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The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation. • The EFK equation is split into three subproblems, then a full-rank splitting scheme is established. The convergence of this scheme is analyzed. • A rank-adaptive low-rank approach is proposed for the EFK equation. To the best of our knowledge, this is new in the literature for the equation. • Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation. [ABSTRACT FROM AUTHOR]

Dynamic mode decomposition (DMD) describes the dynamical system in an equation-free manner and can be used for the prediction and control. It is an efficient data-driven method for the complex systems. In this paper, we extend DMD to the parameterized problems and propose a model reduction method based on DMD to improve the computation efficiency. This method is an offline-online mechanism. In the offline phase, we need to generate the snapshots data by solving the parameterized equations for each parameter in the training set and perform the singular value decomposition (SVD) to get the reduced operator matrices, which would lead to the substantial computation. The weighted & interpolated nearest-neighbors algorithm (wiNN) is adopted to construct the efficient surrogate models of the reduced operator matrices (including the reduced Koopman operator matrix and SVD-modes). In the online phase, for each parameter, we only need to perform the operations based on the low-dimensional matrices to get the parameter DMD solution. Moreover, we choose the least squares radial basis function finite difference method for the spatial discretization. This can make our method more applicable to the parameterized problems defined on complex geometries. At last, the reaction-diffusion, the incompressible miscible flooding and the incompressible Navier-Stokes models defined on the complex geometries are presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Fluid-solid interaction (FSI) phenomena play an important role in many biomedical engineering applications. While FSI techniques and models have enabled detailed computational simulations of flow and tissue motion, the application of FSI can present challenges, particularly when data for constraining models is sparse and/or when fast computational simulations are required for assessment. In this paper, we propose a novel method for flexible-wall fluid dynamics in an ALE framework applicable for cardiovascular applications where adaptive fluid motion that emulates patient data is required. Efficiency and model simplicity are gained by a physics-based reduction to solid membrane formulations at the fluid-tissue interface combined with a Galerkin projection to a subspace spanned by boundary motion modes, leveraging snapshots observed from imaging data by use of Proper Orthogonal Decomposition (POD). The resulting fluid-reduced-solid interaction (FrSI) model is verified for a series of examples, illustrating efficacy and efficiency. Focusing on an idealized left ventricle model, we demonstrate homogenization of transmural active stress along with the capacity to accommodate prestress in the FrSI model, accounting for whole cycle mechanics by coupling to 0D pre- and afterload models (showing end-diastolic and end-systolic projected endocardial surface position errors of less than 1.5% and 3.5%, respectively). Further, we present strategies to compensate for the inherent approximation errors of the FrSI model, allowing for minimizing both the integral and spatial error between reduced and full-order model by re-calibrating parameters that govern diastolic and systolic function. Finally, the ability of FrSI to extrapolate to impaired system states (increased afterload, localized region of infarct) is shown, providing a simple yet effective strategy to enhance the POD subspace to further reduce errors. These results illustrate the potential of FrSI to streamline the simulation of hemodynamics in the heart and cardiovascular system. • Novel model reduction for cardiovascular fluid-solid interaction (FSI). • Physics- and projection-based approach to address the solid mechanics problem in FSI. • Suitable for modeling large-deformation mechanics using 'fluid-only' discretization. • Full-cycle ventricular mechanics with low spatial and integral errors. • Efficient approach to streamline hemodynamics simulations in the heart and arteries. [ABSTRACT FROM AUTHOR]

Automatic differentiation (AD) is a critical step in physics-informed machine learning, required for computing the high-order derivatives of network output w.r.t. coordinates of collocation points. In this paper, we present a novel and lightweight algorithm to conduct AD for physics-informed operator learning, which we call the trick of Zero Coordinate Shift (ZCS). Instead of making all sampled coordinates as leaf variables, ZCS introduces only one scalar-valued leaf variable for each spatial or temporal dimension, simplifying the wanted derivatives from "many-roots-many-leaves" to "one-root-many-leaves" whereby reverse-mode AD becomes directly utilisable. It has led to an outstanding performance leap by avoiding the duplication of the computational graph along the dimension of functions (physical parameters). ZCS is easy to implement with current deep learning libraries; our own implementation is achieved by extending the DeepXDE package. We carry out a comprehensive benchmark analysis and several case studies, training physics-informed DeepONets to solve partial differential equations (PDEs) without data. The results show that ZCS has persistently reduced GPU memory consumption and wall time for training by an order of magnitude, and such reduction factor scales with the number of functions. As a low-level optimisation technique, ZCS imposes no restrictions on data, physics (PDE) or network architecture and does not compromise training results from any aspect. • We present a novel algorithm to conduct automatic differentiation w.r.t. coordinates for physics-informed operator learning. • Our algorithm can reduce GPU memory and wall time for training physics-informed DeepONets by an order of magnitude. • Our algorithm neither affects training results nor imposes any restrictions on data, physics (PDE) or network architecture. [ABSTRACT FROM AUTHOR]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

