In this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws When a difference scheme introduces artificial numerical diffusion, for example the Lax-Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax-Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock. In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first-order accurate; they are known to be contractive in the sense of any lp norm. Jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes. Here we study similar questions for the Lax-Wendroff scheme without artificial viscosity; this is a nonmonotone, second-order accurate scheme. We prove existence of a one-parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm. The proof relies on studying the linearized Lax-Wendroff scheme. 相似文献
We give a brief discussion of some of the contributions of Peter Lax to Computational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We develop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh’s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented. 相似文献
In this paper we construct a new type of symmetrical dissipative difference scheme. Except discontinuity these schemes have uniformly second-order accuracy. For calculation using these, the simple-wave is very exact, the shock has high resolution, the programing is simple and the CPU time is economical. Since the paper [1] introduced that in some conditions Lax-Wendroff scheme would converge to nonphysical solution, many researchers have discussed this problem. According to preserving the monotonicity of the solution preserving monotonial schemes and TVD schemes have been introduced by Harten, et. According to property of hyperbolic wave propagation the schemes of split-coefficient matrix (SCM) and split-flux have been formed. We emphasize the dissipative difference scheme, and these schemes are dissipative on arbitrary conditions. 相似文献
In this paper, the Lax-Wendroff and “cabaret” schemes for the Buckley-Leverett equation are studied. It is shown that these schemes represent unstable solutions. The choice of an unstable solution depends on the Courant number only. A finite-element version of the “cabaret” scheme is given. 相似文献
We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed estimates of the approximate solutions, compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.
A cubic spline approximation to the wave equation is shown toproduce a fully implicit finite difference representation, themore usual explicit and implicit approximations of this equationbeing shown as special cases of the formulation given here.The truncation error and stability condition are derived forthe scheme produced, and the full computational procedure forthe scheme's solution is developed. 相似文献