# Shallow Water Hydrodynamics

## Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow-water Equations

Within this monograph a comprehensive and systematic knowledge on shallow-water hydrodynamics is presented. A two-dimensional system of shallow-water equations is analyzed, including the mathematical and mechanical backgrounds, the properties of the system and its solution. Also featured is a new mathematical simulation of shallow-water flows by compressible plane flows of a special virtual perfect gas, as well as practical algorithms such as FDM, FEM, and FVM. Some of these algorithms have been utilized in solving the system, while others have been utilized in various applied fields. An emphasis has been placed on several classes of high-performance difference schemes and boundary procedures which have found wide uses recently for solving the Euler equations of gas dynamics in aeronautical and aerospatial engineering.

This book is constructed so that it may serve as a handbook for practicians. It will be of interest to scientists, designers, teachers, postgraduates and professionals in hydraulic, marine, and environmental engineering; especially those involved in the mathematical modelling of shallow-water bodies.

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Published: August 1992

Imprint: Elsevier

ISBN: 978-0-444-98751-8

## Contents

• Background in Mechanics. Physical Objects. System of Fluid-Dynamics Equations for Homogeneous Isotropic Incompressible Flows. Two-Dimensional System of Shallow-Water Equations (2-D SSWE). Physical Meanings of Various Terms in 2-D SSWE. Various Forms of 2-D SSWE. Properties of 2-D SSWE. Conceptual Mechanical Behavior. Dimensional Analysis of 2-D SSWE. Basic Mathematics for Systems of First-Order Quasilinear Hyperbolic Equations. Geometric Theory of Characteristics. Riemann Invariants. Theory of Nonlinear Wave Propagation. Properties of the Solutions of 2-D SSWE. Initial and Boundary Conditions for Well-Posed Problems. Behavior of Solutions. Discontinuous Solutions of SSWE. Isentropic Flow Simulation of SSWE and its Limitations. Discontinuous Solutions of 1-D First-Order Hyperbolic Systems. Introduction to 2-D Discontinuous Solutions. Mathematical Conditions of Shock Waves for 2-D SSWE. Preliminary Review of Finite Difference Methods. General Description. Basic Performance of a Difference Scheme. Basic Difference Schemes for First-Order Hyperbolic Systems in One Space Dimension. FDMs for the Computation of 1-D Unsteady Open Flows. Difference Schemes for 2-D SSWE. FDMs for the Solution of 2-D SSWE in Nonconservative Form. FDMs for the Solution of 2-D SSWE in Conservative Form. Fractional-Step Methods and Splitting-Up Algorithms. Fractional-Step Difference Schemes for 2-D Unsteady Flow Computations. FDMs for Curvilinear Meshes. Finite Volume Method (FVM). Numerical Solutions Using Finite Element Methods. Related Principles in Variational Calculus. Piecewise Approximation of Plane Problems and Convergence of FEM Solutions. FEM for 2-D Unsteady Open Flows. Several Classes of Special FEMs. Techniques for the Implementation of Algorithms. Computational Mesh. Classical Techniques for Improving Computational Stability and Accuracy. New Developments of Difference Schemes for 2-D First-Order Hyperbolic Systems of Equations. General Description. Two-Dimensional Methods of Characteristics. Characteristic-Based Splitting. Riemann Approach. Approximate Factorization of Implicit Schemes. FCT Algorithms and TVD Schemes. Square Conservation and Energy Conservation Schemes. Stability Analysis and Boundary Procedures. Mathematical Definitions of Stability for Difference Schemes. Von Neumann Linear Stability Analysis. Nonlinear Instability. Boundary Procedures and Their Influence on Numerical Solutions. Stability Theory for Mixed Problems. Concluding Remarks. Requirements for an Ideal Finite-Difference Scheme. Comparison of Performance, Merits and Drawbacks Between FDM and FEM. Brief Introduction to Other Algorithms. Towards a Truly 2-D Algorithm. Index.