standard p -adaptive and p -uniform FR discretizations with a fraction of degrees of freedom. Results for a cylinder at Re = 150 showcase speedup factors in excess of 6 for hybridized methods in comparison with p -adaptive standard FR schemes and up to 40 against p -uniform FR discretizations. Similarly, results for a NACA 0012 airfoil at Re = 10 , 000 demonstrate speedup factors close to 6 against p -adaptive FR discretizations and up to 33 against p -uniform conventional FR. Hence, combining hybridization with adaptation yields a significant reduction in computational cost compared with standard implicit discretizations. • We introduce polynomial adaptation in the context of hybridized flux reconstruction schemes. • The proposed formulation generalizes to discontinuous and continuous traces. • The cost of our adaptation procedure is akin to a single time step. • We observed speedup factors over 5 against p-adaptive standard FR schemes. • Speedup factors over 30 against p-uniform standard FR schemes were also observed. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1,2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach. We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository. In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials. • The effectiveness of a multirate time-integration approach based on dynamic partitioning due to AMR is demonstrated. • A nonlinear stability analysis of the P-ERK schemes by means of the fully discretized system is conducted. • A novel approach for estimating the outer eigenvalues of high-dimensional Jacobians is presented. [ABSTRACT FROM AUTHOR]

This paper develops energy-preserving discontinuous Galerkin (DG) methods for the Vlasov-Ampère (VA) system coupled with the Dougherty-Fokker-Planck (DFP) collision operator. While the classical VA system has been extensively studied, the inclusion of the collision operator introduces new challenges in conserving the total energy of the system. To address this, we design two energy-conserving temporal discretization methods: a second-order explicit scheme and a second-order implicit-explicit (IMEX) scheme. These schemes are coupled with the DG method, specifically using the local DG (LDG) method for the DFP part. We prove that the fully discrete schemes conserve the total particle number and total energy of the VA-DFP system at the fully discrete level. We further establish the L 2 stability of the fully discrete explicit scheme. Numerical experiments are conducted to assess the accuracy, conservation property, and performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation. However, this approach usually assumes scale separation and that the heterogeneity of the solution can be approximated by the solution average in each coarse block. For complex multiscale problems, the computed single effective properties/continuum might be inadequate. In this paper, we propose a novel learning-based multi-continuum model to enrich the homogenized equation and improve the accuracy of the single continuum model for multiscale problems with some given data. Without loss of generalization, we consider a two-continuum case. The first flow equation keeps the information of the original homogenized equation with an additional interaction term. The second continuum is newly introduced, and the effective permeability in the second flow equation is determined by a neural network. The interaction term between the two continua aligns with that used in the Dual-porosity model but with a learnable coefficient determined by another neural network. The new model with neural network terms is then optimized using trusted data. We discuss both direct back-propagation and the adjoint method for the PDE-constraint optimization problem. Our proposed learning-based multi-continuum model can resolve multiple interacted media within each coarse grid block and describe the mass transfer among them, and it has been demonstrated to significantly improve the simulation results through numerical experiments involving both linear and nonlinear flow equations. [ABSTRACT FROM AUTHOR]

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O (N 2) to O (N log ⁡ N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs. • An effective approach is presented to establish high-order embedded ERKN pairs. • A special implementation algorithm by diagonalizing the frequency matrix is proposed to reduce the calculation amount. • The application of the embedded ERKN pairs to wave equations prescribed with different boundary conditions is discussed. [ABSTRACT FROM AUTHOR]

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett et al. (2007) [8] , due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques. • We point out that the shape metrics, instead of function norms, should be employed to measure the numerical errors for solving the geometric flows. • By using classical BGN scheme as mesh regularization, we propose a temporal second-order PFEM which has a favorable mesh distribution. • By using the implicit BGN scheme for the trivial flow as mesh regularization, we propose a second-order PFEM which is unconditionally energy stable. [ABSTRACT FROM AUTHOR]

