The Shallow Water Wave Equations: Formulation, Analysis and Application

I. Introduction.- Areas of Application for the Shallow Water Equations.- Finite Element Methods for Solution of the Shallow Water Equations.- Methods for Analyzing Spatial Oscillations in Numerical Schemes.- Methods for Analyzing Stability of Numerical Schemes.- II. Equation Formulation.- Primitive Equation Form.- Wave Equation Form.- Generalized Wave Equation Form.- Linearized Form of the Continuity and Momentum Equations.- III. Fourier Analysis Methods.- Fourier Analysis: An Accuracy Measure.- Amplitude of Propagation Factors Arising from Second Degree Polynomials.- IV. Stability.- General Concepts.- Routh-Hurwitz and Liénard-Chipart.- Routh-Hurwitz and Orlando.- Factorization of Higher Degree Polynomials into Lower Degree Polynomials.- Determination of Stability for a Product of Polynomials.- V. Explicit Methods Using Various Spatial Discretizations.- Equal Node Spacing and Constant Bathymetry in One Dimension.- Application to Unequal Node Spacing.- Applications with Even Node Spacing and Variable Bathymetry.- Application to a Rectangular Grid.- VI. Implicit Methods.- Reducing the Number of Time Dependent Terms in the Matrix for the Wave Equation.- Explicit Treatment of the Coriolis Term in an Implicit Wave Continuity Equation.- Repeated Back Substitutions Replacing Decompositions.- The Generalized Wave Continuity Equation.- VII. Spatial Oscillations.- N-Dimensional Uniform Rectangular Grid.- N-Dimensional Nonuniform Rectangular Grid with Multi-Information Nodes.- Leapfrog Scheme and Wave Equation Formulation on Linear Elements.- Leapfrog Scheme and Wave Equation Formulation on Quadratic Elements.- The Use of Dispersion Analysis in Evaluating Numerical Schemes.- The 2?x Test: Assessing the Ability to Suppress Node-to-Node Oscillations.- VIII. TemporalOscillations.- Numerical Artifacts.- A Different Three Time Level Approximation of the Momentum Equations.- A Two Time Level Approximation of the Momentum Equations.- IX. Applications.- Application to Quarter Circle Harbor.- Application to the Southern Part of the North Sea - I.- Application to the Southern Part of the North Sea - II.- X. Conclusions.- A. Equivalent Formulations of Conditions Which Guarantee Roots of Magnitude Less than Unity.