Computational Algorithms for Shallow Water Equations

Abstract

This BookThis book deals with computational methods for solving systems of shallow water equations within the broader field of geophysical fluid dynamics.Such systems, comprised of hyperbolic partial differential equations, serve as mathematical models for simulating the dynamics of water flows in open channels, rivers, lakes, and oceans.Practical applications range from modelling oceanic tides to the breaking of waves on shallow beaches, roll waves in open channels, flood waves in rivers, dam-break waves, tsunami wave propagation, and various environmental fluid dynamics scenarios.Mathematical models of the shallow-water type are also employed for simulating gas dynamics, such as the dispersion of heavy gases over the Earth's surface and atmospheric dynamics.However, the physical complexity of time-dependent, threedimensional oceanic and atmospheric flows necessitates more comprehensive mathematical models than shallow-water type systems alone.Nevertheless, the basic, onelayer shallow water equations play a crucial role in providing benchmark solutions for assessing methodologies intended for more complex models.Significantly, multilayer shallow water models are currently undergoing major developments in oceanic and atmospheric dynamics, where the underlying concepts and methodologies from the basic one-layer shallow water equations in this book find applicability.This book offers fundamental concepts and methods essential for constructing advanced high-order computational algorithms within the frameworks of finite volume and discontinuous Galerkin finite element methods.The finite volume methodology is developed in full detail, presenting non-linear, fully discrete high-order upwind and centred methods for solving the non-linear, time-dependent two-dimensional shallow water equations.

Topics & Concepts