In this paper, we develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique, where the numerical fluxes in the DG scheme are treated separately. However, if the numerical approximations are close to the steady state, the numerical fluxes are not independent, and it is possible to use the relationship to obtain a new penalty parameter which is zero at the steady state and the full scheme is PP. To be more precise, we first reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods. • Construct positivity-preserving and well-balanced schemes for Euler equations with gravitation. • The penalty is chosen to be zero in the Lax-Friedrichs fluxes at the steady-states. • A novel positivity-preserving technique is developed. • The scheme is straightforward to implement. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical link between numerical shock thickness and shock-capturing dissipation. The link is derived rigorously from the compressible flow governing equations involving explicitly added shock-capturing numerical dissipation terms. The derivation employs only a natural assumption that the shock-capturing dissipation takes its maximum at the maximum velocity gradient location within the numerically diffused shock layer. Therefore, as long as this assumption is satisfied, the shock-capturing dissipation can take an arbitrary distribution in space. The derived theoretical link is verified through the numerical experiment of a 1D normal-shock problem, where the results agree well with the derived theory. Furthermore, by controlling the numerical shock thickness across the hierarchical Cartesian mesh boundary using the derived theoretical link, we demonstrate that the nonphysical errors associated with the mismatch in the shock thickness are significantly reduced. [ABSTRACT FROM AUTHOR]

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future. • We introduce second-derivative summation-by-parts operators for general (non-polynomial) function spaces. • We prove their existence and mimetic properties. • We present a straightforward construction procedure for general multi-dimensional domains/elements/blocks. [ABSTRACT FROM AUTHOR]

In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work (Li et al. (2021) [17]). Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities. • A compact fifth-order ADER scheme is proposed for hyperbolic conservation laws. • A compact reconstruction technique is designed by better utilizing the information in time evolution. • The computational stencil is more compact than the original ADERWENO scheme. • It is more efficient than the original multi-stage RK-HWENO scheme. [ABSTRACT FROM AUTHOR]

In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein–Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (ρ = p = 0) up to t = 1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un–tractable within the Z4 formalism. Due to the above features, we consider that our new algorithm can be particularly beneficial for the numerical study of quasi normal modes of oscillations, both of black holes and of neutron stars. • Strongly hyperbolic first order Z4 formulation of the Einstein-Euler equations. • New and simple well-balanced discontinuous Galerkin schemes for numerical general relativity. • Robust conversion from conservative to primitive variables also in the presence of vacuum. • Stable long-time simulations of black holes and TOV stars in two and three space dimensions. • Head-on collision of two puncture black holes in three space dimensions using the Z4 formulation. [ABSTRACT FROM AUTHOR]

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes—seen as dynamical systems on a commutative ring—can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods. • We study the initialization of general lattice Boltzmann methods introducing an ad hoc modified equation analysis. • We find the constraints to obtain consistent initialization schemes, preserving second-order for the overall method. • We finely describe initial boundary layers due to dissipation mismatches between bulk and initialization schemes. • We introduce the observability of a lattice Boltzmann scheme, characterizing those with easily-mastered initializations. • We test the introduced analytical tools and their effectiveness through several—very conclusive—numerical experiments. [ABSTRACT FROM AUTHOR]

Efficient numerical schemes for solving the Poisson equation with jump conditions are of great interest for a variety of problems, including the modeling of electroporation phenomena and filamentary discharges. In this paper, we propose a modification to a finite volume scheme, namely the discrete dual finite volume method, in order to account for jump conditions with surface charges, i.e. with a source term. Our numerical tests demonstrate second-order convergence even with highly distorted meshes. We then apply the proposed method to model electroporation phenomena in biological cells by proposing a model that considers the thickness of the cell membrane as a separate domain, which differs from the literature. We show the advantages of the proposed method in this context through numerical experiments. • The discrete dual finite volume scheme is extended to solve the Poisson equation with jump conditions and surface charges. • The method is shown to exhibit a second-order convergence through canonical numerical tests. • The method is applied to the electroporation phenomena, where accurate modeling of the potential at the membrane is obtained. • Numerical experiments on the stationary and non-stationary case are provided. [ABSTRACT FROM AUTHOR]

An artificial viscosity-based shock capturing scheme is extended to the context of high-order triangular Spectral Difference method for solving gas dynamics problems featuring discontinuities and shock waves. The equations are regularised thanks to an artificial diffusivity method combined with a shock sensor based on dilatation and vorticity fields valid for any polynomial order of approximation. The methodological novelty of this paper is an L 2 projection based iterative Restriction Prolongation filtering operation introduced for high-order triangular elements. This filtering operation applied to the artificial bulk viscosity and thermal conductivity stabilises the simulations and improves the quality of the shock capturing approach. The methodology is validated on various canonical relevant compressible test cases. Numerical results demonstrate the performance and the flexibility of the triangular high-order Spectral Difference method for solving simultaneously compressible flows and shock waves. [ABSTRACT FROM AUTHOR]

