# Preprint 2013-001

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

## Debora Amadori and Laurent Gosse

**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.

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