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1999 | Buch

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Numerical Methods for Wave Equations in Geophysical Fluid Dynamics

verfasst von: Dale R. Durran

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

Enthalten in: Professional Book Archive

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Mathematics 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.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The possibility of deterministic weather prediction was suggested by Vilhelm Bjerknes as early as 1904. Around the time of the First World War, Lewis Richardson actually attempted to produce such a forecast by manually integrating a finitedifference approximation to the equations governing atmospheric motion. Unfortunately, his calculations did not yield a reasonable forecast. Moreover, the human labor required to obtain this disappointing result was so great that subsequent attempts at deterministic weather prediction had to await the introduction of a high-speed computational aid. In 1950 a team of researchers, under the direction of Jule Charney and John von Neumann at the Institute for Advanced Study, at Princeton, journeyed to the Aberdeen Proving Ground, where they worked for approximately twenty-four hours to coax a one-day weather forecast from the first general-purpose electronic computer, the ENIAC.¹ The first computer-generated weather forecast was surprisingly good, and its success led to the rapid growth of a new meteorological subdiscipline, “numerical weather prediction.” These early efforts in numerical weather prediction also began a long and fruitful collaboration between numerical analysts and atmospheric scientists.² The use of numerical models in atmospheric and oceanic science has subsequently expanded into almost all areas of current research.

Dale R. Durran

2. Basic Finite-Difference Methods

Abstract

As discussed in the preceding chapter, there are two conceptually different ways to represent continuous functions on digital computers: as a finite set of gridpoint values or as a finite set of series-expansion functions. The grid-point approach is used in conjunction with finite-difference methods, which were widely implemented on digital computers somewhat earlier than the series-expansion techniques. In addition, the theory for these methods is somewhat simpler than that for series-expansion methods. We will parallel this historical development by studying finite-difference methods in this chapter and deferring the treatment of series expansion methods to Chapter 4. Moreover, it is useful to understand finitedifference methods before investigating series-expansion techniques because even when series expansions are used to represent the spatial dependence of some atmospheric quantity, the time dependence is almost always discretized and treated with finite differences.

Dale R. Durran

3. Beyond the One-Way Wave Equation

Abstract

The basic properties of finite-difference methods were explored in Chapter 2 by applying each scheme to a simple prototype problem: the one-way wave equation (or, equivalently, the one-dimensional constant-wind-speed advection equation). The equations governing wave-like geophysical flows include additional complexities. In particular, the flow may depend on several unknown functions that are related by a system of partial differential equations, the unknowns may be functions of more than two independent variables, and the equations may be nonlinear. It may also be necessary to account for weak dissipation, sources, and sinks. In this chapter we will examine some of the additional considerations that arise in the design and analysis of finite-difference schemes for the approximation of these more general problems.

Dale R. Durran

4. Series-Expansion Methods

Abstract

Series-expansion methods that are potentially useful in geophysical fluid dynamics include the spectral method, the pseudospectral method, and the finite-element method. The spectral method plays a particularly important role in global atmospheric models, in which the horizontal structure of the numerical solution is often represented as a truncated series of spherical harmonics. Finite-element methods, on the other hand, are not commonly used in multidimensional wave propagation problems because they generally require the solution of implicit algebraic systems and are therefore not as efficient as competing explicit methods. All of these series-expansion methods share a common foundation that will be discussed in the next section.

Dale R. Durran

5. Finite-Volume Methods

Abstract

As demonstrated in the preceding chapters, the errors in most numerical solutions increase dramatically as the physical scale of the simulated disturbance approaches the minimum scale resolvable on the numerical mesh. When solving equations for which smooth initial data guarantees a smooth solution at all later times, such as the barotropic vorticity equation (3.123), any difficulties associated with poor numerical resolution can be avoided by using a sufficiently fine computational mesh. But if the governing equations allow an initially smooth field to develop shocks or discontinuities, as is the case with Burgers’s equation (3.113), there is no hope of maintaining adequate numerical resolution throughout the simulation, and special numerical techniques must be used to control the development of overshoots and undershoots in the vicinity of the shock. Numerical approximations to equations with discontinuous solutions must also satisfy additional conditions beyond the stability and consistency requirements discussed in Chapter 2 to guarantee that the numerical solution converges to the correct solution as the spatial grid interval and the time step approach zero.

Dale R. Durran

6. Semi-Lagrangian Methods

Abstract

Most of the fundamental equations in fluid dynamics can be derived from first principles in either a Lagrangian form or an Eulerian form. Lagrangian equations describe the evolution of the flow that would be observed following the motion of an individual parcel of fluid. Eulerian equations describe the evolution that would be observed at a fixed point in space (or at least at a fixed point in a coordinate system such as the rotating Earth whose motion is independent of the fluid). If S(x, t) represents the sources and sinks of a chemical tracer Ψ(x, t) the evolution of the tracer in a one-dimensional flow field may be alternatively expressed in Lagrangian form as

$$\frac{{d\Psi }}{{dt}} = S$$

(6.1)

, or in Eulerian form as

$$\frac{{\partial \Psi }}{{\partial t}} + u\frac{{\partial \Psi }}{{\partial x}} = S$$

Dale R. Durran

7. Physically Insignificant Fast Waves

Abstract

One reason that explicit time-differencing is widely used in the simulation of wave-like flows is that accuracy considerations and stability constraints often yield similar criteria for the maximum time step in numerical integrations of systems that support a single type of wave motion. Many fluid systems, however, support more than one type of wave motion, and in such circumstances accuracy considerations and stability constraints can yield very different criteria for the maximum time step. If explicit time-differencing is used to construct a straightforward numerical approximation to the equations governing a system that supports several types of waves, the maximum stable time step will be limited by the Courant number associated with the most rapidly propagating wave, yet that rapidly propagating wave may be of little physical significance.

Dale R. Durran

8. Nonreflecting Boundary Conditions

Abstract

If the boundary of a computational domain coincides with a true physical boundary, an appropriate boundary condition can generally be derived from physical principles and can be implemented in a numerical model with relative ease. It is, for example, easy to derive the condition that the fluid velocity normal to a rigid boundary must vanish at that boundary, and if the shape of the boundary is simple, it is easy to impose this condition on the numerical solution. More serious difficulties may be encountered if the computational domain terminates at some arbitrary location within the fluid. When possible, it is a good idea to avoid artificial boundaries by extending the computational domain throughout the entire fluid. Nevertheless, in many problems the phenomena of interest occur in a localized region, and it is impractical to include all of the surrounding fluid in the numerical domain. As a case in point, one would not simulate an isolated thunderstorm with a global atmospheric model just to avoid possible problems at the lateral boundaries of a limited domain. Moreover, in a fluid such as the atmosphere there is no distinct upper boundary, and any numerical representation of the atmosphere’s vertical structure will necessarily terminate at some arbitrary level.

Dale R. Durran

Backmatter

Titel: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
verfasst von: Dale R. Durran
Verlag: Springer New York
Electronic ISBN: 978-1-4757-3081-4
Print ISBN: 978-1-4419-3121-4
DOI: https://doi.org/10.1007/978-1-4757-3081-4

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Basic Finite-Difference Methods

3. Beyond the One-Way Wave Equation

4. Series-Expansion Methods

5. Finite-Volume Methods

6. Semi-Lagrangian Methods

7. Physically Insignificant Fast Waves

8. Nonreflecting Boundary Conditions

Backmatter