Communications on Applied Mathematics and Computation

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Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

Jean-Luc Guermond, Bojan Popov, Laura Saavedra

Communications on Applied Mathematics and Computation 2023, 5 (2): 923-945. DOI: 10.1007/s42967-021-00165-y

Abstract （1200）

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An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving nonlinear hyperbolic systems is developed. The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements. The method is made invariant domain preserving for the Euler equations using convex limiting and is tested on various benchmarks.

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Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

J. C. González-Aguirre, J. A. González-Vázquez, J. Alavez-Ramírez, R. Silva, M. E. Vázquez-Cendón

Communications on Applied Mathematics and Computation 2023, 5 (2): 885-922. DOI: 10.1007/s42967-021-00162-1

Abstract （1141）

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Sediment transport can be modelled using hydrodynamic models based on shallow water equations coupled with the sediment concentration conservation equation and the bed conservation equation. The complete system of equations is made up of the energy balance law and the Exner equations. The numerical solution for this complete system is done in a segregated manner. First, the hyperbolic part of the system of balance laws is solved using a finite volume scheme. Three ways to compute the numerical flux have been considered, the Q-scheme of van Leer, the HLLCS approximate Riemann solver, and the last one takes into account the presence of non-conservative products in the model. The discretisation of the source terms is carried out according to the numerical flux chosen. In the second stage, the bed conservation equation is solved by using the approximation computed for the system of balance laws. The numerical schemes have been validated making comparisons between the obtained numerical results and the experimental data for some physical experiments. The numerical results show a good agreement with the experimental data.

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AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

Eleuterio F. Toro, Andrea Santacá, Gino I. Montecinos, Morena Celant, Lucas O. Müller

Communications on Applied Mathematics and Computation 2023, 5 (2): 776-852. DOI: 10.1007/s42967-021-00147-0

Abstract （1134）

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In this paper, we present a novel spatial reconstruction scheme, called AENO, that results from a special averaging of the ENO polynomial and its closest neighbour, while retaining the stencil direction decided by the ENO choice. A variant of the scheme, called m-AENO, results from averaging the modified ENO (m-ENO) polynomial and its closest neighbour. The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system, in conjunction with the fully discrete, high-order ADER approach implemented up to fifth order of accuracy in both space and time. The results, as compared to the conventional ENO, m-ENO and WENO schemes, are very encouraging. Surprisingly, our results show that the L₁-errors of the novel AENO approach are the smallest for most cases considered. Crucially, for a chosen error size, AENO turns out to be the most efficient method of all five methods tested.

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Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Paula Luna, Arturo Hidalgo

Communications on Applied Mathematics and Computation 2023, 5 (2): 946-964. DOI: 10.1007/s42967-022-00200-6

Abstract （1133）

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Two-phase flow in porous media is a very active field of research, due to its important applications in groundwater pollution, CO₂ sequestration, or oil and gas production from petroleum reservoirs, just to name a few of them. Fractional flow equations, which make use of Darcy's law, for describing the movement of two immiscible fluids in a porous medium, are among the most relevant mathematical models in reservoir simulation. This work aims to solve a fractional flow model formed by an elliptic equation, representing the spatial distribution of the pressure, and a hyperbolic equation describing the space-time evolution of water saturation. The numerical solution of the elliptic part is obtained using a finite-element (FE) scheme, while the hyperbolic equation is solved by means of two different numerical approaches, both in the finite-volume (FV) framework. One is based on a monotonic upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, whereas the other makes use of a weighted essentially non-oscillatory (ENO) reconstruction. In both cases, a first-order centered (FORCE)-α numerical scheme is applied for intercell flux reconstruction, which constitutes a new contribution in the field of fractional flow models describing oil-water movement. A relevant feature of this work is the study of the effect of the parameter α on the numerical solution of the models considered. We also show that, in the FORCE-α method, when the parameter α increases, the errors diminish and the order of accuracy is more properly attained, as verified using a manufactured solution technique.

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Multi-dimensional Simulation of Phase Change by a 0D-2D Model Coupling via Stefan Condition

Adrien Drouillet, Romain Le Tellier, Raphaël Loubère, Mathieu Peybernes, Louis Viot

Communications on Applied Mathematics and Computation 2023, 5 (2): 853-884. DOI: 10.1007/s42967-021-00157-y

