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 [12]. 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 [9,10,11, 14], 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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Ndjinga, M., Ait-Ameur, K. (2020). A New Class of \(L^2\)-Stable Schemes for the Isentropic Euler Equations on Staggered Grids. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_39
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