A Stable Adaptive Numerical Scheme for Hyperbolic Conservation Laws

A new adaptive finite-difference scheme for scalar hyperbolic conservation laws is introduced. A key aspect of the method is a new automatic mesh selection algorithm for problems with shocks. We show that the scheme is L1-stable in the sense of Kuznetsov, and that it generates convergent approximations for linear problems. Numerical evidence is presented that indicates that if an error of size ε is required, our scheme takes at most O(ε-3) operations. Standard monotone difference schemes can take up to O(ε-4) calculations for the same problems.