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We highlight some problems in this paper. They lead to question the model itself. [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 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]

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]

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]

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 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]

Abstract: The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property. [Copyright &y& Elsevier]

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ -based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ -based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties. [ABSTRACT FROM AUTHOR]

This article presents a general reduced order model (ROM) framework for addressing fluid dynamics problems involving time-dependent geometric parametrisations. The framework integrates Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) hyper-reduction techniques to effectively approximate incompressible computational fluid dynamics simulations. To demonstrate the applicability of this framework, we investigate the behaviour of a planar contraction-expansion channel geometry exhibiting bifurcating solutions known as the Coanda effect. By introducing time-dependent deformations to the channel geometry, we observe hysteresis phenomena in the solution. The paper provides a detailed formulation of the framework, including the stabilised finite elements full order model (FOM) and ROM, with a particular focus on the considerations related to geometric parametrisation. Subsequently, we present the results obtained from the simulations, analysing the solution behaviour in a phase space for the fluid velocity at a probe point, considered as the Quantity of Interest (QoI). Through qualitative and quantitative evaluations of the ROMs and hyper-reduced order models (HROMs), we demonstrate their ability to accurately reproduce the complete solution field and the QoI. While HROMs offer significant computational speedup, enabling efficient simulations, they do exhibit some errors, particularly for testing trajectories. However, their value lies in applications where the detection of the Coanda effect holds paramount importance, even if the selected bifurcation branch is incorrect. Alternatively, for more precise results, HROMs with lower speedups can be employed. • General reduced order model (ROM) framework for time-dependent geometric parametrisations. • Study of the hysteresis of the Coanda Effect for a contraction-expansion channel in a ROM context. • Presentation of the empirical cubature method (ECM) hyper-reduction algorithm for elements selection. [ABSTRACT FROM AUTHOR]

This paper presents a model order reduction method to accelerate broadband topology optimization of structural-acoustic interaction systems by coupling Finite Element Methods and Boundary Element Methods. The finite element method is used for simulating thin-shell vibration and the boundary element method for exterior acoustic fields. Moreover, the finite element and boundary element methods are implemented in the context of isogeometric analysis, whereby the geometric accuracy and high order continuity of Kirchhoff-Love shells can be guaranteed and meantime no meshing is necessary. The topology optimization method takes continuous material interpolation functions in the density and bulk modulus, and adopts adjoint variable methods for sensitivity analysis. The reduced order model is constructed based on second-order Arnoldi algorithm combined with Taylor's expansions which eliminate the frequency dependence of the system matrices. Numerical results show that the proposed algorithm can significantly improve the efficiency of broadband topology optimization analysis. • Broadband topology optimization of 3D structural-acoustic interaction with reduced order isogeometric FEM/BEM. • Model order reduction is conducted for topology optimization of structural-acoustic interaction by coupling FEM/BEM. • The isogeometric analysis is applied to the topology optimization framework with reduced order FEM/BEM. [ABSTRACT FROM AUTHOR]

This paper presents a boundary element method (BEM) for computing the energy transmittance of a singly-periodic grating in 2D for a wide frequency band, which is of engineering interest in various fields with possible applications to acoustic metamaterial design. The proposed method is based on the Padé approximants of the response. The high-order frequency derivatives of the sound pressure necessary to evaluate the approximants are evaluated by a novel fast BEM accelerated by the fast-multipole and hierarchical matrix methods combined with the automatic differentiation. The target frequency band is divided adaptively, and the Padé approximation is used in each subband so as to accurately estimate the transmittance for a wide frequency range. Through some numerical examples, we confirm that the proposed method can efficiently and accurately give the transmittance even when some anomalies and stopband exist in the target band. • A novel boundary element method for sweeping the acoustic transmittance of a singly-periodic slab is developed. • The proposed method is based on the FMM and the hierarchical methods to compute the frequency derivatives of sound pressure. • An adaptive subdivision for the frequency band of interest is also presented to investigate a reliable fast frequency sweep. [ABSTRACT FROM AUTHOR]

