## 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 Adams-Bashforth advection equation algorithm aliasing error amplification factor amplitude boundary condition Boussinesq Boussinesq equations Burgers's equation centered coefficients computational mode conservation law constant coordinate Courant number damping defined determined difference differencing differential-difference diffusion discrete dispersion relation domain evaluated expansion functions filter finite finite-difference approximation finite-difference scheme first-order flow fluid flux flux-limited formula fourth-order governing equations gravity waves grid points group velocity higher-order horizontal hydrostatic hyperbolic implicit initial condition integration Lax-Wendroff method leapfrog scheme limiter linear matrix mesh nodes nonlinear numerical approximation numerical solution obtained one-dimensional ordinary differential equations partial differential equation perturbations phase speed phase-speed error polynomial preceding problem propagation pseudospectral Rossby waves Runge-Kutta satisfy second-order semi-implicit semi-Lagrangian shown in Fig simulation spatial derivatives spectral method spherical harmonics stable time step Suppose third-order time-differencing tion trajectory trapezoidal true solution truncation error two-dimensional unstable upstream variables velocity field vertical wave number wavelengths yields zero