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted. • Innovative Transparent Boundary Conditions: A novel transparent boundary condition is designed to enhance the accuracy and efficiency of simulations. • Enhanced Numerical Quadratures: Highly efficient and accurate numerical quadrature techniques are developed to ensure reliable computations. • Comprehensive Numerical Examples: An array of carefully chosen numerical examples are presented to demonstrate the method's effectiveness and versatility. [ABSTRACT FROM AUTHOR]

The so-called Faster is Slower (FIS) effect is observed in some particular real-life or experimental situations. In the context of an evacuation process, it expresses that increasing the speed (or, more generally, the competitiveness) of individuals may induce a reduction of the flow through the exit door. We propose here a parameter-free model to reproduce and investigate this effect (more precisely its backward "Slower is Faster" equivalent). In spite of its non-smooth character, which makes it difficult to analyze, this granular approach is based on very basic ingredients in terms of behavior. In its native, purely asocial version, individuals are represented by hard-discs, each of which has a desired velocity, and the actual velocity is built as the projection of this field on the set of admissible velocities (which respect the non-overlapping constraints). We implement the slower effect by introducing here an extra step to account for the fact that individuals refrain from pushing, and therefore tend to reduce their desired velocity accounting for the velocities of people upfront. The present paper has two objectives: establish the relevance of this model by showing that it satisfactorily reproduces various empirical effects in highly crowded evacuations with various levels of competitiveness, and explore how it can be implemented to recover and explain the FIS effect. In this spirit, we confront this Inhibition-Based (IB) model to experimental data, focusing on the Faster is Slower effect. We show in particular that this approach makes it possible to accurately recover the effect of competitiveness upon power-law distributions of time lapses which have been experimentally observed. We also study the effect of mixed behaviors, by introducing a two-population model using both approaches. We investigate in particular the effect upon evacuation efficiency of the ratio between competitive agents and non-competitive ones. In a similar context, we investigate the role of an obstacle placed upstream the exit upon evacuation efficiency. • Numerical investigation of the Faster is Slower effect based on a parameter free model. • Integration of an inhibition tendency of polite pedestrians. • Numerical evidence of the fluidizing role of an obstacle upstream the exit during evacuation. • Numerical recovering of an experimental power law for time lapses. [ABSTRACT FROM AUTHOR]

This paper contains the foundation for a new Particle-In-Cell model for gas discharges, based on Îto diffusion and Kinetic Monte Carlo (KMC). In the new model the electrons are described with a microscopic drift-diffusion model rather than a macroscopic one. We discuss the connection of the Îto-KMC model to the equations of fluctuating hydrodynamics and the advection-diffusion-reaction equation which is conventionally used for simulating streamer discharges. The new model is coupled to a particle description of photoionization, providing a non-kinetic all-particle method with several attractive properties, such as: 1) Taking the same input as a fluid model, e.g. mobility coefficients, diffusion coefficients, and reaction rates. 2) Guaranteed non-negative densities. 3) Intrinsic support for reactive and diffusive fluctuations. 4) Exceptional stability properties. The model is implemented as a particle-mesh model on cut-cell grids with Cartesian adaptive mesh refinement. Positive streamer discharges in atmospheric air are considered as the primary application example, and we demonstrate that we can self-consistently simulate large discharge trees. • New, non-kinetic Particle-In-Cell method for electric discharges. • Self-consistent and stochastic evolution of streamer discharge trees. • Compatible with adaptive mesh refinement (AMR) and parallel computations. [ABSTRACT FROM AUTHOR]

In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H 1 -norm, with a 2nd-order convergence, and in the L 2 -norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L ∞ -norm on several distorted meshes. [ABSTRACT FROM AUTHOR]

• A second order intersection-based remapping method for cell-centered quantities in two dimensional cylindrical coordinates is proposed. • The remapping method is capable to preserve spherical symmetry when the mesh and mesh moving are symmetric. • The remapping method has good properties such as conservation for mass, momentum and total energy. • The symmetry preserving property and second order accuracy are strictly proved and demonstrated by numerical tests. The importance of preserving physical symmetry in arbitrary Lagrangian-Eulerian (ALE) simulations is well recognized. In the past decades, first order spherical-symmetry-preserving Lagrangian and ALE methods in two-dimensional cylindrical coordinates were developed. And there are also several works about second order spherical-symmetry-preserving cell-centered Lagrangian schemes. The goal of this paper is to develop Second order spherical-symmetry-preserving intersection-based remapping method in two-dimensional cylindrical coordinates. The method is capable to preserve one-dimensional spherical symmetry when the old grid is equiangular polar mesh and the rezoning algorithm only moves points in the radial directions by the same amount for all points with the same spherical radius. The remapping method has second order accuracy almost uniformly (the accuracy drops to first order close to the revolution axis), has good properties such as conservation for mass, momentum and total energy. The symmetry-preserving property and second order accuracy of our new remapping method are proved mathematically and demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit methods require impractically small time steps, while implicit methods, which offer stability, come with the challenge to guarantee the convergence of nonlinear iterative solvers. To overcome these challenges, we propose a predictor-corrector approach and design proper implicit-explicit time discretizations. In the predictor step, the system is reformulated into a nonconservative form and linear diffusion terms are introduced as a penalization to mitigate strong nonlinearities. We then employ a Picard iteration to secure convergence in handling the nonlinear aspects. The corrector step guarantees the conservation of total energy, which is vital for accurately simulating the speeds of propagating sharp fronts in this system. For spatial approximations, we utilize local discontinuous Galerkin finite element methods, coupled with positive-preserving and TVB limiters. We validate the orders of accuracy, conservation properties, and suitability of using large time steps for our proposed methods, through numerical experiments conducted on one- and two-dimensional spatial problems. In both homogeneous and heterogeneous non-equilibrium radiation diffusion problems, we attain a time stability condition comparable to that of a fully implicit time discretization. Such an approach is also applicable to many other reaction-diffusion systems. • High order conservative LDG-IMEX methods for non-equilibrium radiation diffusion problems are developed. • Degeneracy, strong nonlinearity and stiff source are all taken into account. • A predictor-corrector procedure based on an IMEX scheme is proposed. • Only mildly nonlinear system needs to be solved and is robust for convergence. • The scheme is of high order, conservative, positivity-preserving and allow large time steps. [ABSTRACT FROM AUTHOR]