Abstract （1085）

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Considering phase changes associated with a high-temperature molten material cooled down from the outside, this work presents an improvement of the modelling and the numerical simulation of such processes for an application pertaining to the safety of light water nuclear reactors. Postulating a core meltdown accident, the behaviour of the core melt (aka corium) into a steel vessel is of tremendous importance when evaluating the vessel integrity. Evaluating correctly the heat fluxes requires the numerical simulation of the interaction between the liquid material and its solid counterpart which forms during the solidification process, but also may melt back. To simulate this configuration, encountered in various industrial applications, one considers a bi-phase model constituted by a liquid phase in contact and interaction with its solid phase. The liquid phase may solidify in presence of low energetic source, while the solid phase may melt due to an intense heat flux from the high-temperature liquid. In the frame of the in-house legacy code, several simplifying assumptions (0D multi-layer discretization, instantaneous heat transfer via a quadratic temperature profile in solids) are made for the modelling of such phase changes. In the present work, these shortcomings are illustrated and further overcome by solving a 2D heat conduction model in the solid by a mixed Raviart-Thomas finite element method coupled to the liquid phase due to heat and mass exchanges through Stefan condition. The liquid phase is modeled with a 0D multi-layer approach. The 0D-liquid and 2D-solid models are coupled by a Stefan like phase change interface model. Several sanity checks are performed to assess the validity of the approach on 1D and 2D academical configurations for which exact or reference solutions are available. Then more advanced situations (genuine multi-dimensional phase changes and an "industrial-like scenario") are simulated to verify the appropriate behavior of the obtained coupled simulation scheme.

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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

R. Abgrall, J. Nordström, P. Öffner, S. Tokareva

Communications on Applied Mathematics and Computation 2023, 5 (2): 573-595. DOI: 10.1007/s42967-020-00086-2

Abstract （1066）

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In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series[6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

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A Posteriori Stabilized Sixth-Order Finite Volume Scheme with Adaptive Stencil Construction: Basics for the 1D Steady-State Hyperbolic Equations

Gaspar J. Machado, Stéphane Clain, Raphaël Loubère

Communications on Applied Mathematics and Computation 2023, 5 (2): 751-775. DOI: 10.1007/s42967-021-00140-7

Abstract （1047）

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We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High accuracy (up to the sixth-order presently) is achieved, thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation. The stencil is shifted away from troubles (shocks, discontinuities, etc.) leading to less oscillating polynomial reconstructions. Experimented on linear, Bürgers', and Euler equations, we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations. Moreover, we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.

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Construction of Conservative Numerical Fluxes for the Entropy Split Method

Björn Sjögreen, H. C. Yee

Communications on Applied Mathematics and Computation 2023, 5 (2): 653-678. DOI: 10.1007/s42967-020-00111-4

Abstract （1019）

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The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations. The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts (SBP) difference boundary closure of (Gerritsen and Olsson in J Comput Phys 129:245-262, 1996; Olsson and Oliger in RIACS Tech Rep 94.01, 1994; Yee et al. in J Comp Phys 162:33-81, 2000). Sjögreen and Yee (J Sci Comput https://doi.org/10.1007/s10915-019-01013-1) recently proved that the entropy split method is entropy conservative and stable. Standard high-order spatial central differencing as well as high order central spatial dispersion relation preserving (DRP) spatial differencing is part of the entropy stable split methodology framework. The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives. Due to the construction, this conservative numerical flux requires higher operations count and is less stable than the original semi-conservative split method. However, the Tadmor entropy conservative (EC) method (Tadmor in Acta Numerica 12:451-512, 2003) of the same order requires more operations count than the new construction. Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative, a modified nonlinear filter approach of (Yee et al. in J Comput Phys 150:199-238, 1999, J Comp Phys 162:3381, 2000; Yee and Sjögreen in J Comput Phys 225:910934, 2007, High Order Filter Methods for Wide Range of Compressible flow Speeds. Proceedings of the ICOSAHOM09, June 22-26, Trondheim, Norway, 2009) is proposed in conjunction with the entropy split method as the base method for problems containing shock waves. Long-time integration of 2D and 3D test cases is included to show the comparison of these new approaches.

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A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

Matteo Bergami, Walter Boscheri, Giacomo Dimarco

Communications on Applied Mathematics and Computation 2023, 5 (2): 596-637. DOI: 10.1007/s42967-020-00093-3

Abstract （1013）

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In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.

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Preface to the Focused Issue on High-Order Numerical Methods for Evolutionary PDEs

Arturo Hidalgo, Michael Dumbser, Eleuterio F. Toro

Communications on Applied Mathematics and Computation 2023, 5 (2): 529-531. DOI: 10.1007/s42967-022-00229-7

Abstract （1000）

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Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids

Wasilij Barsukow

Communications on Applied Mathematics and Computation 2023, 5 (2): 638-652. DOI: 10.1007/s42967-020-00094-2

Abstract （1000）

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Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

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A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Johannes Markert, Gregor Gassner, Stefanie Walch

Communications on Applied Mathematics and Computation 2023, 5 (2): 679-721. DOI: 10.1007/s42967-021-00120-x

Abstract （992）

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In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

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A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Jonas Zeifang, Andrea Beck

Communications on Applied Mathematics and Computation 2023, 5 (2): 722-750. DOI: 10.1007/s42967-021-00137-2

Abstract （989）

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Considering droplet phenomena at low Mach numbers, large differences in the magnitude of the occurring characteristic waves are presented. As acoustic phenomena often play a minor role in such applications, classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction. In this work, a novel scheme based on a specific level set ghost fluid method and an implicit-explicit (IMEX) flux splitting is proposed to overcome this timestep restriction. A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface. In this part of the domain, the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases. It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method. Applications to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.

