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Computing qualitatively correct approximations of balance laws. Exponential-fit, well-balanced and asymptotic-preserving. (English) Zbl 1272.65065

SIMAI Springer Series 2. Milano: Springer (ISBN 978-88-470-2891-3/hbk; 978-88-470-2892-0/ebook). xix, 340 p. (2013).
This book under review gives a systematized and impressive account of the research of the author during the last some fifteen years, written in a well understandable English with some Romanic flavour and featured by a large amount of physical details in this mathematical text, and of pictures showing computing results, often in colours.
The long title already shows what is to be awaited but the content is divided into two main parts: hyperbolic quasi-linear balance laws (you understand: not just conservation laws but equations with the named differential operator and a right hand side: a source term), and weakly nonlinear kinetic equations.
After an introductory chapter there follow: Lifting of a non-resonant scalar balance law (the condition of non-resonance rather often being posed to be able to prove convergence and accuracy results); Lyapunov functional for linear error estimates (here also a convincing example is given that the customary time splitting, for an accretive source term, is not leading to an acceptable error growth like \(O(\sqrt{t})\) but to exponential growth!); Early well balanced derivations for various systems, Viscosity solutions and large-time behavior for non-resonant balance laws; Kinetic scheme with reflections and linear geometric optics; Material variables, strings and infinite domains, whereas the second part contains the chapters: The special case of 2-velocity kinetic models, Elementary solutions and analytical discrete-ordinates for radiative transfer; Aggregation phenomena with kinetic models of chemotaxis dynamics (this being one of the important applications of the theory); Time-stabilization on flat currents with non-degenerate Boltzmann-Poisson models (semiconductor equations are another important application); Klein-Kramers equation and Burgers/Fokker-Planck model of spray; A model for scattering of forward-peaked beams; Linearized BGK model of heat transfer; Balances in two dimensions: kinetic semiconductor equations again; Conclusion: Outlook and shortcomings.
Whereas the first chapter already clarifies that there will be no mathematical definition of the term “well balanced” which goes back to J. M. Greenberg and A.-Y. Le Roux [SIAM J. Numer. Anal. 33, No. 1, 1–16 (1996; Zbl 0876.65064)], and everybody knows what is meant like in the case of the term “stiff”, Chapters 2 and 3 show the general procedure: the 1D balance equation is written as a \(2\times 2\) homogeneous Temple system, the former source is approximated by a weighted sum of Dirac delta functions (for this approach, convergence is proved in the weak-* topology). For numerical computing, this boils down to an enlarged Godunov scheme (the seminal paper of whom is cited only in Chapter 3 though it well could be subsumed under the title “well balanced”) in which there is, however, an additional step resulting the values at the jumps (with subsequent solution of Riemann problems). For those latter values, the steady-state equation is solved remembering the Scharfetter-Gummel approach for semiconductor equations, or that of R. Eymard, J. Fuhrmann and K. Gärtner [Numer. Math. 102, No. 3, 463-495 (2006; Zbl 1116.65101)], for 2D nonlinear parabolic equations. Of course, these additional computations need their attention and computing time and, ideally, lead to small linear equations (like in the numerous physical examples of Part II). An error estimate for the proposed “non-resonant Godunov scheme” is given in Chapter 3. Further theoretical results are obtained, e.g., in Chapter 5: Existence, uniqueness, \(L^1\)-stability and large-time behaviour for bounded variation viscosity solutions of balance equations.
In the second part, often there are 2D linear equations for which two approaches are shown: either the differential (or integral) operator is developed in terms of an appropriate function system or the 2D operator is discretized à la P. L. Roe and D. Sidilkofer [SIAM J. Numer. Anal. 29, No. 6, 1542–1568 (1992; Zbl 0765.65093)]. Then the well balanced method of the first part is applied.
There follow three appendices: Non-conservative products and locally Lipschitzian paths; A tiny step toward hypocoercivity estimates for well-balanced schemes on \(2\times 2\) models; and Preliminary analysis of the errors for Vlasov-BGK.
Every chapters has its own references and often ends with notes reviewing the results reached and discussing their relation to works by other authors.
Interesting is, e.g. a discussion in the Notes at the end of Chapter 15: “It may be worth insisting on the fact that sometimes, reducing a 2D model into a tricky one-dimensional formulation can constitute an interesting strategy …” -followed by examples. Well, sometimes, but 2D models should reveal more details and insights. The 1D models considered in the book are nonlinear, the 2D equations are mainly linear (and approximated by the optimum positive scheme of Roe and Sidilkover) and usually are less fast convergent than discrete equations stemming from 1D, necessitating perhaps finer discretizations. Even then, there should not be a problem in solving them using Matlab’s \(x=A\backslash b\), and if nonlinear, in employing some iteration. Finally, why generalizing to 2D only by looking for appropriate places to put point-Dirac delta functions onto lines of an equidistant grid? Is there no possibility to employ at least Dirac functions concentrated on pieces of lines?
Also, not too much words are lost on the handling of boundary conditions; mostly, specular conditions are assumed but for Boltzmann-Poisson equations (Chapter 11) there is a special paragraph on these matters. What concerns the numerous figures, it seems more a decision of the editing house to give them a measure which forces the reader to use a magnifying glass.
All in all, a book interesting for mathematical researchers in partial differential equations and for physicist, bringing a wealth of fresh ideas and methods much improving the time splitting approach.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35L72 Second-order quasilinear hyperbolic equations
35Q82 PDEs in connection with statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D37 Statistical mechanics of semiconductors
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q20 Boltzmann equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q83 Vlasov equations

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