We investigate the coupling of the Multi-dimensional Optimal Order Detection (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 [1, 2, 3] 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 decrementing 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 [4, 5]. 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.
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 partial differential equation.
If time permits we will present the extension of the a posteriori treatment to construct a subcell limiter for Discontinuous Galerkin methods of high accuracy (polynomial degree 9).
References:
[1] S. Clain, S. Diot, and R. Loubère, A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028 – 4050, 2011.
[2] S. Diot, S. Clain, and R. Loubère, Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids, 64:43 – 63, 2012.
[3] S. Diot, R. Loubère, and S. Clain, The MOOD method in the three-dimensional case: Very-high-order finite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids, 73:362–392, 2013.
[4] M. Dumbser, Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier- Stokes equations. Computers and Fluids, 39:60–76, 2010.
[5] M. Dumbser, M. Castro, C. Parés, and E.F. Toro, ADER schemes on unstructured meshes for non-conservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731 – 1748, 2009.