The recently-developed general synthetic iterative scheme (GSIS) is efficient in simulating multiscale rarefied gas flows due to the coupling of mesoscopic kinetic equation and macroscopic synthetic equation: for linearized Poiseuille flow where the boundary flux is fixed at each iterative step, the steady-state solutions are found within dozens of iterations in solving the gas kinetic equations, while for general nonlinear flows the iteration number is increased by about one order of magnitude, caused by the incompatible treatment of the boundary flux for the macroscopic synthetic equation. In this paper, we propose a generalized boundary treatment (GBT) to further accelerate the convergence of GSIS. The main idea is, the truncated velocity distribution function at the boundary, similar to that used in the Grad 13-moment equation, is reconstructed by the macroscopic conserved quantities from the synthetic equation, as well as the high-order correction of non-equilibrium stress and heat flux from the kinetic equation; therefore, in each inner iteration solving the synthetic equation, the explicit constitutive relations facilitate real-time updates of the macroscopic boundary flux, driving faster information exchange in the flow field, and consequently achieving faster convergence. Moreover, the high-order correction derived from the kinetic equation can compensate the approximation by the truncation and ensure the boundary accuracy. The one-dimensional Fourier flow, two-dimensional hypersonic flow around a cylinder, three-dimensional pressure-driven pipe flow and the flow around the hypersonic technology vehicle are simulated. The accuracy of GSIS-GBT is validated by the direct simulation Monte Carlo method, the previous versions of GSIS, and the unified gas-kinetic wave-particle method. For the efficiency, in the near-continuum flow regime and slip regime, GSIS-GBT can be faster than the conventional iteration scheme in the discrete velocity method and the previous versions of GSIS by two- and one-order of magnitude, respectively. • The generalized boundary treatment (GBT) is proposed to further accelerate the general synthetic iterative scheme (GSIS). • In both low-speed internal flows and hypersonic flows, GSIS-GBT can achieve convergence within 100 iterations. • GSIS-GBT is respectively faster than previous version of GSIS/conventional iterative scheme by one/two orders of magnitude. [ABSTRACT FROM AUTHOR]

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

In this paper, we develop the first entirely graphics processing unit (GPU) based h -adaptive flux reconstruction (FR) method with linear trees. The adaptive solver fully operates on the GPU hardware, using a linear quadtree for two dimensional (2D) problems and a linear octree for three dimensional (3D) problems. We articulate how to efficiently perform tree construction, 2:1 balancing, connectivity query, and how to perform adaptation for the flux reconstruction method on the GPU hardware. As a proof of concept, we apply the adaptive flux reconstruction method to solve the inviscid isentropic vortex propagation problem on 2D and 3D meshes and transitional flow over a 3D square cylinder at Re = 400 to demonstrate the efficiency of the developed adaptive FR method on a single GPU card. Depending on the computational domain size, acceleration of one or two orders of magnitude can be achieved compared to uniform meshing for long distance transportation. The total computational cost of adaption, including tree manipulations, connectivity query and data transfer, compared to that of the numerical solver, is insignificant. It can be less than 2% of the total wall clock time for 3D problems when we perform adaptation every 10-40 time steps with an explicit 3-stage Runge–Kutta time integrator. • Entirely GPU based h -adaptation via linear trees. • One or two orders of magnitude speedup can be achieved. • Adaptation takes as little as two to three percent of the total runtime for 3D simulations. [ABSTRACT FROM AUTHOR]

