论文标题
有限差近似值的强度一致性和托马斯分解与部分微分方程的系统
Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations
论文作者
论文摘要
对于部分偏微分方程的一类多项式非线性系统，我们建议采用一种算法方法，该方法结合了差分和差异代数以分析有限差近似值的S（TRONG） - 一致性。我们的方法适用于常规溶液网格。对于这种类型的网格，我们给出了有限差近似值的S-一致性的新定义，这概括了我们先前针对笛卡尔网格的定义。本文中介绍的S-一致性的算法验证是基于差异和差异分解的使用。首先，我们将差分分解应用于输入系统，从而导致其解决方案空间的分区。然后，对于包含感兴趣的解决方案的输出子系统，我们应用了差分托马斯分解的差异类似物，该类似物允许检查S-符合性。对于线性和某些准线性差异系统，也可以将差异\ GR碱基应用于S符合性分析。我们通过许多示例来说明我们的方法和算法，其中包括用于粘性不可压缩流的Navier-Stokes方程。
For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which generalizes our definition given earlier for Cartesian grids. The algorithmic verification of s-consistency presented in the paper is based on the use of both differential and difference Thomas decomposition. First, we apply the differential decomposition to the input system, resulting in a partition of its solution space. Then, to the output subsystem that contains a solution of interest we apply a difference analogue of the differential Thomas decomposition which allows to check the s-consistency. For linear and some quasi-linear differential systems one can also apply difference \Gr bases for the s-consistency analysis. We illustrate our methods and algorithms by a number of examples, which include Navier-Stokes equations for viscous incompressible flow.
