On the convergence of Godunov scheme for nonlinear hyperbolic systems
On the convergence of Godunov scheme for nonlinear hyperbolic systems
复制标题
非线性双曲系统Godunov格式的收敛性
DOI：
10.1142/s0252959900000303
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
H. Jenssen
中科院分区：
文献类型：
--
作者：
A. Bressan;H. Jenssen
作者考虑了以下形式的系统：假设矩阵 A(u) 是严格双曲的，并且具有特征向量场的积分曲线是直线的特性。对于此类系统，我们可以定义一个自然黎曼求解器，从而定义一个戈杜诺夫方案，它概括了保守系统的标准黎曼求解器和戈杜诺夫方案。本文展示了该方案应用于总变异较小的数据时的收敛性和 L1 稳定性。证明的主要步骤是估计由于二次耦合项而导致方案产生的总变异的增加。使用杜哈梅尔原理，问题被简化为两个格林核乘积的估计，代表离散随机游走的概率密度。然后，耦合总量由平均速度严格不同的两次随机游走之间的预期交叉数量决定。这提供了[3,9]中与连续随机过程相关的论证的离散模拟。
The authors consider systems of the form where the matrix A(u) is assumed to be strictly hyperbolic and with the property that the integral curves of the eigenvector fields are straight lines. For this class of systems one can define a natural Riemann solver, and hence a Godunov scheme, which generalize the standard Riemann solver and Godunov scheme for conservative systems. This paper shows convergence and L1 stability for this scheme when applied to data with small total variation. The main step in the proof is to estimate the increase in the total variation produced by the scheme due to quadratic coupling terms. Using Duhamel's principle, the problem is reduced to the estimate of the product of two Green kernels, representing probability densities of discrete random walks. The total amount of coupling is then determined by the expected number of crossings between two random walks with strictly different average speeds. This provides a discrete analogue of the arguments developed in [3,9] in connection with continuous random processes.