Linear Differential Ideals and Generation of Difference Schemes for PDEs
In this talk we present an algorithmic approach outlined in  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  of Janet-like Gr\"obner bases  and their implementation in Maple .