This paper proposes a new Helmholtz decomposition based windowed Green function (HD-WGF) method for solving the time-harmonic elastic scattering problems on a half-space with Dirichlet boundary conditions in both 2D and 3D. The Helmholtz decomposition is applied to separate the pressure and shear waves, which satisfy the Helmholtz and Helmholtz/Maxwell equations, respectively, and the corresponding boundary integral equations of type (I + T) ϕ = f , that couple these two waves on the unbounded surface, are derived based on the free-space fundamental solution of Helmholtz equation. This approach avoids the treatment of the complex elastic displacement tensor and traction operator that involved in the classical integral equation method for elastic problems. Then a smooth "slow-rise" windowing function is introduced to truncate the boundary integral equations and a "correction" strategy is proposed to ensure the uniformly fast convergence for all incident angles of plane incidence. Numerical experiments for both two and three dimensional problems are presented to demonstrate the accuracy and efficiency of the proposed method. • A novel Helmholtz decomposition based windowed Green function method is proposed for solving the elastic half-space problem. • New boundary integral equations based on Helmholtz decomposition are derived. • The new approach avoids the treatment of the full elastic fundamental tensor and traction operator. • Numerical examples demonstrate the uniform fast convergence for all incident angles of plane incidence. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and requires the knowledge of the boundary location. [ABSTRACT FROM AUTHOR]

This paper presents a model for simulating a two-phase flow involving cavitation and condensation, focusing on its application to turbopump flows. The model is derived by extending Kapila's approach, which incorporates thermal and free energy relaxation terms, to compressible multiphase flow equations in a rotating reference frame. The numerical method employed is a finite volume approach based on the Godunov scheme, ensuring pressure equilibrium and total energy conservation through a relaxation step. The model utilizes separate Equations of State (EOS) for the liquid and vapor phases, with the experimental saturation curve, expressed as p sat (T) , being the only parameter needed for cavitation modeling. To broaden its applicability beyond axial pumps, a specific Riemann solver is developed to couple rotating and non-rotating regions within the flow domain. The thermodynamic phase change process is validated using simple cases, and the model is further utilized to simulate water flows in a turbopump inducer. The characteristic curve of the pump is obtained in the non-cavitating regime for several mass flow rates conditions, and the model is successfully assessed in severe cavitation conditions, demonstrating the performance breakdown caused by vapor pockets. The modeling also provides detailed flow descriptions, including analysis of vapor pockets and their impact on blade pressure loading. • Modeling multiphase flow in rotating reference frame. • A new Riemann problem to connect fixed and rotating flow regions. • Phase change modeling by a relaxation method from pure phase to mixture. • Cavitating flow in turbopumps without parameter for the growth of the cavitation pockets. [ABSTRACT FROM AUTHOR]

The present paper focuses on lattice Boltzmann method (LBM) simulations of turbulent flows in combination with a Zonal Detached Eddy Simulation (ZDES) approach, specifically its mode 2 (2020). A common issue of the concomitant use of LBM and DES type models is the systematic modeled-stressed depletion (MSD) triggered by fine isotropic grids imposed by the lattice. This issue is responsible for erroneous surface coefficients that can eventually lead to grid induced separation (GIS). In this work, it is shown that the enhanced boundary layer protection brought by ZDES mode 2 (2020) is sufficient to allow the combination of LBM and a DES type method for high Reynolds number with equal accuracy as its Navier-Stokes counterpart. The emphasis is put on the ability of ZDES to recover its main properties within the numerics of lattice Boltzmann formalism. This is achieved on multiple test-cases namely the turbulent flow over a flat plate, a backward facing step and a three-element airfoil. • Salient combination of lattice Boltzmann Method and Zonal Detached Eddy Simulation. • RANS shielding of turbulent boundary layers ensured even on fine LBM-induced grids. • Consistency between the model and its Navier-Stokes counterpart. • Model assessment on flat plate, backward-facing step and three-element airfoil test-cases. • Robust combination promising for industrial applications. [ABSTRACT FROM AUTHOR]

The simulation of plasma physics is computationally expensive because the underlying physical system is of high dimensions, requiring three spatial dimensions and three velocity dimensions. One popular numerical approach is Particle-In-Cell (PIC) methods owing to its ease of implementation and favorable scalability in high-dimensional problems. An unfortunate drawback of the method is the introduction of statistical noise resulting from the use of finitely many particles. In this paper we examine the application of the Smoothness-Increasing Accuracy-Conserving (SIAC) family of convolution kernel filters as denoisers for moment data arising from PIC simulations. We show that SIAC filtering is a promising tool to denoise PIC data in the physical space as well as capture the appropriate scales in the Fourier space. Furthermore, we demonstrate how the application of the SIAC technique reduces the amount of information necessary in the computation of quantities of interest in plasma physics such as the Bohm speed. • SIAC filters are effective denoisers for moment data arising from PIC simulations. • SIAC captures appropriate Fourier signal information. • SIAC reduces amount of information necessary in computation of quantities of interest. • SIAC is useful in Bohm speed calculation. [ABSTRACT FROM AUTHOR]

