High order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes

Abstract

In this work, a family of high order accurate Central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear kinetic equation of relaxation type is presented. After discretization of the velocity space by using a discrete ordinate approach, the space reconstruction is realized by integration over conformal arbitrary shaped closed space control volumes in a CWENO fashion. Compared to other WENO methods on unstructured meshes, in the method here presented, the total stencil size is the minimum possible and the linear weights can be arbitrarily chosen. These two aspects make their use for kinetic equations and the practical implementation on general unstructured meshes particularly interesting. The full discretization is then obtained by combining the previous phase-space approximation with an Implicit-Explicit Runge Kutta high order time discretization which guarantees stability, accuracy and preservation of the asymptotic state. In particular, to guarantee in the finite volume framework space accuracy higher than two, a new class of IMEX methods has been set into place and its properties have been studied. The formal order of accuracy is numerically measured for different regimes, computational performances of the proposed class of methods are tested on several standard two dimensional benchmark problems for kinetic equations. The novel methods are finally applied to a prototype engineering problem consisting in a supersonic flow around a NACA 0012 airfoil. In our computations we employ up to ≈325 millions of degrees of freedom and 256 GB of RAM run on 128 cores with Fortran-MPI providing evidence that the above schemes are suitable for implementation on parallel distributed memory supercomputers.