A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws (Communications in Computational Physics, 2014)
In this paper, we investigate the coupling of the Multi-dimensional OptimalOrder Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)approach in order to design a new high order accurate, robust and computationallyefficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two andthree space dimensions, respectively. The Multi-dimensional Optimal Order Detection(MOOD) method for 2D and 3D geometries has been introduced in a recent series ofpapers for mixed unstructured meshes. It is an arbitrary high-order accurate FiniteVolume scheme in space, using polynomial reconstructions with a posteriori detectionand polynomial degree decrementing processes to deal with shock waves and otherdiscontinuities. In the following work, the time discretization is performed with anelegant 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 reachedon smooth solutions, while spurious oscillations near singularities are prevented. TheADER technique not only reduces the cost of the overall scheme as shownon a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENOschemes and the new ADER-MOOD approach has been carried out for high-orderschemes in space and time in terms of cost, robustness, accuracy and efficiency. Themain finding of this paper is that the combination of ADER with MOOD generallyoutperforms 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 givengrid resolution. A large suite of classical numerical test problems has been solvedon unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equationsof ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolicpartial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.