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We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws