A new class of $L^2$ -stable schemes for the isentropic Euler equations on staggered grids
Abstract
Staggered schemes for compressible flows are highly non linear and the stability analysis has historically been performed with a heuristic approach and the tuning of numerical parameters . We investigate the $L^2$-stability of staggered schemes by analysing their numerical diffusion operator. The analysis of the numerical diffusion operator gives new insight into the scheme and is a step towards a proof of linear stability or stability for almost constant initial data. For most classical staggered schemes, we are able to prove the positivity of the numerical diffusion only in specific cases (constant sign velocities). We then propose a class of linearly $L^2$-stable staggered schemes for the isentropic Euler equations based on a carefully chosen numerical diffusion operator. We give an example of such a scheme and present some first numerical results on a Riemann problem.
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