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A Very High-Order Finite Volume method on 2D unstructured meshes: Multi-dimensional Optimal Order Detection

We will present an original Very High-Order Finite Volume method for conservation laws on unstructured meshes called Multi-dimensional Optimal Order Detection (MOOD) and its extension to the Euler system for hydrodynamics. In order to deal with problems generated by high-order approximations, the proposed method consists of detecting problematic situations after each time update of the solution and of reducing the local polynomial degree before recomputing the solution.
As multi-dimensional MUSCL methods, the concept is simple and independent of mesh structure but the MOOD method is more easily able to take physical constraints such as density and pressure positivity into account through the ?a posteriori? detection. Moreover implementation of the method into a existing first-order code does not involve important difficulties since no modification of the solver is needed.
Numerical results on classical and demanding test cases for advection and Euler system will be presented on polygonal meshes to support the promising potential of this approach.
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