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), meaning they consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by the BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system, which is the relativistic analogue of Godunov's symmetrizable form of the non-relativistic MHD system (Godunov, 1972 [19]). A key novelty in our schemes is the meticulous discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. Notably, we provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework (Wu and Shu, 2023 [49]), which equivalently converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. Finally, we implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios like ultra-relativistic blasts and jets in strongly magnetized environments. • Proposed bound-preserving, locally divergence-free CDG methods for RMHD with general EOS. • Rigorously proved bound preservation using geometric quasilinearization (GQL). • Established theoretical links between bound-preserving and discrete divergence-free properties on overlapping mesh. • Stably discretized modified RMHD system, analogous to Godunov's symmetrizable non-relativistic MHD system. • Enhanced efficiency with a novel 2D cell average decomposition on overlapping meshes. [ABSTRACT FROM AUTHOR]

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions. • High order accuracy fast sweeping method on unstructured meshes is developed for the first time. • Multiple reference points on the computational domain are introduced to order all the cells and form sweeping directions. • The local solver is an unstructured WENO scheme with unequal-sized sub-stencils for the absolute convergence. • The residue of the new high order unstructured fast sweeping method converges to round off errors / machine zero. • By comparing with the other popular schemes, the proposed new scheme is very efficient. [ABSTRACT FROM AUTHOR]

Traditional approaches for transport in porous media rely on empirical hydrodynamic/transport parameters to quantify the interfacial exchange terms between fluid and solid, which leads to considerable uncertainties and also to a narrow application range of each method for realistic investigations of porous media. A parameter-free approach for flow, transport, and chemical reactions in porous media is presented in this paper. This approach adopts a novel, single-domain set of equations that are rigorously derived based on the elementary conservation laws describing the physical, continuous domain. The impact of the solid on nearby fluid being directly quantified in terms of the quantity to be solved, this avoids the introduction of empirical parameters. Various transport problems including external, conjugate, and reactive processes are properly formulated in this context, with the coupling between fluid and porous structures at different scales naturally solved in a monolithic manner. All these equations remain simple in terms of their numerical implementation, with solid structures represented by the porosity field on a background mesh. The application of the present method in both pore-scale and representative elementary volume (REV-scale) simulations are checked by performing a series of test cases, including interfacial momentum/heat/mass transport, conjugate heat/mass transfer, chemical reactions, and two practical applications containing reacting interfaces. An overall first-order convergence rate has been observed in both spatial and temporal accuracy tests. • Transport processes in porous media are modeled without empirical parameters. • The coupling between fluid and solid/porous regions is intrinsically solved in a monolithic manner. • The approach is suitable for both pore-scale and REV-scale simulations. • Various external, conjugate, and reactive transport problems are successfully solved with the approach. [ABSTRACT FROM AUTHOR]

This paper presents a new methodology allowing the discretisation of phase specific transport equations within the Volume-of-Fluid interface capturing method framework. The method uses a sharp interface algorithm to compute the transport of species concentration. The interface reconstruction and advection is provided by the geometric advection scheme isoAdvector [1] implemented in the OpenFOAM® library. When discretising the transport equation for species, lack of consistency with the free surface advection scheme can lead to numerical errors, causing conservation or boundedness issues. This work addresses the issue of consistency in convective transport of species and is divided in two parts. First, a new interpolation procedure is used to compute face values from cell-centered values. Then, the diffusive operator of the transport equation is corrected. Finally, a set of test cases are presented to validate the transport equation's consistency with the free surface advection. Species transport across the interface is not part of the scope of this article, however, this methodology can further be used to study mass transfer at the gas-liquid interface using additional mass source terms that are not discussed here. • A new methodology to discretize a transport equation for species concentration has been described. • This methodology is consistent with the geometric Volume-of-Fluid method isoAdvector which is available in OpenFOAM. • The developments made in this work avoid artificial mass transfer between the phases. • Diffusive fluxes were also corrected to avoid non-physical fluxes using the species concentration inside the phase. [ABSTRACT FROM AUTHOR]

