Difference Schemes with Operator Factors

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Springer Science & Business Media, Aug 31, 2002 - Mathematics - 384 pages
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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.
 

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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
Difference Schemes for Nonstationary Convection
177
Kortewegde Vries Equation
193
SCHEMES ON ADAPTIVE GRIDS
235
DIFFERENCE SCHEMES OF DOMAIN
321
Regionally Additive Schemes of TwoComponent Splitting
329
References
367
Index
383
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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".

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About the author (2002)

Alexander A. Samarskii and Peter N. Vabishchevich, Russian Academy of Sciences, Moscow, Russia.

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