Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Finite Differences And Partial Differential Equations
Finite Differences And Partial Differential Equations
Admissible orderings and finiteness criteria for differential standard bases
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra
Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Consistency analysis of finite difference approximations to PDE systems
MMCP'11 Proceedings of the 2011 international conference on Mathematical Modeling and Computational Science
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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.