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Conservative finite-difference methods on general grids. Ed. by Stanly Steinberg. Incl. 1 disk. (English) Zbl 0844.65067
Boca Raton, FL: CRC Press (ISBN 978-0-8493-7375-6/print; 978-0-367-44874-5/pbk; 978-1-315-14020-9/ebook). 359 p. (1996).
The author belongs to the scientific school of A. A. Samarskij at the Keldysh Institute of Applied Mathematics in Moscow. The main concept of this book is close to the books of Samarskij, published in Russian (the material in the book arose from the author’s original research!). This book is unique, because it is the first book not in Russian presenting the work of the Moscow group of Samarskij, together with a lot of carefully prepared references to the original literature (mostly in Russian).
The subject of the book is the “support operators” method (or, more appropriate, fundamental, basic, or reference operators method) for constructing finite difference schemes that mimic fundamental properties of continuum physical systems. First-order differential operators are the objects in vector and tensor analysis. They satisfy integral identities, as those of Green, Gauss, and Stokes, closely related to the conservation laws of the continuum models.
Thus, to obtain high-quality finite difference schemes, it is important to construct discrete analogues of the continuous operators divergence, gradient, and curl, which will satisfy discrete analogues of the integral identities.
Hence, the text starts with the construction of systems of consistent difference operators discretizing the (continuous) operators divergence and gradient, given by their invariant definitions via integrals. An appropriate definition of (discrete approximations of) inner products ensures the validity of conservation laws (i.e. of integral relations) for the discrete operator, too. The discrete operators, found in this way, are applied to construct finite difference schemes of the Dirichlet and the Robin boundary value problems for elliptic equations, further for the usual initial-boundary value problems for the heat equation (consequently, with special emphasis to reproducing the conservation laws), and finally on more than 100 pages, but of an outstanding lucidity and consistency, the finite difference approximation of Lagrangian gas dynamics. (A pleasure to read this fifth chapter!) – A floppy diskette is attached to the book containing numerous FORTRAN program files with tests and examples for all of the content of the monograph: see the directories: eleq, heateq, gasdeq.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35K05 Heat equation
35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
76N15 Gas dynamics (general theory)
35-04 Software, source code, etc. for problems pertaining to partial differential equations
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