Numerical grid generation. Foundations and applications.

*(English)*Zbl 0598.65086
New York-Amsterdam-Oxford: North-Holland. XV, 483 p. $ 34.95; Dfl. 150.00 (1985).

The book deals with the numerical solution of partial differential equations (PDE) at its first stage generation of the grids. In few words any arbitrary shaped region is firstly transformed onto a standard (rectangular or composed of rectangles) computational region where the correspondingly transformed PDE is discretized on a square grid. The last is a field well studied in any textbook of numerical solution of PDE’s. The aim of the book is to provide a necessary basic information with some mathematical background and from the point of coding implementations for numerical solutions of PDE’s to be constructed on general regions. The first chapter is an introductory one. In chapter II various configurations of curvilinear coordinate systems are discussed, as a rectangular computational field (or such composed of rectangles) may be obtained by a wide variety of configurations. Also any subrectangle (in the computational field) may be generated by a separate coordinate system. Special attention is paid to the implementation tools. In the third chapter the basic transformation relations are studied applicable to the use of general curvilinear coordinate systems. These relations are used in the next chapter in order to derive approximation of the transformed PDE. Discretizations by finite differences and the behaviour of the truncation error are studied in chapters V and VI.

In the book two ways of generating curvilinear coordinate systems are distinguished: i) by numerical solutions of PDE’s; ii) construction by algebraic interpolation. i) includes solutions of elliptic and parabolic or hyperbolic PDE’s. In the elliptic case conformal (two-dimensional) mapping which is orthogonal is included. In general, orthogonal or nearly orthogonal coordinate systems are studied in chapter IX. Algebraic ways of constructing curvilinear coordinate systems are studied in chapter VIII. They include simple normalization, transfinite interpolation, in general, any interpolation technique as Lagrange interpolation, Hermite interpolation, splines, tension splines, B-splines etc.

In the last chapter (XI) the field of dynamically adaptive grids is considered. The problem of grid adaptation is formulated as a variational one, i.e., as a problem of finding a mininum of a variational functional, where the weight function is some measure of the error, or the solution variation. Upon this condition the grid points are closely distributed where the solution is smooth. Some particular choices of weight functions are discussed.

The book is illustrated by a set of examples after each chapter which makes the text easier to understand. The text is addressed to graduate students and may be used as a base of course in the field of numerical grid generation and in deriving coles in solving PDE’s on arbitrary regions. A minimum knowledge in the field of differential geometry is provided as an appendix. Thus the book as a selfclosed system which is another positive property.

In the book two ways of generating curvilinear coordinate systems are distinguished: i) by numerical solutions of PDE’s; ii) construction by algebraic interpolation. i) includes solutions of elliptic and parabolic or hyperbolic PDE’s. In the elliptic case conformal (two-dimensional) mapping which is orthogonal is included. In general, orthogonal or nearly orthogonal coordinate systems are studied in chapter IX. Algebraic ways of constructing curvilinear coordinate systems are studied in chapter VIII. They include simple normalization, transfinite interpolation, in general, any interpolation technique as Lagrange interpolation, Hermite interpolation, splines, tension splines, B-splines etc.

In the last chapter (XI) the field of dynamically adaptive grids is considered. The problem of grid adaptation is formulated as a variational one, i.e., as a problem of finding a mininum of a variational functional, where the weight function is some measure of the error, or the solution variation. Upon this condition the grid points are closely distributed where the solution is smooth. Some particular choices of weight functions are discussed.

The book is illustrated by a set of examples after each chapter which makes the text easier to understand. The text is addressed to graduate students and may be used as a base of course in the field of numerical grid generation and in deriving coles in solving PDE’s on arbitrary regions. A minimum knowledge in the field of differential geometry is provided as an appendix. Thus the book as a selfclosed system which is another positive property.

Reviewer: R.D.Lazarov

##### MSC:

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65D15 | Algorithms for approximation of functions |