This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invariance, conservation of momentum, and conservation of energy. We verify these advantages of the FSIB method numerically both for the Stokes equations and for the Navier-Stokes equations in both two and three space dimensions. The FSIB method converges faster than the IB method. In particular, we observe second-order convergence in various problems for the Navier-Stokes equations in three dimensions. The FSIB method is also computationally efficient with complexity of O (N 3 log ⁡ (N)) per time step for N 3 Fourier modes in three dimensions. • The FSIB method is gridless and has exact translation invariance. • A divergence-free interpolated velocity field that conserves volume. • Equivalent to the use of a new 'sinc' kernel in the framework of IB method. • Improved boundary resolution and faster convergence rate than the IB method. • Conservation of momentum and conservation of energy. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels by Tlupova and Beale 2019 (given in [34]). We also use partition-of unity based finite differences that are required for the computation of interfacial forces. Given an N -point surface discretization, our numerical scheme has fourth-order accuracy and O (N) asymptotic complexity, which is an improvement over the O (N 2 log ⁡ N) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures by Veerapaneni et al. 2011 [36]. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art. • Develops a scheme for deformable capsules in Stokesian fluid. • Employs integral equations with regularized Stokes kernels. • Utilizes atlas-based finite differences for shape derivatives. • Achieves O (N) complexity, surpassing O (N 2 log ⁡ N). • Leverages GPU for dynamic, high-resolution simulations of complex shapes. • Employs integral equations with regularized Stokes kernels. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios. [ABSTRACT FROM AUTHOR]

In this paper we propose an improved three-dimensional immersed boundary method coupled with a finite-difference code to simulate self-propelled phoretic particles in viscous incompressible flows. We focus on the phenomenon of diffusiophoresis which, using the driving of a concentration gradient, can generate a slip velocity on a surface. In such a system, both the Dirichlet and Neumann boundary conditions are involved. In order to enforce the boundary conditions, we propose two improvements to the basic direct-forcing immersed boundary method. The main idea is that the immersed boundary terms are corrected by adding the force of the previous time step, in contrast to the traditional method which relies only on the instantaneous forces in each time step. For the Neumann boundary condition, we add two auxiliary layers inside the body to precisely implement the desired concentration gradient. To verify the accuracy of the improved method, we present problems of different complexity: The first is the pure diffusion around a sphere with Dirichlet and Neumann boundary conditions. Then we show the flow past a fixed sphere. In addition, the motion of a self-propelled Janus particle in the bulk and the spontaneously symmetry breaking of an isotropic phoretic particle are reported. The results are in very good agreements with the data that are reported in previously published literature. • We correct immersed boundary terms through the addition of forces from previous time steps. • The approach departs from traditional methods that rely solely on instantaneous forces, enhancing simulation accuracy. • The study presents a range of validations, from pure diffusion around a sphere to flow past a fixed sphere. • The motion of self-propelled Janus particles and the symmetry breaking of isotropic phoretic particles are explored. • Phoretic particle is a new field that immersed boundary method can be applied to. [ABSTRACT FROM AUTHOR]

• A unified formulation of artificial compressibility and local low Mach number preconditioning methods is presented. • The formulation enables a single solver can be used to analyze problems from incompressible fluid flows to compressible flows. • A new artificial compressibility method is proposed for the unified system. • Roe's approximate Riemann solver, and JST artificial dissipation method are used with the formulation. This paper presents a unified approach to artificial compressibility and local low Mach number preconditioning methods. The numerical formulation is presented, which provides a seamless transition between incompressible and compressible systems. A single preconditioner that can act as the artificial compressibility or the local preconditioning under a given Mach number condition is derived. An artificial compressibility method is proposed for the unified system. The unified preconditioner is applied to the upwind approach, Roe's approximate Riemann solver, and the JST artificial dissipation approach. A Roe average, which can be applied to the unified systems in a single form is derived. The preconditioning matrix is applied to a single solver to analyze problems ranging from incompressible fluid flows to compressible flows with shocks. [ABSTRACT FROM AUTHOR]

