A Stable Adaptive Numerical Scheme for Hyperbolic Conservation Laws

Abstract

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 $L^1 $-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 $\varepsilon $ is required, our scheme takes at most $O(\varepsilon ^{ - 3} )$ operations. Standard monotone difference schemes can take up to $O(\varepsilon ^{ - 4} )$ calculations for the same problems.