The reservoir technique: a way to make Godunov-type schemes zero or very low diffuse. Application to Colella–Glaz solver

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Although it is commonly thought that first order schemes are not accurate enough to approximate nonlinear hyperbolic problems, we here explore a conservative time integration with global time steps but local updates (see [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires, C. R. Math. Acad. Sci. Paris, Ser. I 335 (7) (2002) 627–632. [1]]; [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir scheme for systems of conservation laws, in: Finite Volumes for Complex Applications, III, Porquerolles, 2002, Lab. Anal. Topol. Probab. CNRS, Marseille, 2002, pp. 247–254 (electronic). [2]]). This overall conservative method can be interpreted as a system of reservoirs at cell interfaces that fill up and empty when local CFL conditions are reached. For Euler equations, particularly good results are obtained when one uses this technique together with the Riemann solver proposed by Colella and Glaz.