In this paper, a new type of increasingly high-order modified multi-resolution weighted essentially non-oscillatory (MMR-WENO) schemes with new troubled cell indicators is designed in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. It is a first time to design new troubled cell indicators that based on two high-degree reconstruction polynomials, which are defined on the three-point, five-point, seven-point, and nine-point spatial stencils, respectively. Such new troubled cell indicators can automatically identify the discontinuous solutions without manually adjusting the parameters related to the problems. Subsequently, a series of MMR-WENO schemes are designed by using these new troubled cell indicators, which use the simple linear upwind schemes in smooth areas and the sophisticated MR-WENO schemes in discontinuous areas, thus achieving the goal of inheriting the excellent characteristics of original MR-WENO schemes while reducing computational costs. The new modified methodology is divided into two parts: if all extreme points of two reconstruction polynomials defined on the big spatial stencil are outward the smallest interval [ x i − 1 / 2 , x i + 1 / 2 ] , the numerical flux is straightforwardly approximated by a high-degree reconstruction polynomial and the approximation has a high-order accuracy. Otherwise, the high-order MR-WENO spatial reconstruction procedures are adopted. The main benefits of new MMR-WENO schemes are their robustness and efficiency in comparison to original MR-WENO schemes, since the MMR-WENO schemes could save at most 63%-79% CPU time than the same-order MR-WENO schemes do for some numerical examples. • A new type of high-order finite difference modified MR-WENO schemes is proposed for solving hyperbolic conservation laws. • A simple troubled cell indicator is designed based on two reconstruction polynomials. • The new troubled cell indicator works in the target cell without any artificially adjustable parameters. • The modified MR-WENO schemes are more robust and efficient in comparison to the original MR-WENO schemes. [ABSTRACT FROM AUTHOR]

In this paper a highly parallel method is developed for simulating the elastodynamics of a four-chamber human heart with patient-specific geometry. The heterogeneous hyperelastic model is discretized by a finite element method in space and a fully implicit adaptive method in time, and the resulting nonlinear algebraic systems are solved by a scalable domain decomposition algorithm. The deformations of the cardiac muscles are quite complex due to the realistic geometry, the heterogeneous hyperelasticity of the cardiac tissue, and the myocardial fibers with active stresses. Moreover, the deformations in different chambers and at different phases of the cardiac cycle are very different. To simulate all the muscle movements including the atrial diastole, the atrial systole, the isovolumic contraction, the ventricular ejection, the isovolumic relaxation, and the ventricular filling, the temporal-spatial mesh needs to be sufficiently fine, but not too fine so that the overall computing time is manageable, we introduce a baseline mesh in space and a two-level time stepping strategy including a uniform baseline time step size to obtain the desired time accuracy and an adaptive time stepping method within a baseline time step to guarantee the convergence of the nonlinear solver. Through numerical experiments, we investigate the performance of the proposed method with respect to the material coefficients, the fiber orientations, as well as the mesh sizes and the time step sizes. For an unstructured tetrahedral mesh with more than 200 million degree of freedoms, the method scales well for up to 16,384 processor cores for all steps of an entire cardiac cycle. • Introduce a highly parallel method for simulating elastodynamics of a human heart. • A 4-chamber patient-specific heart is modeled by heterogeneous material with fibers. • A fully unstructured finite element method is used for the complex geometry. • A nonlinear domain decomposition algorithm is developed for the algebraic systems. • A 2-level adaptive time integration scheme is introduced for accuracy and robustness. [ABSTRACT FROM AUTHOR]

