Implementing Spectral Methods for Partial Differential Equations

This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.

This book offers a systematic and self-contained approach to solve partial differential equations numerically using single and multidomain spectral methods. It contains detailed algorithms in pseudocode for the application of spectral approximations to both one and two dimensional PDEs of mathematical physics describing potentials, transport, and wave propagation. David Kopriva, a well-known researcher in the field with extensive practical experience, shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries. The book addresses computational and applications scientists, as it emphasizes the practical derivation and implementation of spectral methods over abstract mathematics. It is divided into two parts: First comes a primer on spectral approximation and the basic algorithms, including FFT algorithms, Gauss quadrature algorithms, and how to approximate derivatives. The second part shows how to use those algorithms to solve steady and time dependent PDEs in one and two space dimensions. Exercises and questions at the end of each chapter encourage the reader to experiment with the algorithms.

First book to cover multidomain spectral methods for the numerical solution of time-dependent 1D and 2D partial differential equations Presented without too much abstract mathematics and minutae Contains a set of basic examples as building blocks for solving complex PDEs in realistic geometries With exercises and questions at the end of each chapter Both an introduction for graduate students and a reference for computational scientists working on numerical solutions of PDEs Includes supplementary material: sn.pub/extras

Autorentext

David Kopriva is Professor of Mathematics at the Florida State University, where he has taught since 1985. He is an expert in the development, implementation and application of high order spectral multi-domain methods for time dependent problems. In 1986 he developed the first multi-domain spectral method for hyperbolic systems, which was applied to the Euler equations of gas dynamics.

Klappentext

This book offers a systematic and self-contained approach to solve partial differential equations numerically using single and multidomain spectral methods. It contains detailed algorithms in pseudocode for the application of spectral approximations to both one and two dimensional PDEs of mathematical physics describing potentials, transport, and wave propagation. David Kopriva, a well-known researcher in the field with extensive practical experience, shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries. The book addresses computational and applications scientists, as it emphasizes the practical derivation and implementation of spectral methods over abstract mathematics. It is divided into two parts: First comes a primer on spectral approximation and the basic algorithms, including FFT algorithms, Gauss quadrature algorithms, and how to approximate derivatives. The second part shows how to use those algorithms to solve steady and time dependent PDEs in one and two space dimensions. Exercises and questions at the end of each chapter encourage the reader to experiment with the algorithms.

Inhalt
Approximating Functions, Derivatives and Integrals.- Spectral Approximation.- Algorithms for Periodic Functions.- Algorithms for Non-Periodic Functions.- Approximating Solutions of PDEs.- Survey of Spectral Approximations.- Spectral Approximation on the Square.- Transformation Methods from Square to Non-Square Geometries.- Spectral Methods in Non-Square Geometries.- Spectral Element Methods.- Erratum.- Erratum.