CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX
CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX
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DOI：
10.1142/s0252959904000299
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发表时间：
2004-07
期刊：
影响因子：
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通讯作者：
K. Karlsen;John D. Towers
中科院分区：
文献类型：
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作者：
K. Karlsen;John D. Towers
本文给出了非凸真非线性守恒律方程的Lax-Friedrichs差分格式的第一个收敛性证明，其中允许系数k（x，t）在（x，t）平面上沿沿着曲线不连续.与大多数现有的文献中关于不连续系数的问题相反，这里的收敛性证明不是基于奇异映射方法，而是基于div-curl引理（但不是Young测度）和Lax型熵估计，该估计对于k（x，t）的正则性是鲁棒的。在[14]的基础上，作者提出了熵解的定义，将经典的Kružkov定义推广到k（x，t）在（x，t）平面上分段Lipschitz连续的情形，并证明了其稳定性假设通量函数满足所谓的交叉条件，并且解的强迹沿着曲线存在，其中k（x，t）是不连续的。它示出的收敛子序列的近似产生的拉克斯-弗里德里希计划收敛到这样的熵的解决方案，这意味着整个计算序列收敛。
The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form where the coefficient k(x,t) is allowed to be discontinuous along curves in the (x,t) plane. In contrast to most of the existing literature on problems with discontinuous coefficients, here the convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of k(x,t). Following [14], the authors propose a definition of entropy solution that extends the classical Kružkov definition to the situation where k(x,t) is piecewise Lipschitz continuous in the (x,t) plane, and prove the stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x,t) is discontinuous. It is shown that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.