Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

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.

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

December 9,2022 is the 90th birthday of Tong Zhang,a mathematician in Institute of Mathematics,Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life.To celebrate his 90th birthday and great contributions to this specifc feld,we organize this focused issue in the journal Communications on Applied Mathematics and Computation,since the Riemann problem has been proven to be a building block in all felds of theory,numerics and applications of hyperbolic conservation laws.

Transversal effects of high order numerical schemes for compressible fluid flows

Finite volume schemes for the two-dimensional(2D) wave system are taken to demonstrate the role of the genuine dimensionality of Lax-Wendroff flow solvers for compressible fluid flows. When the finite volume schemes are applied, the transversal variation relative to the computational cell interfaces is neglected, and only the normal numerical flux is used, thanks to the Gauss-Green formula. In order to offset such defects, the Lax-Wendroff flow solvers or the generalized Riemann problem(GRP) solvers are adopted by substituting the time evolution of flows into the spatial variation. The numerical results show that even with the same convergence rate, the error by the GRP2D solver is almost one ninth of that by the multistage Runge-Kutta(RK) method.

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

This paper establishes the resolution finite volume scheme with slope entropy convergence of a new two-value high relaxation for conservation laws. This scheme, motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities, unifies and evolves slopes directly with a slope relaxation equation that governs the evolution of slopes in both smooth and discontinuous regions. Proper choices of slopes are realized adaptively via a relaxation parameter. The scheme is shown to be total-variation-bounded （TVB） stable and satisfies cell-entropy inequalities.

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

With increasing engineering demands,there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct“physics”.There are two families of high order methods:One is the method of line,relying on the Runge-Kutta(R-K)time-stepping.The building block is the Riemann solution labeled as the solution element“1”.Each step in R-K just has first order accuracy.In order to derive a fourth order accuracy scheme in time,one needs four stages labeled as“1111=4”.The other is the one-stage Lax-Wendroff(LW)type method,which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems.In recent years,the pair of solution element and dynamics element,labeled as“2”,are taken as the building block.The direct adoption of the dynamics implies the inherent temporal-spatial coupling.With this type of building blocks,a family of two-stage fourth order accurate schemes,labeled as“22=4”,are designed for the computation of compressible fluid flows.The resulting schemes are compact,robust and efficient.This paper contributes to elucidate how and why high order accurate schemes should be so designed.To some extent,the“22=4”algorithm extracts the advantages of the method of line and one-stage LW method.As a core part,the pair“2”is expounded and LW solver is revisited.The generalized Riemann problem(GRP)solver,as the discontinuous and nonlinear version of LW flow solver,and the gas kinetic scheme(GKS)solver,the microscopic LW solver,are all reviewed.The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed.Besides,the computational performance and prospective discussions are presented.

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

The adaptive generalized Riemann problem(GRP)scheme for 2-D compressible fluid flows has been proposed in[J.Comput.Phys.,229(2010),1448–1466]and it displays the capability in overcoming difficulties such as the start-up error for a single shock,and the numerical instability of the almost stationary shock.In this paper,we will provide the accuracy study and particularly show the performance in simulating 2-D complex wave configurations formulated with the 2-D Riemann problems for compressible Euler equations.For this purpose,we will first review the GRP scheme briefly when combined with the adaptive moving mesh technique and consider the accuracy of the adaptive GRP scheme via the comparison with the explicit formulae of analytic solutions of planar rarefaction waves,planar shock waves,the collapse problem of a wedge-shaped dam and the spiral formation problem.Then we simulate the full set of wave configurations in the 2-D four-wave Riemann problems for compressible Euler equations[SIAM J.Math.Anal.,21(1990),593–630],including the interactions of strong shocks(shock reflections),vortex-vortex and shock-vortex etc.This study combines the theoretical results with the numerical simulations,and thus demonstrates what Ami Harten observed"for computational scientists there are two kinds of truth:the truth that you prove,and the truth you see when you compute"[J.Sci.Comput.,31(2007),185–193].

Remapping-Free Adaptive GRP Method for Multi-Fluid Flows I:One Dimensional Euler Equations

In this paper,a remapping-free adaptive GRP method for one dimensional(1-D)compressible flows is developed.Based on the framework of finite volume method,the 1-D Euler equations are discretized on moving volumes and the resulting numerical fluxes are computed directly by the GRP method.Thus the remapping process in the earlier adaptive GRP algorithm[17,18]is omitted.By adopting a flexible moving mesh strategy,this method could be applied for multi-fluid problems.The interface of two fluids will be kept at the node of computational grids and the GRP solver is extended at the material interfaces of multi-fluid flows accordingly.Some typical numerical tests show competitive performances of the new method,especially for contact discontinuities of one fluid cases and the material interface tracking of multi-fluid cases.