## Numerical Methods for Wave Equations in Geophysical Fluid DynamicsMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as weIlas the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in AppliedMathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and rein force the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and en courage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the AppliedMathematical Sei ences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface This book is designed to serve as a textbook for graduate students or advanced undergraduates studying numerical methods for the solution of partial differen tial equations goveming wave-like flows. Although the majority of the schemes presented in this text were introduced ineither the applied-rnathematics or atmos pheric-science literature, the focus is not on the nuts-and-bolts details of various atmospheric models but on fundamental numerical methods that have applications in a wide range of scientific and engineering disciplines. |

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### Contents

Introduction | 1 |

Basic FiniteDifference Methods | 35 |

Beyond the OneWay Wave Equation | 107 |

SeriesExpansion Methods | 173 |

Problems | 234 |

Finite Volume Methods | 241 |

### Other editions - View all

Numerical Methods for Wave Equations in Geophysical Fluid Dynamics Dale R. Durran Limited preview - 2013 |

Numerical Methods for Wave Equations in Geophysical Fluid Dynamics Dale R. Durran No preview available - 2013 |

### Common terms and phrases

accuracy accurate advection equation amplification amplitude appear applications approach approximation associated becomes boundary condition centered coefficients computational conservation consider constant coordinate correct damping defined dependence derivatives determined difference differencing discrete discussed dispersion relation domain efficiency energy evaluated exact example expansion functions expressed factor field FIGURE filter finite finite-difference finite-element flow fluid flux formula Fourier function governing grid grid points horizontal identical initial integration interval involving leapfrog limiter linear maximum mesh method models modes nonlinear Note numerical solution obtained operator partial differential equation periodic physical preceding pressure problem produced propagation reduces reflection region represent resolved satisfy scheme second-order shallow-water shown in Fig shows simulation space spatial spectral stability step Suppose time-differencing tion transform trapezoidal true truncation error two-dimensional unstable upstream variables velocity vertical wave number yields zero