Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

Overview & Synopsis

In this paper we propose a third order accurate finite volume scheme based on polynomial reconstruction along with a posteriori limiting within an Adaptive-Mesh-Refinement (AMR) simulation code for hydro-dynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge-Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a non-oscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using this time a more dissipating scheme but only locally to these cells. By decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate scheme. Ultimately some troubled cells may be updated with a first order accurate scheme for instance close to steep gradients. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed but only locally with such refined mesh. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology not only can maintain optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.