scholarly journals Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

2012 ◽  
Vol 64 ◽  
pp. 43-63 ◽  
Author(s):  
S. Diot ◽  
S. Clain ◽  
R. Loubère
Keyword(s):  
Unstructured Meshes ◽  
High Order ◽  
Optimal Order ◽  
Very High

A simple construction of very high order non-oscillatory compact schemes on unstructured meshes

2009 ◽  
Vol 38 (7) ◽  
pp. 1314-1323 ◽  
Author(s):  
R. Abgrall ◽  
A. Larat ◽  
M. Ricchiuto ◽  
C. Tavé
Keyword(s):  
Unstructured Meshes ◽  
High Order ◽  
Compact Schemes ◽  
Simple Construction ◽  
Very High

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

2014 ◽  
Vol 16 (3) ◽  
pp. 718-763 ◽  
Author(s):  
Raphaël Loubère ◽  
Michael Dumbser ◽  
Steven Diot
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Hyperbolic Conservation Laws ◽  
Unstructured Meshes ◽  
High Order ◽  
Mhd Equations ◽  
Optimal Order ◽  
Finite Volume Scheme ◽  
Hyperbolic Conservation ◽  
2D And 3D

AbstractIn this paper, 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 for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions witha posterioridetection 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. 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 partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.

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