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. Temporal Oscillations -- 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.