This paper proposes a deep-learning-based method for recovering a signed distance function (SDF) of a given hypersurface represented by an implicit level set function. Using the flexibility of constructing a neural network, we use an augmented network by defining an auxiliary output to represent the gradient of the SDF. There are three advantages of the augmented network; (i) the target interface is accurately captured, (ii) the gradient has a unit norm, and (iii) two outputs are approximated by a single network. Moreover, unlike a conventional loss term which uses a residual of the eikonal equation, a novel training objective consisting of three loss terms is designed. The first loss function enforces a pointwise matching between two outputs of the augmented network. The second loss function leveraged by a geometric characteristic of the SDF imposes the shortest path obtained by the gradient. The third loss function regularizes a singularity of the SDF caused by discontinuities of the gradient. Numerical results across a wide range of complex and irregular interfaces in two and three-dimensional domains confirm the effectiveness and accuracy of the proposed method. We also compare the results of the proposed method with physics-informed neural networks approaches and the fast marching method. • A deep-learning-based method is proposed to recover a signed distance function (SDF) of a level set function. • An augmented neural network structure with the gradient of the SDF as an auxiliary output is designed. • From geometric properties, new loss functions are proposed that impose global relations and alleviate the singularity of SDF. • Several numerical tests on irregular interfaces in 2D and 3D validate the effectiveness and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method. • Multiscale reduction methods are efficient and accurate to solve problems with high-contrast media. • The presence of high-contrast coefficients reduces the stability region of time discretization. • Implicit methods must solve nonlinear equations, which can be a bottleneck computation. • Exponential integrators are alternative techniques and use robust time-stepping. [ABSTRACT FROM AUTHOR]

In the paper we construct one extended Filon method for the multivariate oscillatory integral on a regular simplex avoiding the derivatives. Based on the proposed geometric configuration, the error estimate of the multivariate Lagrange interpolation on a simplex is established. Based on the multivariate interpolating, the extended Filon method is presented to reduce the integration error for the whole regime of the oscillatory parameter. In particular, the explicit formula of the moment is also provided. Numerical results illustrate that the error decreases with increasing oscillatory parameters and more interpolating nodes. [ABSTRACT FROM AUTHOR]

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder. • An algorithm for the evaluation of oscillatory integrals with polynomial phase. • The algorithm automates the numerical steepest descent method. • Our implementation is robust and efficient across all frequencies. [ABSTRACT FROM AUTHOR]

This paper investigates an inverse scattering problem of time-harmonic elastic waves incident on a rigid random periodic structure. A novel and efficient numerical method is proposed to quantify the randomness, i.e., to reconstruct key statistical properties of random structures from boundary measurements of the scattering data. This method consists of two ingredients, the Monte Carlo technique for sampling the probability space and a continuation method with respect to the wavenumber. Unlike existing methods, this approach does not require a priori knowledge of the randomness and can handle a wide range of random structures, including non-Gaussian processes with extremely high levels of complexity. To illustrate the accuracy and effectiveness of the proposed method, numerical examples involving Gaussian and non-Gaussian processes with various covariances are provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose and numerically solve a model considering confined flow in fractured porous media coupled with free flow in conduits. Such a situation arises, for example, for fluid flows in fractured vuggy carbonate reservoirs and caves/conduits, where a matrix and fractures are the main spaces for fluid storage and flow. The flow in caves and conduits is governed by the Stokes equation; the flow in fractured porous media, which consists of both a matrix and fractures, is described by a coupled Darcy–Darcy (bulk–fracture) model. Then, the two models are coupled through suitable (physically consistent) interface conditions on the interface between fractured porous media and caves/conduits, playing a key role in a physically faithful simulation that achieves high accuracy. The construction of all the interface conditions of the coupled model is based on the fundamental properties of the traditional bulk–fracture model and the well-known Stokes–Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is theoretically analyzed. We also propose a polyhedral discontinuous Galerkin finite element approximation for the Stokes flow in conduits and Darcy flow in the porous matrix, which is coupled with a conforming finite element scheme for the flow in the fracture. Optimal error estimates in a suitable energy norm are obtained. Two-dimensional numerical experiments are conducted to validate the proposed model and to demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution; the detail flow characteristics around fractures, the matrix, and conduits; and the applicability to real-world problems. [ABSTRACT FROM AUTHOR]

Accurate and efficient numerical simulations of soil water movement, as described by the highly nonlinear Richards' equation, often require local refinement near recharge or sink/source terms. In this paper, we present a novel numerical scheme for solving Richards' equation. Our approach is based on the vertex-centered finite volume method (VCFVM) and can be easily adapted to locally refined meshes. The proposed scheme offers some key features, including the definition of all unknowns over vertices of the primary mesh, expression of flux crossing dual edges as combinations of hydraulic heads at the vertices of the primary cell, and the capability to handle nonmatching meshes in the presence of local mesh refinement. For performance evaluation, soil water content and soil water potential simulated by the proposed scheme are benchmarked against results produced from HYDRUS (a widely used soil water numerical model) and the observed values in four test cases, including a convergence test case, a synthetic case, a laboratory experiment case and a field experiment case. The comparison results demonstrate the effectiveness and applicability of our scheme across a wide range of soil parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

