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Handbook of Sinc numerical methods. With CD-ROM. (English) Zbl 1208.65143

Chapman & Hall/CRC Numerical Analysis and Scientific Computing. Boca Raton, FL: CRC Press (ISBN 978-1-4398-2158-9/hbk; 978-1-4398-2159-6/ebook). xx, 463 p. (2011).
This book deals with the approximation by sinc functions and the application to the numerical solution of partial differential equations. It contains five chapters. 1. One dimensional Sinc theory – 2. Sinc convolution-boundary integral equation methods for partial differential equations (PDEs) and integral equations – 3. Explicit 1-D program solutions via Sinc-Pack – 4. Explicit program solutions of PDEs via Sinc-Pack – 5. Directory of programs. – Most sections of chapters 1 to 3 contain several problems at their end.
Chapter 1 presents the theory of one dimensional Sinc methods and is based on the (cardinal) interpolation of a given function \(f\):
\[ \sum_{k=-n}^n f(kh) S(k,h)(x) \]
where
\[ S(k,h)(x):= \text{sinc}\left({x \over h}-k\right), \]
\[ \text{sinc}(z):= {\sin(\pi z) \over \pi z}. \]
The truncation of the (originally) infinite series, specified by \(n\), depends on the rate of the assumed exponential decay of \(f(x)\) for \(x\to \pm\infty\), and the step size \(h\) on the assumed exponential decay of the Fourier or Laplace transform. The latter is related to the analytic continuation into the complex plane. The author chooses here the presentation with the transforms above while in other publications he uses the presentation with complex variables as well. A typical result for the approximation on infinite intervals is established in Section 1.3. The error of Sinc approximation is of order \({\mathcal O}(n^{1/2}\exp(-cn^{1/2}))\), where \(c\) depends on the decay properties above and \(h={\mathcal O}(n^{-1/2})\).
Quadrature formulas, Fourier transforms, Hilbert transforms, convolution formulas, and other useful tools for sinc functions are presented. It is described how the required exponential decay is achieved in several cases by transformations of the variable \(x\). They include finite and semifinite intervals and arcs.
Chapter 2 may be characterized by the first lines of its abstract: This chapter applies a separation of variables method to finding approximate solutions to virtually any linear PDE in two or more dimensions with given boundary and/or initial conditions. The method also applies to a wide array of (possibly non-linear) differential and integral equations whose solutions are known to exist through theoretical considerations [End of citation]. The separation of variables means that multidimensional Sinc procedures consist of repeated one dimensional procedures on tensor products.
For this purpose, the knowledge of Green’s function is assumed, and the solution of the PDE is determined by convolution or by the corresponding boundary integral equation. The Sinc methods are based on the statement that PDEs that arise in the real world, exist in a small number of regions on each of which the given data and the solutions are analytic. This (shortened) statement is related to a comment on p. 211, from which a few more lines are cited: “It is perhaps unfortunate that Sobolev spaces have become popular both for constructing solutions using finite difference methods, and for constructing finite element type solutions. Whereas most PDE problems in applications have solutions that are analytic, Sobolev spaces hold a much larger class of functions than is necessary for applied problems. In short, methods based on this approach converge considerably more slowly than Sinc methods.”
The numerical solution of PDE’s by Sinc methods is therefore considered in spaces of analytic functions. A typical error is \({\mathcal O}(\exp(-c\sqrt n))\) and is better than the \({\mathcal O}(n^{\alpha})\) behavior of the \(h\)-version of finite elements. It is outperformed by the \({\mathcal O}(\exp(-c n))\) behavior of spectral elements in connection with analytic solutions. The \(hp\) version of finite elements and the boundary element method are not regarded in this context.
Chapter 3 provides Sinc programs written in MATLAB that illustrate and implement the techniques of Chapter 1. There are fully documented programs for solving elementary and advanced one dimensional problems. They include Sinc quadrature, Sinc indefinite convolution, Laplace transform inversion, Hilbert and Cauchy transforms. Ordinary differential equations are first transformed into Volterra integral equations, and then Picard iteration is used. There is no hint that (nonlinear) stiff equations cannot be dealt with by this concept. An implicit scheme is offered for linear equations. All these programs and those in the next Chapter are contained in Sinc-Pack a package of programs in MATLAB.
Chapter 4 provides Sinc programs for elliptic, hyperbolic, and parabolic PDEs that are also fully documented. The elliptic equations refer to the Laplace equation and to the Poisson equation with known analytic solutions. There is an example with a reentrant corner, but the singularities/discontinuities are contained already in the given boundary values and are not induced by the generic singularity of a reentrant corner. A wave equation is followed by two examples of parabolic equations. One of them refers to the Navier-Stokes equation and is solved by a Picard iteration. This concept is probably restricted to very small Reynolds numbers.
The chapter concludes with a performance comparison of the Sinc method and the \(h\) version of the finite element method. The Sinc method was faster by a factor between 4 and 100. The reported memory constraint for the finite element method due to the main memory of 0.5 GB shows that preconditioned conjugate gradient methods or multigrid methods were not used. Moreover, it is known that finite elements with polynomials of higher degree or the \(hp\) version are substantially more efficient than linear elements, when we have smooth solutions as in the examples here.
The book is a handbook as the author puts it in the title – and not a textbook. There exist many PDE problems which satisfy the cited assumptions although the portion is certainly not as large as postulated here. The fully documented programs make it useful for readers who know whether their numerical problems can be efficiently solved by Sinc methods.

MSC:

65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35-04 Software, source code, etc. for problems pertaining to partial differential equations
65Y15 Packaged methods for numerical algorithms
65D32 Numerical quadrature and cubature formulas
65R20 Numerical methods for integral equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
35L05 Wave equation
35Q30 Navier-Stokes equations

Software:

Sinc-Pack; Matlab
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