Communications on Applied Mathematics and Computation

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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 （996）

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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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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 （960）

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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 （956）

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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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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 （955）

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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 （948）

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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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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 （891）

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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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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 （883）

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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 （877）

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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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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 （861）

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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 （859）

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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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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 （843）

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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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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 （837）

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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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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 （836）

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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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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 （835）

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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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Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

Jiequan Li, Wancheng Sheng, Chi-Wang Shu, Ping Zhang, Yuxi Zheng

Communications on Applied Mathematics and Computation 2023, 5 (3): 965-966. DOI: 10.1007/s42967-022-00230-0

Abstract （117）

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Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

Yanbo Hu, Xingxing Li

Communications on Applied Mathematics and Computation 2023, 5 (3): 1053-1096. DOI: 10.1007/s42967-021-00185-8

Abstract （103）

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This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section. Under the conical assumption, the three-dimensional (3-D) steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles. Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane, we construct a classical conical-supersonic solution near the curve under some reasonable assumptions. To overcome the difficulty caused by the parabolic degeneracy, we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane. The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully. We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

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Global Well-Posedness for Aggregation Equation with Time-Space Nonlocal Operator and Shear Flow

Binbin Shi, Weike Wang

Communications on Applied Mathematics and Computation 2023, 5 (3): 1274-1288. DOI: 10.1007/s42967-022-00214-0

Abstract （101）

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In this paper, we consider the two-dimensional aggregation equation with the shear flow and time-space nonlocal attractive operator. Without the advection, the solution of the aggregation equation may blow up in finite time. We show that the shear flow can suppress the blow-up.

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Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law

Gui-Qin Qiu, Gao-Wei Cao, Xiao-Zhou Yang, Yuan-An Zhao

Communications on Applied Mathematics and Computation 2023, 5 (3): 1180-1234. DOI: 10.1007/s42967-022-00245-7

Abstract （94）

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For the two-dimensional (2D) scalar conservation law, when the initial data contain two different constant states and the initial discontinuous curve is a general curve, then complex structures of wave interactions will be generated. In this paper, by proposing and investigating the plus envelope, the minus envelope, and the mixed envelope of 2D non-selfsimilar rarefaction wave surfaces, we obtain and the prove the new structures and classifications of interactions between the 2D non-selfsimilar shock wave and the rarefaction wave. For the cases of the plus envelope and the minus envelope, we get and prove the necessary and sufficient criterion to judge these two envelopes and correspondingly get more general new structures of 2D solutions.

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Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

Geng Lai, Yingchun Shi

Communications on Applied Mathematics and Computation 2023, 5 (3): 1235-1246. DOI: 10.1007/s42967-022-00197-y

Abstract （92）

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We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data. Using the characteristic decomposition method proposed by Li et al. (Commun Math Phys 267: 1–12, 2006), we derive a group of characteristic decompositions for the system. Using these characteristic decompositions, we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.

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On the Vortex Sheets of Compressible Flows

Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan

Communications on Applied Mathematics and Computation 2023, 5 (3): 967-986. DOI: 10.1007/s42967-022-00191-4

Abstract （91）

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This paper provides a review of the recent results on the stability of vortex sheets in compressible flows. Vortex sheets are contact discontinuities of the underlying flows. The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis. The formulation of the vortex sheet problem will be introduced. The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized. The linear stability of vortex sheets for the three-dimensional elastic flows is also presented. The difficulties of the vortex sheet problems and the ideas of proofs are discussed.

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