The algebraic volume-of-fluid (VOF) method based on the construction strategy of switching technique has received extensive attention recently. However, all the previous algebraic VOF methods were constructed by combining different methods of compressible differencing scheme (CDS) and the high resolution scheme (HRS), which would switch back and forth in most cases. Thus the final algebraic VOF method usually becomes to be complicated, and the numerical instabilities would be increased. In this paper, a novel algebraic VOF method, without the operations of switching back and forth, is constructed within a unified framework of the steepness-adjustable harmonic (SAH) scheme (He et al. (2022) [33]). A thorough validation of the present method is conducted, examining the pure advection of the interface indicator function. The results indicate that the present method can resolve the interface capturing with substantially low numerical diffusion and low numerical oscillations. • The same expression of SAH scheme is utilized for high resolution and compressible differencing scheme with only different steepness parameter. • The switching technique in the classical algebraic VOF method is extended to get the final steepness parameter based on the infimum and supremum. • A strategy based on least squares approximation parameter is proposed for determining the supremum of the steepness. [ABSTRACT FROM AUTHOR]

• We proposed variable linear transformation improved physics-informed neural networks (VLT-PINNs) to solve thin-layer flow forward and inverse problems. • Three principles for determining the VLT parameters were identified for three types of variables in PINNs respectively. • The VLT-PINNs with the principle-suggested parameters show excellent performance over the ten test cases. • Only under the constraints of the VLT principles can we obtain satisfactory results, while traditional linear transformation may yield totally wrong results. • VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity. Physics-informed neural networks (PINNs) have attracted wide attention due to their ability to seamlessly embed the learning process with physical laws and their considerable success in solving forward and inverse differential equation (DE) problems. While most studies are improving the learning process and network architecture of PINNs, less attention has been paid to the modification of the DE system, which may play an important role in addressing some limitations of PINNs. One of the simplest modifications that can be implemented to all DE systems is the variable linear transformation (VLT). Therefore, in this work, we propose the VLT-PINNs that solve the DE systems of the linear-transformed variables instead of the original ones. To clearly illustrate the importance of prior knowledge in determining the VLT parameters, we choose the thin-layer flow problems as our focus. Ten related cases were tested, including the jet flows, wake flows, mixing layers, boundary layers and Kovasznay flows. Based on the principle of normalization and for a better match of the DE system to the preference of NNs, we identify three principles for determining the VLT parameters: magnitude normalization for dependent variables (principle 1), local normalization for independent variables (principle 2), and appropriate scaling for physics-related parameters in inverse problems (principle 3). The VLT-PINNs with the VLT parameters suggested by the proposed principles show excellent performance over all the test cases, while the results are quite poor with the VLT parameters suggested by traditional linear transformations, such as nondimensionalization and global normalization. Comparison studies also show that only under the constraints of the VLT principles can we obtain satisfactory results. Besides, we find tanh is more appropriate as the activation function than sin for thin-layer flow problems, from both posteriori results and priori analyses with physical intuition. We highlight that our VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity, and hope that it can provide new insights into the better integration of prior knowledge, physical intuition and the nature of NNs. The code for this paper is available on https://github.com/CAME-THU/VLT-PINN. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

Fourier analysis based optimization techniques have been developed for numerical schemes on structured grids and result in significant performance improvements. However, there lacks an effective optimization technique for numerical schemes on general unstructured grids, due to the mesh geometry complexity that makes Fourier transformation infeasible. To address this issue, an optimization framework based on machine learning is developed in this paper for numerical schemes on unstructured grids. The optimization of a compact high-order variational reconstruction on triangular grids is performed to illustrate the framework. An artificial neural network (ANN) is used to predict the optimal values of the derivative weights on cell interfaces that are the free parameters of the variational reconstruction, given the local geometry as input. The ANN is trained by minimizing the solution errors of a linear advection equation on a set of generated compact stencils, with a set of sampled sine waves as initial conditions. The developed optimization framework is applicable to general numerical schemes on unstructured grids as the training does not require explicit computation of dispersion or dissipation. Numerical results for inviscid flow problems show that the optimized variational finite volume scheme is remarkably more accurate and efficient than its non-optimized counterpart, demonstrating the effectiveness of the machine learning optimization. • A machine learning optimization framework for numerical schemes on unstructured grids. • A feedforward neural network to predict optimal parameter values of numerical schemes. • Training of the network by minimizing solution errors of a linear advection equation. • A machine learning optimized compact high-order variational reconstruction. • Generalization of the optimized variational reconstruction to nonlinear problems. [ABSTRACT FROM AUTHOR]

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings. [ABSTRACT FROM AUTHOR]

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting model is comprised of up to three neural networks, modelling terms representing conservation, dissipation and external forces, and discrete convolution operators that can either be learned or be given as input. We demonstrate numerically the superior performance of PHNN compared to a baseline model that models the full dynamics by a single neural network. Moreover, since the PHNN model consists of three parts with different physical interpretations, these can be studied separately to gain insight into the system, and the learned model is applicable also if external forces are removed or changed. • We introduce pseudo-Hamiltonian neural networks for partial differential equations. • The models can be separated into different parts, with physical interpretations. • Models learned on a disturbed system can predict future states without disturbances. • Assumptions on geometric structures can be imposed on the models. • Our new models consistently outperform the baseline neural network model. [ABSTRACT FROM AUTHOR]

