Error estimates on the random projection methods for hyperbolic conservation laws with stiff reaction terms

In this paper we give error estimates on the random projection methods, recently introduced by the authors, for numerical simulations of the hyperbolic conservation laws with stiff reaction terms: u(t) + f (u)(x) = -1/epsilon(u - alpha)(u(2) - 1), -1 < alpha < 1. In this problem, the reaction time a is small, making the problem numerically stiff. A classic spurious numerical phenomenon-the incorrect shock speed-occurs when the reaction time scale is not properly resolved numerically. The random projection method, a fractional step method that solves the homogeneous convection by any shock capturing method, followed by a random projection for the reaction term, was introduced in [J. Comput. Phys. 163 (2000) 216-248] to handle this numerical difficulty. In this paper, we prove that the random projection methods capture the correct shock speed with a first order accuracy, if a monotonicity-preserving method is used in the convection step. We also extend the random projection method for more general source term -1/epsilong(u), which has finitely many simple zeroes and satisfying ug(u) > 0 for large \u\. (C) 2002 IMACS. Published by Elsevier Science B.V All rights reserved.