Gröbner基与偏微分方程差分格式的生成
Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations
摘要：
本文提出了一种生成线性偏微分方程全守恒差分格式的算法。该方法是基于扩大的方程在其积分守恒律形式的未知函数和其衍生物之间的额外的积分关系，并离散所获得的系统。离散系统的结构取决于对扩大系统中出现的积分的数值逼近方法。作为离散化的结果，推导出未知函数及其偏导数的线性多项式差分方程系统。通过消去所有的偏导数构造了一个差分格式。通过选择适当的消去秩和计算由离散系统中的多项式生成的线性差分理想的Gröbner基，可以实现消去。为了这些目的，我们使用Janet-like Gröbner基的差分形式及其在Maple中的实现。作为上述方法和算法的例证，我们构造了Burgers方程和Falkowich-Karman方程的若干差分格式，并讨论了它们的数值性质。
英文摘要：
In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.
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