Linear Differential Ideals and Generation of Difference Schemes for PDEs

Abstract:
In this talk we present an algorithmic approach outlined in [1] to 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 Gr\"obner basis of the linear differential ideal generated by the polynomials in the discrete system. For these purposes we use the difference form [2] of Janet-like Gr\"obner bases [3] and their implementation in Maple [4].

1. V.V. Mozzhilkin and Yu.A. Blinkov, Methods of Constructing Difference Schemes in Gas Dynamics (in Russian). Transactions of Saratov University, 1(2), 2001, 145-156.
2. V.P. Gerdt, On Computation of Gr\"obner Bases for Linear Difference Systems, Submitted to CASC-05 (Kalamata, Greece, September 12-16, 2005).
3. V.P. Gerdt and Yu.A. Blinkov, Janet-like Gr\"obner Bases, Submitted to CASC-05 (Kalamata, Greece, September 12-16, 2005).
4. V.P. Gerdt, D. Robertz, A Maple Package for Computing Gr\"obner bases for Linear Recurrence Relations, Submitted to ACAT-05 (DESY-Zeuthen, Germany, May 22-27, 2005).