Preprint 1999-042

Error Estimates on the Random Projection Methods for Hyperbolic Systems with Stiff Reaction Terms

Abstract: 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=-\fl{1}{\vep}(u-\ap)(u^2-1), \qquad -1<\ap<1.

In this problem, the reaction time $\vep$ 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 \cite{Bao} 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 $-\fl{1}{\vep}g(u)$, which has finitely many simple zeroes and satisfying $ug(u)>0$ for large $|u|$.

Paper:

Available as PostScript.

Author(s):

Weizhu Bao

Shi Jin, <jin@math.gatech.edu>

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Submitted by:

<jin@math.gatech.edu> December 6 1999.