Large-scale cosmological simulations are an indispensable tool for modern cosmology. To enable model-space exploration, fast and accurate predictions are critical. In this paper, we show that the performance of such simulations can be further improved with time-stepping schemes that use input from cosmological perturbation theory. Specifically, we introduce a class of time-stepping schemes derived by matching the particle trajectories in a single leapfrog/Verlet drift-kick-drift step to those predicted by Lagrangian perturbation theory (LPT). As a corollary, these schemes exactly yield the analytic Zel'dovich solution in 1D in the pre-shell-crossing regime (i.e. before particle trajectories cross). One representative of this class is the popular ' FastPM ' scheme by Feng et al. 2016 [1] , which we take as our baseline. We then construct more powerful LPT-inspired integrators and show that they outperform FastPM and standard integrators in fast simulations in two and three dimensions with O (1 − 100) timesteps, requiring fewer steps to accurately reproduce the power spectrum and bispectrum of the density field. Furthermore, we demonstrate analytically and numerically that, for any integrator, convergence is limited in the post-shell-crossing regime (to order 3 2 for planar-wave collapse), owing to the lacking regularity of the acceleration field, which makes the use of high-order integrators in this regime futile. Also, we study the impact of the timestep spacing and of a decaying mode present in the initial conditions. Importantly, we find that symplecticity of the integrator plays a minor role for fast approximate simulations with a small number of timesteps. • We construct new efficient time integrators for cosmological simulations. • For their construction, we use results from Lagrangian perturbation theory. • Our integrators produce the exact solution in 1D before particle trajectories cross. • The crossing of particle trajectories limits the convergence order of integrators. • Numerical experiments demonstrate the effectiveness of our integrators. [ABSTRACT FROM AUTHOR]

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method. • Novel density approximations are proposed for learning SDE via maximum likelihood estimation. • The proposed density approximation allows for multiple time steps, effectively reducing discretization errors. • The proposed method exhibits superior accuracy in learning the governing functions and computing the invariant distribution. • The proposed method is capable of handling trajectory data with low time resolution and variable time step sizes. [ABSTRACT FROM AUTHOR]

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting). • We propose a new neural operator model for partial differential equation (PDE) solving and realize an effective linear prediction of non-linear dynamics via Koopman operator. • Our approach achieves robust and accurate modeling of the long-term non-linear dynamics of PDE families or real systems. • Our model may serve as a basic unit for constructing new frameworks of PDE solving and dynamic system modeling. [ABSTRACT FROM AUTHOR]

Small adhesive particles with non-spherical shapes are common in natural and industrial processes. However, it is challenging to accurately simulate the dynamic behaviors of non-spherical particulate systems, not only due to the complex local geometry at the contact region but also to the non-linear coupling between the van der Waals adhesion and the elastic deformation. In this paper, we develop a discrete element method (DEM) framework for small non-spherical particles. Particularly, we present a simple algorithm for solving the implicit equations of the dimensionless generalized JKR model, to accurately resolve the local surface geometry at the contact region and the strong coupling between the elastic deformation and the surface adhesion. New explicit expressions are established for the pull-off force, overlap and minor-to-major ratio of the contact ellipse, to naturally capture the pull-off point upon collision. Then the actual contact parameters including the contact radius, curvatures and overlap are used to modify the interparticle interactions, including the normal elastic-adhesive-dissipative force and sliding-rolling-twisting resistances. Finally, the framework is quantitatively validated by four random packing problems, including adhesive spheres, ellipsoids, spherocylinders and dimers, and the representative non-adhesive ellipsoids. Both the microscopic and macroscopic properties of packings agree well with a large number of classic theories, experiments and simulations, showing the satisfactory accuracy and effectiveness. The newly developed adhesive DEM framework has the full capability to dynamically simulate the collision-related behaviors of complex shaped particles in the presence of van der Waals adhesion and frictional forces. • A simple algorithm is presented for solving the implicit equations of the generalized JKR model. • The non-linear coupling between surface adhesion and elastic deformation is accurately resolved. • New criterion is established to naturally capture the pull-off point upon collision. • Microscopic and macroscopic properties of simulated packing problems agree well with reported results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new set of midpoint-based high-order discretization schemes for computing straight and mixed nonlinear second derivative terms that appear in the compressible Navier-Stokes equations. Firstly, we detail a set of conventional fourth and sixth-order baseline schemes that utilize central midpoint derivatives for the calculation of second derivatives terms. To enhance the spectral properties of the baseline schemes, an optimization procedure is proposed that adjusts the order and truncation error of the midpoint derivative approximation while still constraining the same overall stencil width and scheme order. A new filter penalty term is introduced into the midpoint derivative calculation to help achieve high wavenumber accuracy and high-frequency damping in the mixed derivative discretization. Fourier analysis performed on the both straight and mixed second derivative terms show high spectral efficiency and minimal numerical viscosity with no odd-even decoupling effect. Numerical validation of the resulting optimized schemes is performed through various benchmark test cases assessing their theoretical order of accuracy and solution resolution. The results highlight that the present optimized schemes efficiently utilize the inherent viscosity of the governing equations to achieve improved simulation stability - a feature attributed to their superior spectral resolution in the high wavenumber range. The method is also tested and applied to non-uniform structured meshes in curvilinear coordinates, employing a supersonic impinging jet test case. [ABSTRACT FROM AUTHOR]

