Transient L^1 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 [14] (see also the anterior WB Glimm scheme in [7]). This paper aims at showing, by means of rigorous C0t(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 C0t(L1x) error of conventional fractional-step [42] 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. Numerical results on several test-cases of increasing difficulty (including the classical LeVeque–Yee’s benchmark problem [31] in the non-stiff case) confirm the analysis.