The Method of Characteristics with Applications to Conservation Laws

Scott A. Sarra (e-mail)

Marshall University

Abstract

In this article and its accompanying applet, I introduce the method of characteristics for solving first order partial differential equations (PDEs). First, the method of characteristics is used to solve first order linear PDEs. Next, I apply the method to a first order nonlinear problem, an example of a conservation law, and I discuss why the method may break down for nonlinear problems. I examine difficulties that appear in the nonlinear case, and I introduce the mathematical resolutions of these problems. Concepts dealing with the solutions of nonlinear hyperbolic conservation laws are introduced and explored graphically. However, detailed explanations of such concepts may take up entire textbooks, and the concepts are not explored in detail or with rigor in this short paper. I include References to more detailed material.

Intended Audience

This material is appropriate for undergraduate students in a partial differential equations class, as well as for undergraduate (or graduate) students in mathematics or other sciences who desire a brief and graphical introduction to the solutions of nonlinear hyperbolic conservation laws or to the method of characteristics for first order hyperbolic partial differential equations.

Contents

1 - The Method of Characteristics

1.1 - General Strategy

2 - Special Case: b(xt) = 1 and c(xt) = 0

2.1 - Constant Coefficient Advection Equation

2.2 - Variable Coefficient Advection Equation

3 - Conservation Laws

3.1 - Inviscid Burgers' Equation

3.2 - Numerical Methods for Conservation Laws

3.3 - Inviscid Burgers' equation example problems

3.3.1 - Riemann problem - shock

3.3.2 - Riemann problem - rarefaction

3.3.3 - Sine

3.3.4 - N wave

3.3.5 - Single hump

4 - The Shock Applet

5 - References and Links


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