AMath 574
Winter Quarter, 2009

Conservation Laws and Finite Volume Methods

See the course webpage for more information about the class.

Instructor: Professor Randall J. LeVeque
Guggenheim 415C
(206) 685-3037
rjl at washington dot edu
Time: Mondays and Wednesdays, 3:30 - 4:50 pm

Room: Guggenheim 204

Text: Finite Volume Methods for Hyperbolic Problems, by R. J. LeVeque, Cambridge University Press, 2003

Hyperbolic conservation laws are partial differential equations that arise in modeling phenomena involving wave propagation or fluid flow, including shock waves in nonlinear situations.

Applications are abundant and widespread. A few examples include

While these applications vary greatly, there is a common mathematical structure to the equations which arise from all of these problems. A powerful set of computational techniques have been developed over the past several decades to compute accurate solutions to these problems. In this course we will explore the mathematics of these problems and how it is used to develop finite-volume methods for solving them. In particular we will study Godunov's method and high-resolution extensions such as TVD (total variation diminishing) methods.

We will first study linear problems in detail. These arise in solving many acoustic, elastic, and electromagnetic wave propatation problems. We will consider waves in homogeneous and heterogeneous media. The linear theory is simpler and is fundamental to understanding the nonlinear theory, but still contains many of the essential features. Then we will turn to nonlinear hyperbolic conservation laws, such as the shallow water equations and the Euler equations of gas dynamics. These equations can have shock wave solutions, and introduce a variety of new mathematical and computational difficulties. Both one-dimensional and multidimensional problems will be considered.

The CLAWPACK software will be used to allow students to compute interesting solutions for various applications and as a platform for developing and testing various methods. This software was originally developed in the context of teaching this course at UW and is now widely used in teaching and research elsewhere. See the CLAWPACK applications page for some sample applications.

Prerequisites: AMath 586 or comparable background in numerical methods for differential equations. Please contact the instructor if you want more information.