A Moving Mesh Numerical Method for Hyperbolic Conservation Laws

We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in ${L^1}({\mathbf {R}})$ to within $O({N^{ - 2}})$ by a piecewise linear function with $O(N)$ nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to $O({N^{ - 1}})$. These numerical methods for conservation laws are the first to have proven convergence rates of greater than $O({N^{ - 1/2}})$.