偏微分方程组差分逼近的强相合性和Thomas分解
Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations
摘要：
对于一类广泛的多项式非线性系统的偏微分方程，我们提出了一种算法的方法，结合微分和差分代数分析的s（强）-一致性的有限差分近似。我们的方法是适用于定期解决方案的网格。对于这种类型的网格，我们给出了一个新的定义的s-一致性有限差分近似，推广了我们的定义前面给出的笛卡尔网格。本文提出的s-一致性的算法验证是基于使用差分和差分托马斯分解。首先，我们将微分分解应用于输入系统，导致其解空间的划分。然后，输出子系统，包含一个解决方案的兴趣，我们应用差分模拟的差分托马斯分解，允许检查的s-一致性。对于线性和某些拟线性微分系统，也可以应用差分\Gr基进行s-一致性分析。我们说明我们的方法和算法的一些例子，其中包括粘性不可压缩流的Navier-Stokes方程。
英文摘要：
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.
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