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to Babuška and Lipton (2011) [30] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximizing model order reduction. The implementation of the framework in the Distributed and Unified Numerics Environment (DUNE) software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimization or uncertainty quantification on challenging problems in composite aero-structures. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domains in the Euclidean sphere of R 3 with Neumann boundary conditions. We address two approaches: the first one is a generalization of the initial problem leading to a density method and the other one is a shape optimization procedure using the level-set method. The original goal of those methods was to investigate the conjecture according to which the geodesic ball was optimal for the first non-trivial eigenvalue under certain conditions. These computations provide strong insight into the optimal shapes of those eigenvalue problems and show a rich variety of geometries regarding the proportion of the surface area of the sphere occupied by the domain. In the last part, the same algorithms are used to carry out the same investigations on a torus. [ABSTRACT FROM AUTHOR]

Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI's key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O (100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications. • Proposed a gradient-free inference algorithm for Bayesian physics informed neural networks (BPINNs). • Reframed BPINNs as a classic inverse problem to utilize Ensemble Kalman Inversion (EKI). • Demonstrated our method achieves informative uncertainty estimates at lower cost. [ABSTRACT FROM AUTHOR]

In this paper, we propose an elegant methodology to treat sharp interfaces that are implicitly defined which does not require (a) enrichment functions, (b) additional linear and bilinear terms such as the inter-element penalty terms as in Nitsche's method, or use of multipliers like Lagrange multiplier, in the weak form for enforcing the jump conditions across the interface, and (c) modification to the standard virtual element method solution space. The background mesh consists of structured quadrilateral elements with each element consisting of eight nodes, namely, the four vertices and the mid-points of four edges. A simple and efficient idea to generate an interface-fitted mesh is discussed where the number of nodes remains invariant, esp., for moving boundary problems. A linear virtual element method approximation is assumed on the fitted mesh. The efficiency and accuracy of the presented technique is demonstrated by solving and verifying the rate of convergence in both L 2 norm and H 1 semi-norm, for the benchmark problems with interfaces of various geometries and moving interfaces. • Proposes an elegant way to treat embedded interface problems. • Does not need massive local remeshing. • Dirichlet boundary conditions can be imposed directly on the interface. • Number of nodes remain constant even as the interface evolves. • Less sensitive to mesh distortion. [ABSTRACT FROM AUTHOR]

We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems. • Proposed a new numerical method for modeling SDEs with observation data. • Established a mathematical framework that utilizes stochastic flow map for effective modeling. • Developed novel stochastic flow map modeling and its DNN structure. [ABSTRACT FROM AUTHOR]

This paper presents a Gaussian Process (GP) framework, a non-parametric technique widely acknowledged for regression and classification tasks, to address inverse problems in mean field games (MFGs). By leveraging GPs, we aim to recover agents' strategic actions and the environment's configurations from partial and noisy observations of the population of agents and the setup of the environment. Our method is a probabilistic tool to infer the behaviors of agents in MFGs from data in scenarios where the comprehensive dataset is either inaccessible or contaminated by noises. [ABSTRACT FROM AUTHOR]

Free-surface, mobile-bed, shallow-flow models may present Hybrid hyperbolic systems of partial differential equations characterised by conservative and non-conservative fluxes that can only be expressed in primitive variables. This paper presents the effort we made to derive DOT-type schemes (Osher-type schemes derived by Dumbser and Toro, 2011 [1]) for these kinds of systems formulated for the one-dimensional case. Firstly, for a Hybrid system, we managed to write a quasi-linear form characterised by the presence of a matrix, expressed as a function of the primitive variables, that multiplies the spatial derivative of the conserved variables. Next, we derived the first numerical flux by adapting the approach of Leibinger et al., 2016 [2] to this quasi-linear form. We called this result DOT HCP flux. To achieve a faster algorithm, instead of using an integration path in the space of conserved variables, as in the previous case, we employed a path in the space of primitive variables. We called this second formulation DOT HPP flux. Subsequently, we managed to account for certain physical constraints arising from the generalised Rankine-Hugoniot relations in the expression of one term of the previous flux formulation, thus obtaining the DOT HZR flux. Finally, we showed that these methods can also be applied to Combined systems characterised by conservative and non-conservative fluxes expressed in conserved variables. Several tests show the characteristics and good performances of the proposed methods when applied to Riemann problems of Hybrid and Combined systems deriving from free-surface models. Finally, thanks to the general formulation of the proposed DOT-type fluxes, these can also be applied to Hybrid and Combined hyperbolic systems deriving from different physical problems. [ABSTRACT FROM AUTHOR]

