Transient L1 error estimates for well-balanced schemes on non-resonant scalar balance laws

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 C0 t (L1x ) 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 C0 t (L1x ) error of conventional fractional-step (Tang and Teng, 1995 [45]) numerical approximations grows exponentially in time like exp(max(g )t) ? x (as a consequence of the use of Gronwall's 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 testcases of increasing difficulty (including the classical LeVeque-Yee's benchmark problem (LeVeque and Yee, 1990 [34]) in the non-stiff case) confirm the analysis.