Convergence of approximate solutions to conservation laws

The general setting considered is a system of n conservation laws in one space dimension. It includes a smooth nonlinear mapping, which is strictly hyperbolic in the sense that its Jacobian has n real and distinct eigenvalues. In a study of suitable approaches for an approximation, attention is given to associated parabolic systems and finite difference schemes which are conservative according to the definitions employed by Lax and Wendroff (1960). In the context of conservation laws, the maximum norm and the total variation norm provide a natural pair of metrics for the investigation of stab ility. The maximum norm serves as a measure of the amplitude of the solution, while the total variation norm represents a measure of the gradient of the solution. Attention is also given to finite difference schemes in the strong topology, the finite scale features of the solution, and the problem of providing stability in the total variation norm.