AN ADAPTIVE VERSION OF GLIMM'S SCHEME Dedicated to Professor James Glimm on the occasion of his 75th birthday

Abstract

This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L-1(R) for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad-hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.