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 ε (u−α) u 2−1 , −1<α<1. In this problem, the reaction time ε 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 ε g(u) , which has finitely many simple zeroes and satisfying ug( u)>0 for large | u|.
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