• Not applicable. This is not a research paper but one on the history of CFD. The genesis and contents of the 2010 poster "History of CFD Part II" are described. [ABSTRACT FROM AUTHOR]

Triangular surface-based 3D IIM (Immersed Interface Method) algorithms face major challenges due to the need to calculate surface derivative of jumps. This paper proposes a fast, easy-to-implement, surface-derivative-free of jumps, augmented IIM (AIIM) with triangulated surfaces for 3D Helmholtz interface problems for the first time, which combines the simplified AIIM with domain decomposed and embedding techniques. The computational domain is divided into sub-domains along the interface and the solutions of sub-domains are continuously extended into larger regular domains by embedding. The jumps in normal derivative of solution along the interfaces in the extended domains are introduced as unknowns to impose the original jump relations. The original problem is simplified into Helmholtz interface problems with constant coefficients by coupling them with the augmented equation, which is then solved using fast simplified AIIM. This approach eliminates the need to compute surface derivatives of jumps, making implementation of 3D IIM based on triangulated surfaces fairly simple. Numerical results demonstrate that the algorithm is efficient and can achieve the overall second-order accuracy. • A novel, easy-to-implement and fast augmented simplified IIM is proposed for 3D Helmholtz interface problems. • It seems to be the first work on IIM with the triangular surface mesh indeed. • The method provides a fairly simple way to compute the correction terms without computing surface derivatives of jumps. • Various numerical examples verify the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding eigenfunctions is one of the basic tasks in mathematical and computational physics. With the help of tensor neural network, the high dimensional integrations included in the loss functions of the machine learning process can be computed with high accuracy. The high accuracy of high dimensional integrations can improve the accuracy of the machine learning method for computing multi-eigenpairs of high dimensional eigenvalue problems. Here, we introduce the tensor neural network and design the machine learning method for computing multi-eigenpairs of the high dimensional eigenvalue problems. The proposed numerical method is validated with plenty of numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized Riemann problem (GRP) scheme for a laminar two-phase flow model. The model takes into account the distinctions between different densities and velocities, and is obtained by averaging vertical velocities across each layer for the two-phase flows. The rarefaction wave and the shock wave are analytically resolved by using the Riemann invariants and Rankine-Hugoniot condition, respectively. The source term is incorporated into the resolution of the GRP method. We further extend the GRP method to the two-dimensional (2-D) system, which is non-conservative. The Strang splitting method is applied, but it still can not provide explicit Riemann invariants and shock relations, which prevent us to apply the GRP method directly. Another splitting technique is also applied to the 2-D case, such that each split subsystem contains only one family of waves. Numerical experiments on some typical problems show that the proposed method achieves good performance. [ABSTRACT FROM AUTHOR]

