Modeling Shallow Water Flows Using the Discontinuous Galerkin Method  book cover
1st Edition

Modeling Shallow Water Flows Using the Discontinuous Galerkin Method

ISBN 9781138076464
Published November 22, 2017 by CRC Press
215 Pages 190 B/W Illustrations

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Book Description


Replacing the Traditional Physical Model Approach

Computational models offer promise in improving the modeling of shallow water flows. As new techniques are considered, the process continues to change and evolve. Modeling Shallow Water Flows Using the Discontinuous Galerkin Method examines a technique that focuses on hyperbolic conservation laws and includes one-dimensional and two-dimensional shallow water flows and pollutant transports.

Combines the Advantages of Finite Volume and Finite Element Methods

This book explores the discontinuous Galerkin (DG) method, also known as the discontinuous finite element method, in depth. It introduces the DG method and its application to shallow water flows, as well as background information for implementing and applying this method for natural rivers. It considers dam-break problems, shock wave problems, and flows in different regimes (subcritical, supercritical, and transcritical).

Readily Adaptable to the Real World

While the DG method has been widely used in the fields of science and engineering, its use for hydraulics has so far been limited to simple cases. The book compares numerical results with laboratory experiments and field data, and includes a set of tests that can be used for a wide range of applications.

  • Provides step-by-step implementation details
  • Presents the different forms in which the shallow water flow equations can be written
  • Places emphasis on the details and modifications required to apply the scheme to real-world flow problems

This text enables readers to readily understand and develop an efficient computer simulation model that can be used to model flow, contaminant transport, and other aspects in rivers and coastal environments. It is an ideal resource for practicing environmental engineers and researchers in the area of computational hydraulics and fluid dynamics, and graduate students in computational hydraulics.

Table of Contents


A historical overview

Organization of the book


General formulation of the discontinuous Galerkin method

Conservation form of equations

Shape functions

Isoparametric mapping

Numerical integration

Approximate Riemann solvers

Time integration


Discontinuous Galerkin method for one-dimensional nonconservative equations

Discontinuous Galerkin method for ordinary differential equations

1D Linear convection

1D Transient diffusion

1D Steady diffusion


One-dimensional conservation laws

Burgers’ equation

Total variation diminishing slope limiter

Shallow water flow equations in rectangular channels

DG method for shallow water flow equations

Numerical tests


One-dimensional shallow water flow in nonrectangular channels

General form of the Saint Venant equations

Discontinuous Galerkin method for general Saint Venant equations

Numerical tests


Two-dimensional conservation laws

Pure convection in 2D

Governing equation of convection in 2D

Discontinuous Galerkin formulation for 2D convection

Slope limiters

Numerical tests


Two-dimensional shallow water flow in channels with horizontal beds

Two-dimensional shallow water flow equations for a horizontal bed channel

Numerical flux

Dry bed treatment

Numerical tests


Two-dimensional shallow water flow in channels with bed variations

Two-dimensional shallow water flow equations for natural channels

Numerical flux and source term treatment

Numerical tests in channels with irregular beds


Pollutant transport

Pollutant transport in 1D

Pollutant transport in 2D


Concluding remarks


Current research topics



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Abdul A. Khan, Ph.D., is an associate professor in the Glenn Department of Civil Engineering at Clemson University (South Carolina). He received a Ph.D. from the University of Alberta, Edmonton, Canada. After his Ph.D., Dr. Khan worked at the National Center for Computational Hydroscience and Engineering, University of Mississippi, before moving to Clemson University. He has been working in the area of computational modeling of river hydraulics, dam-break flows, and sediment transport for the past 20 years. He has published close to 50 journal articles related to his research work and several papers on river flood and dam-break flow modeling.

Wencong Lai, Ph.D., earned a Ph.D. (2012) and an M.S. (2010) in civil engineering from Clemson University, in the area of applied fluid mechanics, and a B.E. (2008) in water conservancy and hydropower engineering from Huazhong University of Science and Technology, China. He is currently a postdoctorate research associate at the University of Wyoming and a member of the CI-WATER’s High-Resolution Multi-Physics Watershed Modeling team. Dr. Lai’s research focuses on computational hydraulics and hydrology. He has developed numerical models for shallow water flows in natural rivers and watersheds using the discontinuous Galerkin finite element method.