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Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 1, Pages 118–130 (Mi zvmmf4815)  

This article is cited in 9 scientific papers (total in 9 papers)

Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics

P. N. Vabishchevich

Institute of Mathematical Modeling, Russian Academy of Sciences, Miusskaya pl. 4a, Moskow, 125047 Russia

Abstract: In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.

Key words: Cauchy problem for a first-order evolutionary equation, operator-difference schemes, stability of finite difference scheme.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:1, 112–123

Bibliographic databases:

UDC: 519.63
Received: 01.04.2009

Citation: P. N. Vabishchevich, “Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics”, Zh. Vychisl. Mat. Mat. Fiz., 50:1 (2010), 118–130; Comput. Math. Math. Phys., 50:1 (2010), 112–123

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. N. Vabishchevich, “Factorized SM-stable two-level schemes”, Comput. Math. Math. Phys., 50:11 (2010), 1818–1824  mathnet  crossref  adsnasa  isi
    2. P. P. Matus, “Well-posedness of difference schemes for semilinear parabolic equations with weak solutions”, Comput. Math. Math. Phys., 50:12 (2010), 2044–2063  mathnet  crossref  adsnasa
    3. P. N. Vabishchevich, “Two-level schemes of higher approximation order for time-dependent problems with skew-symmetric operators”, Comput. Math. Math. Phys., 51:6 (2011), 1050–1060  mathnet  crossref  mathscinet  isi
    4. Vabishchevich P.N., “SM stability for time-dependent problems”, Numerical methods and applications, Lecture Notes in Computer Science, 6046, 2011, 29–40  crossref  mathscinet  zmath  isi  scopus
    5. P. N. Vabishchevich, “Construction of splitting schemes based on transition operator approximation”, Comput. Math. Math. Phys., 52:2 (2012), 235–244  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. P. N. Vabishchevich, “SM-stability of operator-difference schemes”, Comput. Math. Math. Phys., 52:6 (2012), 887–894  mathnet  crossref  mathscinet  adsnasa  isi  elib  elib
    7. Axelsson O., Blaheta R., Kohut R., “Preconditioning Methods For High-Order Strongly Stable Time Integration Methods With An Application For a Dae Problem”, Numer. Linear Algebr. Appl., 22:6, SI (2015), 930–949  crossref  mathscinet  zmath  isi  elib  scopus
    8. Vabishchevich P.N., “Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems”, Comput. Methods Appl. Math., 17:2 (2017), 323–335  crossref  mathscinet  zmath  isi  scopus
    9. Vabishchevich P.N., “Fundamental Mode Exact Schemes For Unsteady Problems”, Numer. Meth. Part Differ. Equ., 34:6 (2018), 2301–2315  crossref  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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