This paper presents high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights to solve hyperbolic conservation laws on triangular meshes. They are abbreviated as the ALW-MR-WENO schemes. The novel third-order, fourth-order, and fifth-order ALW-MR-WENO schemes are designed by applying two unequal-sized hierarchical central stencils in comparison to the classical WENO schemes which utilize many equal-sized upwind biased/central stencils. With the application of one simple condition, only two linear weights are automatically adjusted to be positive values on condition that their summation is one. The novel finite volume ALW-MR-WENO schemes could maintain the designed order of accuracy in smooth areas and reduce to the first-order accuracy so as to keep essentially non-oscillatory properties around strong discontinuities. So it is the first time that any high-order WENO schemes with the application of only two unequal-sized stencils are obtained on triangular meshes. And the major benefits are their efficiency, compactness, and simplicity in large scale engineering applications on unstructured meshes. Finally, several tests are used to indicate the effectiveness of these new finite volume WENO schemes. • We design high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights on triangular meshes. They use only two unequal-sized central spatial stencils to obtain arbitrarily high-order spatial approximations in smooth regions. • Two linear weights that sum to one can be automatically adjusted to be any positive values with one simple condition on triangular meshes. • Their major benefits are the high efficiency, compactness, robustness, and simplicity in large scale engineering applications on triangular meshes. [ABSTRACT FROM AUTHOR]

The direct simulation Monte Carlo (DSMC) method is widely used for numerical solutions of the Boltzmann equation. However, the associated computational cost becomes prohibitive in the near-continuum regime. To address this limitation, the particle-based Fokker–Planck (FP) method has been extensively studied in the past decade. The FP equation, which describes Brownian motion, does not require resolution of the collisional time and length scales. While several monatomic FP models have been proposed, the modeling of diatomic gases within the FP framework has received limited attention. In this paper, we propose a new diatomic kinetic model, named the Fokker–Planck–Master (FPM) model, which can accurately describe energy exchanges between translational-rotational and translational-vibrational modes. The FPM model combines the FP equation to describe the evolution of translational and rotational modes, and the master equation to describe the evolution of the vibrational modes. The numerical test cases include relaxation problems, Couette flows, and hypersonic flows past a vertical flat plate. The results demonstrate that the FPM model shows good agreement with both analytical and DSMC solutions. • A new diatomic kinetic model based on the Fokker–Planck and equations. • The correct Prandtl number, accurate relaxation rates for internal energies, and the H-theorem are included in the model. • The development of a conservative scheme for robust numerical simulations. • The application of the model to rarefied and hypersonic flows. [ABSTRACT FROM AUTHOR]

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]

Developing surrogate models for costly mathematical models representing physical systems is challenging since it is typically not possible to generate large training data sets, i.e. to create a large experimental design. In such cases, it can be beneficial to constrain the surrogate approximation to adhere to the known physics of the model. This paper presents a novel methodology for the construction of physics-informed polynomial chaos expansions (PCE) that combines the conventional experimental design with additional constraints from the physics of the model represented by a set of differential equations and specified boundary conditions. A computationally efficient means of constructing physically constrained PCEs, termed PC2, are proposed and compared to the standard sparse PCE. Algorithms are presented for both full-order and sparse PC2 expansions and an iterative approach is proposed for addressing nonlinear differential equations. It is shown that the proposed algorithms lead to superior approximation accuracy and do not add significant computational burden over conventional PCE. Although the main purpose of the proposed method lies in combining training data and physical constraints, we show that the PC2 can also be constructed from differential equations and boundary conditions alone without requiring model evaluations. We further show that the constrained PCEs can be easily applied for uncertainty quantification through analytical post-processing of a reduced PCE by conditioning on the deterministic space-time variables. Several deterministic examples of increasing complexity are provided and the proposed method is demonstrated for uncertainty quantification. • PC2 – a novel framework for physically constrained polynomial chaos expansions is proposed. • An efficient algorithm based on constrained least squares and sparse solver is developed. • Analytical uncertainty quantification of approximated PDEs by PC2 is performed. [ABSTRACT FROM AUTHOR]

The Immersed Boundary (IB) method of Peskin (1977) [1] is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions. • Reformulated Immersed Boundary method for double layer integral equations. • Efficient IB constraint method for PDEs with prescribed boundary values. • Well-conditioned IB method resulting from second-kind integral equation formulation. • IB method for Neumann boundary conditions. [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]

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 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]

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]

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]

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]