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Neural Network-Based Limiter with Transfer Learning

Rémi Abgrall, Maria Han Veiga

Communications on Applied Mathematics and Computation 2023, 5 (2): 532-572. DOI: 10.1007/s42967-020-00087-1

Abstract （959）

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Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes (Morgan et al. in A machine learning approach for detecting shocks with high-order hydrodynamic methods. https://doi.org/10.2514/6.2020-2024; Ray et al. in J Comput Phys 367:166-191. https://doi.org/10.1016/j.jcp.2018.04.029, 2018; Veiga et al. in European Conference on Computational Mechanics and VII European Conference on Computational Fluid Dynamics, vol. 1, pp. 2525-2550. ECCM. https://doi.org/10.5167/uzh-16853 8, 2018). Following this trend, we train a neural network to serve as a shock-indicator function using simulation data from a Runge-Kutta discontinuous Galerkin (RKDG) method and a modal high-order limiter (Krivodonova in J Comput Phys 226:879-896. https://doi.org/10.1016/j.jcp.2007.05.011, 2007). With this methodology, we obtain one- and two-dimensional black-box shock-indicators which are then coupled to a standard limiter. Furthermore, we describe a strategy to transfer the shock-indicator to a residual distribution (RD) scheme without the need for a full training cycle and large dataset, by finding a mapping between the solution feature spaces from an RD scheme to an RKDG scheme, both in one- and two-dimensional problems, and on Cartesian and unstructured meshes. We report on the quality of the numerical solutions when using the neural network shock-indicator coupled to a limiter, comparing its performance to traditional limiters, for both RKDG and RD schemes.

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Optimization of Random Feature Method in the High-Precision Regime

Jingrun Chen, Weinan E, Yifei Sun

Communications on Applied Mathematics and Computation 2024, 6 (2): 1490-1517. DOI: 10.1007/s42967-024-00389-8

Abstract （342）

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Machine learning has been widely used for solving partial differential equations (PDEs) in recent years, among which the random feature method (RFM) exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. Potentially, the optimization problem in the RFM is more difficult to solve than those that arise in traditional methods. Unlike the broader machine-learning research, which frequently targets tasks within the low-precision regime, our study focuses on the high-precision regime crucial for solving PDEs. In this work, we study this problem from the following aspects: (i) we analyze the coefficient matrix that arises in the RFM by studying the distribution of singular values; (ii) we investigate whether the continuous training causes the overfitting issue; (iii) we test direct and iterative methods as well as randomized methods for solving the optimization problem. Based on these results, we find that direct methods are superior to other methods if memory is not an issue, while iterative methods typically have low accuracy and can be improved by preconditioning to some extent.

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A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

Young Kyu Lee, Shingyu Leung

Communications on Applied Mathematics and Computation 2024, 6 (2): 1189-1216. DOI: 10.1007/s42967-023-00303-8

Abstract （312）

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We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces. The approach follows an embedding approach for solving the surface eikonal equation. We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood. Our proposed algorithm is easy to implement and efficient. We will give some two- and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

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A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

Baiying Dong, Zhilin Li, Juan Ruiz-álvarez

Communications on Applied Mathematics and Computation 2024, 6 (2): 992-1012. DOI: 10.1007/s42967-023-00281-x

Abstract （311）

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In this paper, a new finite element and finite difference (FE-FD) method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes. In the spatial discretization, the standard P₁ FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite, while near the interface, the maximum principle preserving immersed interface discretization is applied. In the time discretization, a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate. Correction terms are needed when the interface crosses grid lines. The moving interface is represented by the zero level set of a Lipschitz continuous function. Numerical experiments presented in this paper confirm second order convergence.

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On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Antonio Baeza, Rosa Donat, Anna Martínez-Gavara

Communications on Applied Mathematics and Computation 2024, 6 (2): 1319-1341. DOI: 10.1007/s42967-023-00332-3

Abstract （305）

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Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes. Because they do not involve any special data structure, and do not induce savings in memory requirements, they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required. The multilevel technique can also be applied to balance laws, but in this case, numerical errors may be induced by the technique. We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation, and leads to a more robust multilevel code for balance laws, while maintaining the efficiency rates observed for hyperbolic conservation laws.

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An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

Rémi Abgrall, Fatemeh Nassajian Mojarrad

Communications on Applied Mathematics and Computation 2024, 6 (2): 963-991. DOI: 10.1007/s42967-023-00274-w

Abstract （296）

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We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

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Convergent Data-Driven Regularizations for CT Reconstruction

Samira Kabri, Alexander Auras, Danilo Riccio, Hartmut Bauermeister, Martin Benning, Michael Moeller, Martin Burger

Communications on Applied Mathematics and Computation 2024, 6 (2): 1342-1368. DOI: 10.1007/s42967-023-00333-2

Abstract （295）

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The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: one generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

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