Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag 15 I. Kinnmark The Shallow Water Wave Equations: Formulation, Analysis and Application Spri nger-Verlag Berlin Heidelberg New York Tokyo Series Editors C. A. Brebbia . S. A. Orszag Consulting Editors J. Argyris . K.-J. Bathe' A. S. Cakmak . J. Connor' R. McCrory C. S. Desai' K.-P. Holz . F. A. Leckie' G. Pinder' A. R. S. Pont J. H. Seinfeld . P. Silvester' P. Spanos· W. Wunderlich' S. Yip Author Ingemar (Per Erland) Kinnmark Department of Civil Engineering University of Notre Dame Notre Dame, Indiana 46556 USA ISBN-13: 978-3-540-16031-1 e-ISBN-13: 978-3-642-82646-7 001: 10.1007/978-3-642-82646-7 Library of Congress Cataloging in Publication Data Kinnmark, I. (Ingemar) The shallow water wave equations. (Lecture notes in engineering; v. 15) Bibliography: p. 1. Water waves. 2. Wave equation. I. Title. II. Series. TC172.K56 1985 627'.042 85-27662 ISBN-13: 978-3-540-16031-1 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin, Heidelberg 1986 2061/3020-543210 To Beth, Solvig and Tore ABSTRACT THE SHALLOW WATER WAVE EQUATIONS: FORMULATION, ANALYSIS AND APPLICATION by Ingemar Per Erland Kinnmark The shallow water equations are utilized to describe fluid flow in a vertically well-mixed fluid experiencing tidal and atmospheric forc ing. The wave equation form of the continuity equation on finite ele ments is chosen because of its demonstrated ability to suppress 2ax oscillations. It is shown that the wave equation can be .represented in operator form, consisting of a linear combination of the primitive continuity equation and its time derivative as well as spatial deriva tives of the momentum equations. This operator representation natural ly leads to the formulation of a more general wave continuity equa tion. The generalized form offers an implicit time marching procedure with time invariant matrix. This will strongly reduce the computa tional effort for large size applications. An alternative implicit scheme based on a Taylor expansion of an almost time invariant matrix is presented. The method achieves efficiency for large size applica- VI tions by substituting cheaper back substitutions for costly decomposi tions. If a symmetric three time level approximation of the momentum equa tions is solved in conjunction with the wave continui.ty equation, tem poral oscillations develop. A symmetric two time level approximation, except for the non-linear convective term, is shown to remove the nu merical artifacts that cause the temporal oscillations. It has been suggested that a non-zero phase velocity for 2~x waves, or alternatively a monotonic or non-folded dispersion curve, can ac count for the ability of the wave equation. to suppress spurious' node-to-node oscillations. It is demonstrated with Fourier analysis, however, that when the wave equation is utilized with periodically varying node spacing or bathymetry, or in conjunction with quadratic elements it also experiences zero phase velocity for 2~x waves. The application of the Routh-Hurwitz criterion, and extensions thereof, to polynomials of degree higher than two have proven valuable for determining numerical stability. Applications for two separate data sets in the southern part of the North Sea have shown good agreement with earlier computational re sults. ACKNOWLEDGEMENTS I wish to acknowledge Professor Gray for his continuing, strong support and his willingness to devote time, energy and dedication to scrutinizing all aspects of a research result. My appreciation goes to Professor Pinder for his inspiration and enthusiasm. Myron Allen, Dennis McLaughlin and Zbigniew Kabala all contributed interesting and invigorating discussions. Beth did an excellent linguistic revision and typing of the text. Pat Roman did a very good typing of many tables. Tom Agans did excel lent drafting of the figures. I also wish to thank Beth for her lasting and dedicated companion ship. Finally I wish to thank my parents for their commitment to per forming a task energetically and correctly, and for their strong sup port. This research has been supported in part through grant number CME-7921076-01 from the United States National Science Foundation and in part by a Wallace Memorial Honorific Fellowship awarded by the Gtaduate School of Princeton University. CONTENTS LIST OF TABLES ••••.•••••..•••••••••.•••.••••••••••••••••••••••• xi LIST OF FIGURES xiii LIST OF SYMBOLS xvi Chapter I. INTRODUCTION 1 Areas of Application for the Shallow Water Equations 1 Finite Element Methods for Solution of the Shallow Water Equations ..........•..................... 3 Methods for Analyzing Spatial Oscillations in Numerical Schemes ..•..•......•.......••....•••. 8 Methods for Analyzing Stability of Numerical Schemes 10 II. EQUATION FORMULATION •••••••••••••••••••••••••••••••••••• 12 Primitive Equation Form ..•.•.•...••...••.•..•..••...• 12 Wave Equation Form .•.•......••...••..••..........•.•• 14 Generalized Wave Equation Form ••••••••••••••••••••••• 17 Linearized Form of the Continuity and Momentum Equations •..................................... 