Difference Schemes with Operator Factors

1. Front Matter

   Pages i-x

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2. Introduction

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 1-8

3. Two-Level Difference Schemes

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 9-53

4. Difference Schemes with Operator Factors

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 55-78

5. Three-Level Difference Schemes

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 79-120

6. Three-Level Schemes with Operator Factors

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 121-147

7. Difference Schemes for Non-Stationary Equations

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 149-234

8. Schemes on Adaptive Grids

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 235-320

9. Difference Schemes of Domain Decomposition for Non-Stationary Problems

   • A. A. Samarskii, P. P. Matus, P. N. Vabishchevich

   Pages 321-365

10. Back Matter

    Pages 367-384

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Two-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.

• Mathematica
• computational mathematics
• finite element method
• numerical method
• operator
• problem solving
• stability
• partial differential equations

• Institute for Mathematical Modelling, Russian Academy of Sciences, Moscow, Russia

  A. A. Samarskii, P. N. Vabishchevich

• Department of Numerical Simulation, Institute for Mathematics, Minsk, Belarus

  P. P. Matus

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