The MOOD method in the three-dimensional case: Very-High-Order Finite Volume Method for Hyperbolic Systems.

Abstract

The Multi-dimensional Optimal Order Detection (MOOD) method for two-dimensional geometries has been introduced in A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD), J. Comput. Phys. 230 (2011), and enhanced in Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. & Fluids 64 (2012). We present in this paper the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high-order of accuracy is reached on smooth solutions while spurious oscillations near singularities are prevented. At last, the intrinsic positivity-preserving property of the MOOD method is confirmed in 3D and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared to existing high-order Finite Volume methods.