Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
The Method of Collocations and Least Residuals Combining the Integral Form of Collocation Equations and the Matching Differential Relations at the Solution of PDEs. / Shapeev, Vasily P.; Vorozhtsov, Evgenii V.
Computer Algebra in Scientific Computing - 19th International Workshop, CASC 2017, Proceedings. ed. / VP Gerdt; W Koepf; WM Seiler; EV Vorozhtsov. Vol. 10490 LNCS Springer-Verlag GmbH and Co. KG, 2017. p. 346-361 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10490 LNCS).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
}
TY - GEN
T1 - The Method of Collocations and Least Residuals Combining the Integral Form of Collocation Equations and the Matching Differential Relations at the Solution of PDEs
AU - Shapeev, Vasily P.
AU - Vorozhtsov, Evgenii V.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - 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.
AB - 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.
KW - Collocation of integral relations
KW - Computer algebra system
KW - Krylov subspaces
KW - Multigrid
KW - Preconditioner
KW - Symbolic-numerical algorithm
KW - NAVIER-STOKES EQUATIONS
KW - COMPUTER
KW - COMPUTATION
KW - FLOW
UR - http://www.scopus.com/inward/record.url?scp=85029804371&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-66320-3_25
DO - 10.1007/978-3-319-66320-3_25
M3 - Conference contribution
AN - SCOPUS:85029804371
SN - 9783319663197
VL - 10490 LNCS
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 346
EP - 361
BT - Computer Algebra in Scientific Computing - 19th International Workshop, CASC 2017, Proceedings
A2 - Gerdt, VP
A2 - Koepf, W
A2 - Seiler, WM
A2 - Vorozhtsov, EV
PB - Springer-Verlag GmbH and Co. KG
T2 - 19th International Workshop on Computer Algebra in Scientific Computing, CASC 2017
Y2 - 18 September 2017 through 22 September 2017
ER -
ID: 9910205