Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws


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.
 
 

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