Abstract:
We prove the convergence of flux vector
splitting schemes associated to hyperbolic systems of conservation laws
with a single compatible entropy $\eta_c$. We prove estimate on the $L^2$
norm of the gradient of the numerical approximation in the inverse square
root of the space increment $\Delta x$. This estimate is related to the
notion of (strictly) $\eta_c$-dissipativity on $F^+$, $-F^-$ and $Id-\lambda(F^+-F^-)$,
where $F^+$, $F^-$ is the flux-decomposition. The second tool of the proof
is a kinetic formulation of the flux-splitting scheme with three velocities.
Then we get a control for all entropies and apply the compensated compactness
theory.