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

Abstract

To cite this version:Matteo Semplice, Raphaël Loubère. Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions. AbstractIn 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 hydrodynamics 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 nonoscillatory 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.

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

“…Consequently, some form of limitation must be supplemented to stabilize the high order method. In this work we rely on an a posteriori MOOD loop (Multi-dimensional Optimal Order Detection), see [8,12,13,29] and some applications in [16,5,2,9,11,3,33,6,17,41,4,43]. This approach a posteriori checks if the unlimited candidate solution fulfills some validity criteria (computer, physical and numerical admissibility) and accordingly recomputes the current solution locally in space and time with a more robust but less accurate scheme by reducing the polynomial degree of the local reconstruction.…”

“…Here we use the 1st order accurate FV scheme. While the MOOD paradigm is in use in several codes see for instance [16,5,2,9,11,3,33,6,17,41,4,43], some weaknesses can be pointed. First the a posteriori MOOD loop breaks the parallel efficiency because some cells demand more attention than others: they are possibly recomputed several times while (most) others are accepted at the end of the very first MOOD iteration.…”

GP-MOOD: A positive-preserving high-order finite volume method for hyperbolic conservation laws

“…The fundamental principle of the MOOD paradigm centers around three important building blocks, the detection criteria, the safe scheme, and the scheme cascade [28]. We adopt these three basic components to devise the following relaxed version for our GP-MOOD methods:…”

“…Besides, the positivity-preserving property under an admissible CFL (Courant-Fredirichs-Lewy) condition was made explicit by imposing the Physical Admissibility Detection (PAD) that checks the positivity of density and pressure variables on each cell. The MOOD method was further extended to general three-dimensional unstructured meshes with simplifications of the u2 detection [26,27]; to a third-order FV adaptive mesh refinement (AMR) scheme in [28].…”

Numerical Entropy Production as Smoothness Indicator for Shallow Water Equations

“…Numerical entropy production has further been used to develop numerical schemes for conservation laws and balance laws, such as those in the literature [1,15,30,[33][34][35]. Due to their wide range of applications, these laws have been of interest to a number of researchers [6,7,11,24,26,31,32].…”