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 \cite{grl} (see also the anterior WB Glimm scheme in \cite{we}). This paper aims at showing, by means of rigorous $C^0_t(L^1_x)$ 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 $C^0_t(L^1_x)$ error of conventional fractional-step \cite{tt2} numerical approximations grows {\bf exponentially} in time like $\exp(\max(g')t)\sqrt{ \DX}$ (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 {\bf linearly} in time. Numerical results on several test-cases of increasing difficulty (including the classical LeVeque-Yee's benchmark problem \cite{ly} in the non-stiff case) confirm the analysis.
Transient $L^1$ error estimates for well-balanced schemes on non-resonant scalar balance laws