A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Abstract

International audienceIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decre- menting processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic par- tial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements

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

“…In this work we will use an a posteriori stabilization procedure based on a troubled cell detector and subsequent re-computations with a second-or first-order finite volume scheme. This procedure is based on the so called MOOD paradigm, see [CDL11a,DCL12,DLC13,LDD14].…”

“…Next, we will describe first the high order discretization L i (u(t)) of the spatial operator on the right hand side of (5) and then the time discretization will be achieved with a third order time accurate Runge-Kutta (RK3) scheme maintaining, at the same 4 time, better than second order of accuracy in space and time. At a difference from [SCR15,CS15], here stabilization of the high accurate reconstructions is obtained by means of an a posteriori MOOD limiting [CDL11a,DCL12,DLC13,LDD14] under the classical CFL condition of a RK3 scheme. At last an adaptive mesh refinement (AMR) technique is employed [SCR15] to enhance even further the accuracy of the overall scheme.…”

“…MOOD has been designed originally on fixed grids for Euler equations [CDL11a,DCL12,DLC13,CDL11b]. The a posteriori MOOD concept has also been used as a limiter for ADER schemes [LDD14], as a subcell limiter for high accurate Discontinuous Galerkin schemes in [ZDLS14,ZFDH15a] or as a high-order finite volume solver for convection-diffusion problems [CMNP13,CM14] and also to construct all-entropy finite volume schemes [VC13, V.D13]. Most classical limiting procedures (slope limiter, artificial viscosity, WENO, CWENO, etc.)…”

“…All of the above-mentioned techniques, more or less, answer these two questions, sometimes independently. In this work, we rely on the a posteriori MOOD paradigm which was developed initially in [CDL11a,DCL12,DLC13] and further extended to different contexts for instance in [LDD14,ZDLS14,VC13,ZFDH15a,ZFDH15b,NRCL16]. This a posteriori procedure checks at the end of the timestep for troubled cell, and further recomputes them with a more dissipative scheme.…”

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

“…Another point is the capacity of the method to directly take into account the physics of the problem (positivity, entropy production, etc.). The MOOD paradigm has been developed for non-stationary hyperbolic system [51,29,14] and we aim at demonstrating that the methodology is adapted to steady-state situations. The key point is the introduction of an additional unknown vector which represents the maximum admissible polynomial degree on each cell.…”