Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 051, 26 pages      math.RA/0605334      http://dx.doi.org/10.3842/SIGMA.2006.051

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

Vladimir P. Gerdt a, Yuri A. Blinkov b and Vladimir V. Mozzhilkin b
a) Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Russia
b) Department of Mathematics and Mechanics, Saratov University, 410071 Saratov, Russia

Received December 07, 2005, in final form April 24, 2006; Published online May 12, 2006

Abstract
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.

Key words: partial differential equations; conservative difference schemes; difference algebra; linear difference ideal; Gröbner basis; Janet-like basis; computer algebra; Burgers equation; Falkowich-Karman equation.

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