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.

中文翻译：

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偏微分方程组有限差分逼近的强一致性和 Thomas 分解

对于偏微分方程的一大类多项式非线性系统，我们建议采用一种算法方法，该方法结合微分代数和差分代数来分析有限差分近似的 s(trong)-一致性。我们的方法适用于常规解网格。对于这种类型的网格，我们给出了有限差分近似的 s 一致性的新定义，它概括了我们之前为笛卡尔网格给出的定义。论文中提出的 s 一致性的算法验证基于差分和差分 Thomas 分解的使用。首先，我们将微分分解应用于输入系统，从而对其解空间进行划分。然后，对于包含感兴趣的解决方案的输出子系统，我们应用差分 Thomas 分解的差分模拟，它允许检查 s 一致性。对于线性和一些准线性微分系统，还可以应用差分 \Gr 基进行 s 一致性分析。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动的 Navier-Stokes 方程。