20 III. FOURIER ANALYSIS METHODS 27 Introduction ......................................... 27 Fourier Analysis: An Accuracy Measure ••••••••••••••• 28 Amplitude of Propagation Factors Arising from Second Degree Polynomials ••••••••••••••••••••••••••••• 32 IV. STABILITY .••••••.••••••••••••••••••••••••••••.••.•••••.• 38 General Concepts ........•.....•••••..••.••.•.•...•.•. 38 Routh-Hurwitz and Li~nard-Chipart •••••••••••••••••••• 42 IX Routh-Hurwitz and Orlando •••••••••••••••••••••••••••• 52 Factorization of Higher Degree Polynomials into Lower Degree Polynomials ••••••••••••••••••••••••••••• 61 Determination of Stability for a Product of Polynomials .................................... 65 V. EXPLICIT METHODS USING VARIOUS SPATIAL DISCRETIZATIONS •• 68 Introduction .......................•................. 68 Equal Node Spacing and Constant Bathymetry in One Dimension ...................................... 69 Application to Unequal Node Spacing •••••••••••••••••• 79 Applications with Even Node Spacing and Variable Bathymetry .....................•............•.. 86 Application to a Rectangular Grid •••••••••••••••••••• 86 VI. IMPLICIT METHODS •••••••••••••••••••••••••••••••••••••••• 96 Introduction ......................................... 96 Reducing the Number of Time Dependent Terms in the Matrix for the Wave Equation ••••••••••••••••••• 98 Explicit Treatment of the Corio1is Term in an Implicit Wave Continuity Equation •••••••••••••• 100 Repeated Back Substitutions Replacing Decompositions. 105 The Generalized Wave Continuity Equation ••••••••••••• 109 VII. SPATIAL OSCILLATIONS •••••••••••••••••••••••••••••••••••• 115 Introduction ...•......•.••••.••.•..•••.•.••..•..••... 115 N-Dimensional Uniform Rectangular Grid ••••••••••••••• 117 N-Dimensional Nonuniform Rectangular Grid with Multi-Information Nodes •••••••••••••••••••••••• 123 Leapfrog Scheme and Wave Equation Formulation on Linear Elements ................................ 125 Leapfrog Scheme and Wave Equation Formulation on Quadratic Elements •.•••••.•••••.•..••••.••.••.• 128 The Use of Dispersion Analysis in Evaluating Numerical Schemes .•.....••••.•....•.••.•••••••• 134 The 2~ Test: Assessing the Ability to Suppress Node-to-Node Oscillations •••••••••••••••••••••• 138 VIII. TEMPORAL OSCILLATIONS 148 Introduction .••.•••.••..•.••.•••••....•••••.•. 148 8 •••••• Numerical Artifacts •••••••.•••.•.••.•..•.•.....•.••.. 149 A Different Three Time Level Approximation of the Momentum Equations ....•.••••.•.•.•••.••..••••.. 151 A Two Time Level Approximation of the Momentum Equations ••.•.••.....••••..••..........••..•... 157 x IX. APPLICATIONS 159 Introduction .•....•...............•.................. 159 Application to Quarter Circle Harbor ••••••••••••••••• 160 Application to the Southern Part of the North Sea - I 160 Application to the Southern Part of the North Sea - II 164 X. CONCLUSIONS 172 Appendix A. EQUIVALENT FORMULATIONS OF CONDITIONS WHICH GUARANTEE ROOTS OF MAGNITUDE LESS THAN UNITy ••••••••••••••••••• 176 BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••••••••••••••••••• 181 LIST OF TABLES 4.1. Equivalent Formulations of Necessary and Sufficient Conditions for Ixl < 1 for a First Degree Polynomial. 44 4.2. Equivalent Formulations of Necessary and Sufficient Conditions for Ixl < 1 for a Quadratic Polynomial 45 4.3. Equivalent Formulations of Necessary and Sufficient < Conditions for Ixl 1 for a Cubic Polynomial •.•...•• 46 4.4. Equivalent Formulations for Necessary and Sufficient Conditions for Ixl < 1 for a Quartic Polynomial •..••• 48 4.5. Equivalent Formulations for Necessary and Sufficient < Conditions for Ixl 1 for a Quintic Polynomial •.•••• 50 Character of Roots to a Second Degree Polynomial with Non-Negative Pi .......................•....•.•..••.•. 53 4.7. Character of Roots to a Third Degree Polynomial with Non-Negative Pi and ~2 .•••••.•.•.•••••••..•..•••.•••• 55 4.8. Character of Roots to a Fourth Degree Polynomial with Non-Negative Pi and ~3 .•..•••.•••....••.••••.•..•••.. 58 5.1. Stability Criteria for Explicit Wave Equation Schemes Using Different Spatial Discretization on a Uniform One-Dimensional Grid with Constant Bathymetry .••••••. 75 5.2. Stability Criteria for Explicit Leapfrog Schemes Using Different Spatial Discretization on a Uniform Grid with Constant Bathymetry in One Spatial Dimension 76 5.3. A Values for Equi-Spaced Grid Schemes Using the Wave Equation. C = Courant Number, F = T~t/2, c = cos( a~) ••••••••••••••••••••••••••••••••••••••, _...... 77 5.4. A Values for Equi-Spaced Grid Schemes Using the Leap- Frog Approximation. C = Courant Number, F = T~t, c = cos( atsx.) ••.••.••••••••••...•.•••••••••••••••.•••• 78 5.5. Stability Criteria for Explicit Wave Eqation Methods Using Unequal Spatial Grid Steps in One Dimension. Constant Bathymetry. C = ~t {gh/6K. C = CIa ••••••• 83 a A Values for Unequally-Spaced Grid Schemes •••••••••••••• 84 5.7. Stability Criteria for Explicit Wave Equation Methods on a Uniform One-Dimensional Grid with Variable Bathymetry ..•............................•...•..•.... 88