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Consistency of finite difference approximations for linear PDE systems and its algorithmic verification

Authors:

Vladimir P. Gerdt,

Daniel RobertzAuthors Info & Claims

ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

Pages 53 - 59

https://doi.org/10.1145/1837934.1837950

Published: 25 July 2010 Publication History

Abstract

In this paper we consider finite difference approximations for numerical solving of systems of partial differential equations of the form f₁ = · · · = f_p = 0, where F := {f₁, ..., 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 f₁ = · · · = 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 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.

References

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V. P. Gerdt and D. Robertz. A Maple Package for Computing Gröbner Bases for Linear Recurrence Relations. Nuclear Instruments and Methods in Physics Research 559(1), 2006, 215--219. arXiv:cs.SC/0509070. Cf. also http://wwwb.math.rwth-aachen.de/Janet

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V. P. Gerdt and Yu. A. Blinkov. Involution and Difference Schemes for the Navier-Stokes Equations. Computer Algebra in Scientific Computing / CASC 2009, LNCS 5743, Springer-Verlag, Berlin, 2009, pp. 94--105.

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T. Bächler, V. P. Gerdt, M. Lange-Hegermann and D. Robertz. Thomas Decomposition: I. Algebraic Systems; II. Differential Systems. Submitted to Computer Algebra in Scientific Computing / CASC 2010 (September 5--12, 2010, Tsakhkadzor, Armenia).

Cited By

Gomes DKrannich FMajrashi BRibeiro R(2024)Algorithmic Detection of Conserved Quantities for Finite-Difference SchemesMathematics in Computer Science10.1007/s11786-024-00595-w18:4Online publication date: 15-Nov-2024
https://doi.org/10.1007/s11786-024-00595-w
BLINKOV YREBRINA A(2023)INVESTIGATION OF DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS BY USING COMPUTER ALGEBRA ALGORITHMSПрограммирование10.31857/S0132347423010028(32-37)Online publication date: 1-Jan-2023
https://doi.org/10.31857/S0132347423010028
Blinkov YRebrina A(2023)Investigation of Difference Schemes for Two-Dimensional Navier–Stokes Equations by Using Computer Algebra AlgorithmsProgramming and Computing Software10.1134/S036176882301002449:1(26-31)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1134/S0361768823010024
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Index Terms

Consistency of finite difference approximations for linear PDE systems and its algorithmic verification
1. Applied computing
  1. Physical sciences and engineering
    1. Mathematics and statistics
2. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Computer algebra systems
    2. Symbolic and algebraic algorithms

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cover image ACM Other conferences

ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

July 2010

366 pages

ISBN:9781450301503

DOI:10.1145/1837934

Program Chair:
Wolfram Koepf
London, Ontario, Canada

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 25 July 2010

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Russian Foundation for Basic Research

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ISSAC '10

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ISSAC '10: International Symposium on Symbolic and Algebraic Computation

July 25 - 28, 2010

Munich, Germany

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ISSAC '10 Paper Acceptance Rate 45 of 110 submissions, 41%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Gomes DKrannich FMajrashi BRibeiro R(2024)Algorithmic Detection of Conserved Quantities for Finite-Difference SchemesMathematics in Computer Science10.1007/s11786-024-00595-w18:4Online publication date: 15-Nov-2024
https://doi.org/10.1007/s11786-024-00595-w
BLINKOV YREBRINA A(2023)INVESTIGATION OF DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS BY USING COMPUTER ALGEBRA ALGORITHMSПрограммирование10.31857/S0132347423010028(32-37)Online publication date: 1-Jan-2023
https://doi.org/10.31857/S0132347423010028
Blinkov YRebrina A(2023)Investigation of Difference Schemes for Two-Dimensional Navier–Stokes Equations by Using Computer Algebra AlgorithmsProgramming and Computing Software10.1134/S036176882301002449:1(26-31)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1134/S0361768823010024
Levin A(2022)A New Type of Difference Dimension PolynomialsMathematics in Computer Science10.1007/s11786-022-00540-916:4Online publication date: 13-Oct-2022
https://doi.org/10.1007/s11786-022-00540-9
Gerdt VRobertz DDavenport JWang DKauers MBradford R(2019)Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE SystemsProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326255(163-170)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326255
Blinkov YGerdt VPankratov IKotkova E(2019)Construction of a New Implicit Difference Scheme for 2D Boussinesq Paradigm EquationComputer Algebra in Scientific Computing10.1007/978-3-030-26831-2_11(152-163)Online publication date: 26-Aug-2019
https://dl.acm.org/doi/10.1007/978-3-030-26831-2_11
Blinkov YGerdt VLyakhov DMichels D(2018)A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified EquationsDevelopments in Language Theory10.1007/978-3-319-99639-4_5(67-81)Online publication date: 23-Aug-2018
https://doi.org/10.1007/978-3-319-99639-4_5
Blinkov YGerdt VMarinov K(2017)Discretization of quasilinear evolution equations by computer algebra methodsProgramming and Computing Software10.1134/S036176881702004943:2(84-89)Online publication date: 1-Mar-2017
https://dl.acm.org/doi/10.1134/S0361768817020049
Amodio PBlinkov YGerdt VLa Scala R(2017)Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D NavierStokes equationsApplied Mathematics and Computation10.1016/j.amc.2017.06.037314:C(408-421)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1016/j.amc.2017.06.037
Amodio PBlinkov YGerdt VScala R(2013)On Consistency of Finite Difference Approximations to the Navier-Stokes EquationsProceedings of the 15th International Workshop on Computer Algebra in Scientific Computing - Volume 813610.1007/978-3-319-02297-0_4(46-60)Online publication date: 9-Sep-2013
https://dl.acm.org/doi/10.1007/978-3-319-02297-0_4
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