Transient [formula omitted] error estimates for well-balanced schemes on non-resonant scalar balance laws

Abstract

The ability of Well-Balanced (WB) schemes to capture very accurately steady-state regimes of non-resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by Greenberg and LeRoux (1996) [15] (see also the anterior WB Glimm scheme in E, 1992 [8]). This paper aims at showing, by means of rigorous Ct0(Lx1) estimates, that these schemes deliver an increased accuracy in transient regimes too. Namely, after explaining that for the vast majority of non-resonant scalar balance laws, the Ct0(Lx1) error of conventional fractional-step (Tang and Teng, 1995 [45]) numerical approximations grows exponentially in time like exp(max(g′)t)Δx (as a consequence of the use of Gronwallʼs lemma), it is shown that WB schemes involving an exact Riemann solver suffer from a much smaller error amplification: thanks to strict hyperbolicity, their error grows at most only linearly in time (see also Layton, 1984 [30]). Numerical results on several test-cases of increasing difficulty (including the classical LeVeque–Yeeʼs benchmark problem (LeVeque and Yee, 1990 [34]) in the non-stiff case) confirm the analysis.