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM. [ABSTRACT FROM AUTHOR]

This paper presents a novel projection-based embedded discrete fracture model (pEDFM) applicable to general two-phase flow simulation in anisotropic reservoirs for the first time. The novel pEDFM is developed based on hybrid two-point flux approximation and mimetic finite difference (TPFA-MFD) methods, which retains the basic features of two-point flux approximation (TPFA) based generic pEDFM workflow and incorporates new features of mimetic finite difference (MFD) to handle full-tensor permeabilities. This is achieved by introducing the concept of effective connection, formulating additional pressure degrees of freedom for effective connections, assigning effective matrix-matrix (m-m) and matrix-fracture (m-f) connections to the corresponding sides of matrix-cell interfaces, presenting methods corresponding to different types of connections to handle low-conductivity fractures, and further deriving formulas of numerical fluxes for effective connections. The resulting global equation system is solved by a Newton-Raphson nonlinear solver with the temporary discretization of the implicit backward Euler scheme. The 2D and 2.5D numerical examples show the novel pEDFM could achieve the same level of computation accuracy and convergence as the reference discrete fracture model (DFM), while avoiding the gridding difficulty of complex fracture networks. It also significantly outperforms classical EDFM based on MFD and generic pEDFM workflow in the computational accuracy for cases with anisotropic full tensor permeabilities. The novel pEDFM, to our best knowledge, is for the first time proposed and developed for general anisotropic two-phase flow simulation and maybe the numerical simulation framework for multi-phase flow in fractured reservoirs with the best comprehensive performance so far. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical study of droplet impact on solid substrates in the presence of surfactants. We formulate the problem in an energetically variational framework and introduce an incompressible Cahn-Hilliard-Navier-Stokes system for the phase-field modeling of two-phase flows. Flory-Huggins potential and generalized Navier boundary condition are used to account for soluble surfactants and moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The discrete energy dissipation law for the original energy is rigorously proved for the first-order scheme. The numerical methods are implemented using finite difference method in three-dimensional cylindrical coordinates with axisymmetry. Using the proposed methods for this model, we systematically study the impact dynamics of both clean and contaminated droplets (with surfactants) in a series of numerical experiments. In general, the dissipation in the impact dynamics of a contaminated drop is smaller than that in the clean case, and topological changes are more likely to occur for contaminated drops. Adding surfactants can significantly influence the impact phenomena, leading to the enhancement of adherence effect on hydrophilic surfaces and splashing on hydrophobic surfaces. Some qualitative agreements with experiments are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the Petrov-Galerkin immersed finite element method is developed for simulating flow in fractured porous media. We consider the hybrid-dimensional fracture model as an elliptic interface problem with multiple physical coupling interface conditions, and embed the information of low-dimensional fractures into the local functions of interface elements. The immersed finite element space is constructed by explicitly representing the local solution functions in the interface element. This approach addresses the limitation of conforming meshes. More importantly, it can capture both the continuity and discontinuity of pressure, making it effective for both conductive and blocking fractures. We provide a detailed illustration of how to construct local functions on triangular and rectangular meshes. Finally, numerical examples and benchmarks are presented to validate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

This paper addresses the validation and verification of numerical models for complex multiphase flows involving immiscible fluids. Specifically, the focus is on the dynamics of the interface between the fluids, which remains a challenge in many applications. Previous studies have primarily examined simpler flows with limited complexity, while real-world scenarios involve intricate interfaces and turbulent dynamics. To overcome this, a representative test case involving the motion of an oil cube within a water cube proposed by [12] is used. Different computational models using volume-of-fluid (VOF) approaches are validated and compared, with particular attention given to the influence of numerical schemes for convective terms. The impact of these schemes on integral quantities and interface dynamics is analyzed. Particular attention is paid at the surface tension contribution to the energy budgets, which is usually neglected in the available literature. Furthermore, it is shown here that, the model adopted for the surface tension is responsible for an additional contribution to numerical dissipation. To the authors' knowledge, it is the first time that a detailed analysis of such aspects is presented. • Analysis of the advective term discretization in two-phase solver. • Identify optimal numerical schemes for robustness and energy conservation. • The role of surface tension in the energy balance is explored. • Research offers insights on interface dynamics and numerical discretization effects. [ABSTRACT FROM AUTHOR]

This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side. [ABSTRACT FROM AUTHOR]

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method. • Immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method. • Polynomial preserving recovery (PPR) method. • A new mapping array structure for the particle positioning. • A new charge deposition with some special particle positions in the adaptive Cartesian meshes. • A series of numerical examples for validation. [ABSTRACT FROM AUTHOR]

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions. • The first quantum simulation algorithms for PDEs with physical boundary conditions. • The first quantum simulation algorithms for PDEs incorporating physical interface conditions. • The first quantum algorithm for partial transmission and reflections from geometric optics with interfaces. [ABSTRACT FROM AUTHOR]