consistency of finite difference approximations for linear pde systems and its algorithmic verification
In this paper we consider finite difference approximations for numerical solving of systems of partial differential equations of the form f 1 =    = f p = 0, where F := { f 1 , ..., f p } 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 f 1 =    = f p = 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, ..., f p } with the linear subset of the differential ideal ? F ?. The last consistency, which we call s-consistency (strong consistency), admits algorithmic verification via a Grbner 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.
consistency partial differential equations finite difference schemes gr&ouml bner basis applications involutive algorithm