An L1--Error Bound for a Semi-Implicit Difference Scheme Applied to a Stiff System of Conservation Laws

Abstract

A straightforward semi-implicit finite-difference method approximating a system of conservation laws including a stiff relaxation term is analyzed. We show that the error, measured in L 1 , is bounded by O( √ ∆t) independent of the stiffness, where the time step ∆t represents the mesh size. As a simple corollary we obtain that solutions of the stiff system converge toward the solution of an equilibrium model at a rate of O(δ 1/3 ) in L 1 as the relaxation time δ tends to zero.

Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

“…Most of these results deal with either large-time, nonlinear asymptotic stability or the zero relaxation limit for Cauchy problems. Tveito and Winther [18,26] provided an O( 1/3 )-rate of convergence for some relaxation systems with nonlinear convection arising in chromatography. Katsoulakis and Tzavaras [7] introduced a class of relaxation systems, the contractive relaxation systems, and established an O( √ ) error bound in the case that the equilibrium equation is a scalar multidimensional one.…”

“…Katsoulakis and Tzavaras [7] introduced a class of relaxation systems, the contractive relaxation systems, and established an O( √ ) error bound in the case that the equilibrium equation is a scalar multidimensional one. The approaches in [7,18,26] are based on the extensions of Kruzhkov and Kuznetzov-type error estimates [9]. Kurganov and Tadmor [8] studied convergence and error estimates for a class of relaxation systems, including (1.1) and the one arising in chromatography, and concluded an O( ) order of convergence for scalar convex conservation laws.…”

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

“…The approximation of the stiff case was recently studied by several authors (see [2], [6], [8], [13], [15], [17], [19], [25], and [26]), where different methods like asymptotic or splitting methods are used.…”

On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms

“…The approximation of the stiff case was recently studied by several authors (see [2], [5], [7], [9], [10], [12], [16], [18], [19]), where different methods like asymptotic or splitting methods are used.…”