Semi-conservative finite volume schemes for conservation laws

This paper aims to introduce a new class of high order conservative schemes to solve systems of conservation laws. The idea is to couple the conservation form of the system with, possibly simpler, alternative formulations, which can be used to speed up the time update. In this work, we illustrate the procedure for a Runge-Kutta time advancement, but other choices are possible. We show that, as long as the last update is carried out in conservative form, all internal stages can be computed using any consistent nonconservative formulation, still ensuring the propagation of shock waves with the correct speeds. The same procedure can be easily extended to finite difference schemes. Tests from classical and relativistic gas dynamics are carried out to study convergence, numerical robustness and performance.