ON THE CENTRAL RELAXING SCHEME Ⅱ: SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

1. IntroductionWe are interested in construction of the central reltalng sChemes for system of noIilinearhyperbolic conservation lawswith initial data U(0, x) = Uo(x), x = (x1 ? ...! xd), based on the local relaJxation approkimationof Eq.(1.1) [2, 3, 6, 8, 9, 12].To i11ustrate the basic idea of the relaalng schemes, for the sake of simplicity in the presentation, we restrict our attention to onedimensional scalar conservaioll lawsFirst, introduce a linear hyperbollc system with a stiff sourc…     

THE RELAXING SCHEMES FOR HAMILION-JACOBI EQUATIONS

1. IntroductionWe are illterested in the numerical approximation of viscosity solution of the following lirstorder Hamilton-Jacobi eqllationopt + H(&.,, rk..' ...) gb'.) - 0' (1'1)with initial data ac(~, 0) = ado(x). It is well known that the solutions to problem (1.1) typicallyare continuous (typically they are locally Lipschitz continuous) but with discoatinuous derivatives, even though the initial data ado E Coo. The nonuniqueness of such solutions to (1.1) alsonecessitates the introducti…     

Existence and asymptotic stability of relaxation discrete shock profiles

In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in , provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

     

Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L^∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.     

Shock Layers Interactions for a Relaxation Approximation to Conservation Laws

We consider the interaction between shock waves in a semilinear relaxation approximation to N×N systems of conservation laws. This interaction is described by proving the existence of a particular solution to the relaxing system, with a certain asymptotic as Received July 1998     

MULTIDIMENSIONAL RELAXATION APPROXIMATIONS FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.     

Relaxation WENO schemes for multidimensional hyperbolic systems of conservation laws

We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws

This paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws. Under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws. The proof is due to the energy principle, using the weighted norms, the interaction of waves from various modes is treated by imposing suitable weight matrix.     

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

High order Hybrid central—WENO finite difference scheme for conservation laws

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.     

Nonstrictly hyperbolic systems with stiff relaxation terms

In this paper, a zero factor idea is introduced to extend the convergence framework in [G.-Q. Chen, C.D. Levermore, T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994) 787-830] for the singular limits of stiff relaxation from general 2×2 hyperbolic conservation systems to nonstrictly hyperbolic systems and an application of this framework on the so-called system of extended traffic flow is obtained.     

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Summary. This paper concerns the study of a relaxation scheme for hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero. Received September 29, 1998 / Revised version received December 20, 1999 / Published online August 24, 2000     

On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms

The stability of nonlinear explicit difference schemes with not, in general, open domains of the scheme operators are studied. For the case of path-connected, bounded, and Lipschitz domains, we establish the notion that a multi-level nonlinear explicit scheme is stable iff (if and only if) the corresponding scheme in variations is stable. A new modification of the central Lax–Friedrichs (LxF) scheme is developed to be of the second-order accuracy. The modified scheme is based on nonstaggered grids. A monotone piecewise cubic interpolation is used in the central scheme to give an accurate approximation for the model in question. The stability of the modified scheme is investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second-order nonstaggered central scheme based on operator-splitting techniques.     

BV-estimates of Lax–Friedrichs' scheme for hyperbolic conservation laws with relaxation

In this paper, we will give BV-estimates of Lax–Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ε>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence of two-dimensional staggered central schemes on unstructured triangular grids

In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.     

Error analysis of discrete conservation laws for Hamiltonian PDEs under the central box discretizations

The central box scheme has been the most successful of the multisymplectic integrators for Hamiltonian PDEs. In this paper, we investigate conservative properties of the central box scheme for Hamiltonian PDEs and derive the error formulas of discrete local and global conservation laws of energy and momentum. We apply these results to the nonlinear Schrödinger equation and Klein-Gordon equation. Numerical experiments are presented to verify the theoretical predications.     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.     

Solving shallow-water systems in 2D domains using Finite Volume methods and multimedia SSE instructions

The goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. García, J.M. González, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.     

一个解KdV方程的满足两个守恒律的差分格式

Korteweg-de Vries(KdV)方程是人们在研究一些物理问题时得到的非线性波 动方程,其解满足无穷多个守恒律．本文为该方程设计了一种差分格式,其采用的是有限 体积法．但与传统的有限体积法不同的是,它的数值解同时满足两个相关的守恒律．这样 可以更好地保持解的物理上的守恒性质．数值例子表明这一算法是有效的．