Geodesic
ethods of the variational Riemann problem in computational gasdynamics
Matematičeskoe modelirovanie, Tome 19 (2007) no. 6, pp. 86-108.

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The variational Riemann problem is considered, which is to determine the first variation of the solution to Riemann's initial-value problem, also known as the problem of breakup of a discontinuity in a gas, when the initial data undergo small variations. It is shown that the solution to this problem is unique, analytically obtained, and written in a compact explicit form, provided that the baseline Riemann problem solution is available. The obtained solution is then utilized in two numerical applications. The first is the exact linearization of Godunov's numerical flux-function to solve the equations of the Godunov implicit method. Another relates to the approximation of the acoustic numerical flux in a numerical approach for simulating of the propagation of small disturbances on the background of non-uniform basic flows. 
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I. S. Menshov. ethods of the variational Riemann problem in computational gasdynamics. Matematičeskoe modelirovanie, Tome 19 (2007) no. 6, pp. 86-108. http://geodesic.mathdoc.fr/item/MM_2007_19_6_a7/

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