## Difference Schemes with Operator FactorsTwo-and three-level difference schemes for discretisation in time, in conjunction with finite difference or finite element approximations with respect to the space variables, are often used to solve numerically non stationary problems of mathematical physics. In the theoretical analysis of difference schemes our basic attention is paid to the problem of sta bility of a difference solution (or well posedness of a difference scheme) with respect to small perturbations of the initial conditions and the right hand side. The theory of stability of difference schemes develops in various di rections. The most important results on this subject can be found in the book by A.A. Samarskii and A.V. Goolin [Samarskii and Goolin, 1973]. The survey papers of V. Thomee [Thomee, 1969, Thomee, 1990], A.V. Goolin and A.A. Samarskii [Goolin and Samarskii, 1976], E. Tad more [Tadmor, 1987] should also be mentioned here. The stability theory is a basis for the analysis of the convergence of an approximative solu tion to the exact solution, provided that the mesh width tends to zero. In this case the required estimate for the truncation error follows from consideration of the corresponding problem for it and from a priori es timates of stability with respect to the initial data and the right hand side. Putting it briefly, this means the known result that consistency and stability imply convergence. |

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### Contents

TWOLEVEL DIFFERENCE SCHEMES | 9 |

DIFFERENCE SCHEMES WITH OPERATOR FACTORS | 55 |

THREELEVEL DIFFERENCE SCHEMES | 79 |

THREELEVEL SCHEMES WITH OPERATOR FACTORS | 121 |

Schemes with D E + 0 5t2AGi B tAG2 | 133 |

Difference Schemes of Divergent Form | 139 |

DIFFERENCE SCHEMES FOR NONSTATIONARY | 149 |

Problems with Generalized Solutions | 164 |

### Other editions - View all

Difference Schemes with Operator Factors A.A. Samarskii,P.P. Matus,P.N. Vabishchevich Limited preview - 2013 |

Difference Schemes with Operator Factors A.A. Samarskii,P.P. Matus,P.N. Vabishchevich No preview available - 2013 |

Difference Schemes with Operator Factors A.A. Samarskii,P.P. Matus,P.N. Vabishchevich No preview available - 2010 |

### Common terms and phrases

accuracy approximate solution assume boundary value problem canonical form Cauchy inequality Cauchy problem coefficients computational conditions of stability consider constant operators construct convective convergence defined difference solution Differential Equations differential problem domain decomposition dot product energy identity estimates of stability explicit scheme fc=i following a priori following estimate formula grid function Hilbert space initial data inner product Korteweg-de Vries equation Lemma Lipschitz continuity Math mathematical physics Matus necessary and sufficient nodes non-stationary problems norm operator A(t operator inequality order of approximation parabolic equation priori estimate Proof right hand side Russian Samarskii satisﬁed scheme with weights schemes of domain schemes with operator schemes with variable second order self-adjoint operator solution of problem space H splitting stability of scheme stability with respect sub-domains sufficient condition Taking into account Theorem three-level difference schemes three-level schemes tion transl truncation error two-level operator-difference schemes variable weighted variable weighted factors

### Popular passages

Page 374 - P. (1980). Difference Methods for the Solution of Problems of Gas Dynamics. Nauka, Moscow. Sedov, LI (1983). Continuum Mediantes (in Russian). Nauka, Moscow. Smagulov, Sh. (1984). Converging difference schemes for equations of a viscous heat-conducting gas, its properties and error estimates "in the large".