Joint Institute for Nuclear Research,

141980 Dubna,

RUSSIA

and

Saratov University,

410071 Saratov,

RUSSIA

E-mail: gerdt@jinr.ru, BlinkovUA@info.sgu.ru

**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].

- 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.
- V.P. Gerdt, On Computation of Gr\"obner Bases for Linear Difference Systems, Submitted to CASC-05 (Kalamata, Greece, September 12-16, 2005).
- V.P. Gerdt and Yu.A. Blinkov, Janet-like Gr\"obner Bases, Submitted to CASC-05 (Kalamata, Greece, September 12-16, 2005).
- 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).