these notes developed from a course on the numerical solution of conservation laws first taught at the university of washington in the fall of 1988 and then at eth during the following spring.
the overall emphasis is on studying the mathematical tools that are essential in developing, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. a reasonable understanding of the mathematical structure of these equations and their solutions is first required, and part i of these notes deals with this theory. part ii deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. i have stressed the underlying ideas used in various classes of methods rather than presenting the most sophisticated methods in great detail. my aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding.
ⅰ mathematical theory
1 introduction
1.1 conservation laws
1.2 applications
1.3 mathematical difficulties
1.4 numerical difficulties
1.5 some references
2 the derivation of conservation laws
2.1 integral and differential forms
2.2 scalar equations
2.3 diffusion
3 scalar conservation laws
3.1 the linear advection equation
3.2 burgers' equation
3.3 shock formation ⅰ mathematical theory
1 introduction
1.1 conservation laws
1.2 applications
1.3 mathematical difficulties
1.4 numerical difficulties
1.5 some references
2 the derivation of conservation laws
2.1 integral and differential forms
2.2 scalar equations
2.3 diffusion
3 scalar conservation laws
3.1 the linear advection equation
3.2 burgers' equation
3.3 shock formation
3.4 weak solutions
3.5 the riemann problem
3.6 shock speed
3.7 manipulating conservation laws
3.8 entropy conditions
4 some scalar examples
4.1 traffic flow
4.2 two phase flow
5 some nonlinear systems
5.1 the euler equations
5.2 isentropic flow
5.3 isothermal flow
5.4 the shallow water equations
6 linear hyperbolic systems
6.1 characteristic variables
6.2 simple waves
6.3 the wave equation
6.4 linearization of nonlinear systems
6.5 the riemann problem
7 shocks and the hugoniot locus
7.1 the hugoniot locus
7.2 solution of the riemann problem
7.3 genuine nonlinearity
7.4 the lax entropy condition
7.5 linear degeneracy
7.6 the riemann problem
8 rarefaction waves and integral curves
8.1 integral curves
8.2 rarefaction waves
8.3 general solution of the riemann problem
8.4 shock collisions
9 the pdemann problem for the euler equations
9.1 contact discontinuities
9.2 solution to the riemann problem
ⅱ numerical methods
bibliography
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