Consistency of finite difference approximations for linear PDE systems and its algorithmic verification
Vladimir P. Gerdt, Daniel Robertz · 2010
In this paper we consider finite difference approximations for numerical solving of systems of partial differential equations of the form f1 = · · · = fp = 0, where F := {f1, ..., fp} is a set of linear partial differential polynomials over the field of rational functions with rational coefficients. For orthogonal and uniform solution grids we strengthen the generally accepted concept of equation-wise consistency (e-consistency) of the difference equations f1 = · · · = fp = 0 as approximation of the differential ones. Instead, we introduce a notion of consistency of the set of all linear consequences of the difference polynomial set f := {f, ..., fp} with the linear subset of the differential ideal 〈F〉. The last consistency, which we call s-consistency (strong consistency), admits algorithmic verification via a Gröbner basis of the difference ideal 〈f〉. Some related illustrative examples of finite difference approximations, including those which are e-consistent and s-inconsistent, are given.