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces. • Mathematical modeling of the N-component Cahn-Hilliard model on evolving surfaces. • The componential and point-wise mass conservation and their discrete preservations on evolving surfaces. • The stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. • Investigation on the similarities and differences in phase field evolution between evolving and static surfaces. [ABSTRACT FROM AUTHOR]

Variables interpolation is one of the key concepts of MUSCL schemes. Originally developed for one-dimensional frameworks, many improvements have been made over the last decades to extend these methods to general unstructured multi-dimensional meshes. It results that scalar interpolation in a finite volume cell-centered framework is already quite well understood. It is known that linear reconstructions have to be limited in order to prevent non-physical oscillations of the solutions while ensuring a spatial second-order accuracy for smooth solutions. This is done thanks to a limiting function which allows the reconstruction to satisfy a monotonicity property, which then ensure the stability of the scheme. Nevertheless, some difficulties arise when we try to extend this process to vectorial variables. Generally, vectorial reconstructions are done componentwise, but this process reveals to be frame-dependent and leads to a loss of precision due to false detection of extrema. In this paper, we present a new method dealing with vectorial reconstructions in a multislope MUSCL context. • The cell-centered finite volume method on general unstructured meshes is considered. • Alternative formulations of the scalar multislope MUSCL method are introduced. • The flaws of componentwise vectorial reconstructions are illustrated. • A new vectorial multislope MUSCL method is introduced from the scalar procedure. • Numerical results show its accuracy, frame-independence, efficiency and robustness. [ABSTRACT FROM AUTHOR]

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P 1 D G and non conforming P 1 N C Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P 1 N C discrete frequencies compared to the P 1 D G ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P 1 D G and P 1 N C methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P 1 N C approach. • A Fourier analysis of the 2D shallow-water model is performed for the linear DG and NC Galerkin discretizations. • A long time stability result is proven for all schemes and fluxes examined in this paper. • A super-convergent result is demonstrated for the NC frequencies compared to the DG ones. • We show that the Roe flux yields spurious frequencies in O(h) for the Rossby frequency for both the DG and NC methods. • The CFL number of the NC method is 5/3 as large as that of the DG method. [ABSTRACT FROM AUTHOR]

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. • Hybrid BVD schemes are presented to compute gas/vapor interfaces in 6-equation model for multi-phase flows. • Interfaces newly generated between liquid and vapor during the evaporation process are well-reproduced for the first time. • Basic benchmark validations are conducted for two-phase flows associated with phase changes. • The numerical methods presented are well-suited for simulating multi-phase flows involving phase changes. [ABSTRACT FROM AUTHOR]

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10. [ABSTRACT FROM AUTHOR]

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately computing weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms. • Conservation laws. • Neural network solver. • Accurate discontinuity tracking. • Shock wave generation. • Wave interaction. [ABSTRACT FROM AUTHOR]