Abstract
We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by blocks .
Highlights
We are interested in numerical integration of non linear hyperbolic systems of conservation laws of the form. ⎧ ⎨ ⎩ ∂w ∂t + ∂f (w) ∂x =0, w(0, x) = w0(x),(t, x) ∈ R+ × R x ∈ R. (1)Article published by EDP Sciences
Starting from 500 cells, the adaptive schemes lead to very close solutions for each scheme and the numerical density of entropy production vanishes everywhere where the solution is smooth and every solution fit to the reference solution
In the framework of the local time stepping, for almost the same accuracy the AB2M scheme computes 3 times faster than the RK2 which corresponds to a significant gain in time
Summary
We are interested in numerical integration of non linear hyperbolic systems of conservation laws of the form (express here in 1d for the sake of simplicity). Solving Equation (1) with high accuracy is a challenging problem since it is well-known that solutions can and will breakdown at a finite time, even if the initial data are smooth, and develop complex structure (shock wave interactions). In such a situation, the uniqueness of the (weak) solution is lost and is recovered by completing the system (1) with an entropy inequality of the form:. The reader can found more details about the 3d approach in [11, 12]
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