The Method of Collocations and Least Residuals Combining the Integral Form of Collocation Equations and the Matching Differential Relations at the Solution of PDEs

Vasily P. Shapeev, Evgenii V. Vorozhtsov

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

2 Citations (Scopus)

Abstract

To increase the accuracy of computations by the method of collocations and least residuals (CLR) it is proposed to increase the number of degrees of freedom with the aid of the following two techniques: an increase in the number of basis vectors and the integration of the linearized partial differential equations (PDEs) over the subcells of each cell of a spatial computational grid. The implementation of these modifications, however, leads to the necessity of increasing the amount of symbolic computations needed for obtaining the work formulas of the new versions of the CLR method. The computer algebra system (CAS) Mathematica has proved to be successful at the execution of all these computations. It is shown that the proposed new symbolic-numeric versions of the CLR method possess a higher accuracy than the previous versions of this method. Furthermore, the version of the CLR method, which employs the integral form of collocation equations, needs a much lesser number of iterations for its convergence than the “differential” CLR method.

Original languageEnglish
Title of host publicationComputer Algebra in Scientific Computing - 19th International Workshop, CASC 2017, Proceedings
EditorsVP Gerdt, W Koepf, WM Seiler, EV Vorozhtsov
PublisherSpringer-Verlag GmbH and Co. KG
Pages346-361
Number of pages16
Volume10490 LNCS
ISBN (Print)9783319663197
DOIs
Publication statusPublished - 1 Jan 2017
Event19th International Workshop on Computer Algebra in Scientific Computing, CASC 2017 - Beijing, China
Duration: 18 Sep 201722 Sep 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10490 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference19th International Workshop on Computer Algebra in Scientific Computing, CASC 2017
CountryChina
CityBeijing
Period18.09.201722.09.2017

Keywords

  • Collocation of integral relations
  • Computer algebra system
  • Krylov subspaces
  • Multigrid
  • Preconditioner
  • Symbolic-numerical algorithm
  • NAVIER-STOKES EQUATIONS
  • COMPUTER
  • COMPUTATION
  • FLOW

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