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# Transient L¹ 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 [14] (see also the anterior WB Glimm scheme in [7]). This paper aims at showing, by means of rigorous $C_t^0(L^1_x)$ 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 $C_t^0(L^1_x)$ error of conventional fractional-step [42] numerical approximations grows exponentially in time like $\exp(\max(g')t)\sqrt{∆x}$ (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.

References
[7] W. E, Homogenization of scalar conservation laws with oscillatory forcing terms, SIAM J. Appl. Math. 52 (1992), 959–972
[14] J. Greenberg and A.Y. LeRoux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), 1–16
[31] R.J. LeVeque and H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comp. Phys. 86 (1990), 187–210
[42] T. Tang and Z.-H. Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal. 32 (1995) 110–127
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