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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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 （236）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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The Perturbed Riemann Problem for a Geometrical Optics System Shiwei Li, Hanchun Yang Communications on Applied Mathematics and Computation    2023, 5 (3): 1148-1179.   DOI: 10.1007/s42967-022-00192-3 Abstract （267）      PDF       Knowledge map    Save The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented. Reference | Related Articles | Metrics | Comments（0）

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Configurations of Shock Regular Reflection by Straight Wedges Qin Wang, Junhe Zhou Communications on Applied Mathematics and Computation    2023, 5 (3): 1256-1273.   DOI: 10.1007/s42967-022-00207-z Abstract （254）      PDF       Knowledge map    Save We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge. It has been known that patterns of the shock reflection are various and complicated, including the regular and the Mach reflection. Most of the fundamental issues for the shock reflection have not been understood. Recently, there are great progress on the mathematical theory of the shock regular reflection problem, especially for the global existence, uniqueness, and structural stability of solutions. In this paper, we show that there are two more possible configurations of the shock regular reflection besides known four configurations. We also give a brief proof of the global existence of solutions. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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Global Existence and Stability of Solutions to River Flow System Xian-ting Wang, Yun-guang Lu, Naoki Tsuge Communications on Applied Mathematics and Computation    2023, 5 (3): 1247-1255.   DOI: 10.1007/s42967-022-00198-x Abstract （233）      PDF       Knowledge map    Save In this short note, we are concerned with the global existence and stability of solutions to the river flow system. We introduce a new technique to set up a relation between the Riemann invariants and the finite mass to obtain a time-independent, bounded solution for any adiabatic exponent. The global existence of solutions was known long ago [Klingenberg and Lu in Commun. Math. Phys. 187: 327–340, 1997]. However, since the uncertainty of the function b(x), which corresponds physically to the slope of the topography, the L∞ estimates growed larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. By employing a suitable mathematical transformation to control the slope of the topography by the friction and the finite mass, we prove the uniformly bounded estimate with respect to the time variable. This means that our solutions are stable. Reference | Related Articles | Metrics | Comments（0）

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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 （262）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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Recent Progress on Outflow/Inflow Problem for Viscous Multi-phase Flow Fangfang Hao, Hai-Liang Li, Luyao Shang, Shuang Zhao Communications on Applied Mathematics and Computation    2023, 5 (3): 987-1014.   DOI: 10.1007/s42967-022-00194-1 Abstract （229）      PDF       Knowledge map    Save According to the boundary condition with the zero, negative, or positive velocity, the initial boundary problem for compressible multi-phase flow with the Dirichlet-type boundary condition can be classified into three cases: impermeable problem, inflow problem, or outflow problem. In this paper, we review the recent progress on the existence and nonlinear stability of the stationary solution to the outflow/inflow problems for viscous multi-phase flow. Reference | Related Articles | Metrics | Comments（0）

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Singularity Formation for the General Poiseuille Flow of Nematic Liquid Crystals Geng Chen, Majed Sofiani Communications on Applied Mathematics and Computation    2023, 5 (3): 1130-1147.   DOI: 10.1007/s42967-022-00190-5 Abstract （244）      PDF       Knowledge map    Save We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation. In this paper, we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation, extended from an earlier result on a special case. Reference | Related Articles | Metrics | Comments（0）

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Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems Gui-Qiang G. Chen Communications on Applied Mathematics and Computation    2023, 5 (3): 1015-1052.   DOI: 10.1007/s42967-022-00210-4 Abstract （225）      PDF       Knowledge map    Save We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional (2-D) Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations. In particular, we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws Matania Ben-Artzi, Jiequan Li Communications on Applied Mathematics and Computation    2023, 5 (3): 1289-1298.   DOI: 10.1007/s42967-022-00224-y Abstract （195）      PDF       Knowledge map    Save This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）

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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 （240）      PDF       Knowledge map    Save 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. Reference | Related Articles | Metrics | Comments（0）