An all Mach number semi-implicit hybrid Finite Volume/Virtual Element method for compressible viscous flows on Voronoi meshes

TL;DR

This paper introduces a high-order semi-implicit hybrid finite volume/virtual element method for compressible viscous flows on Voronoi meshes, enabling efficient simulations across all Mach numbers with energy conservation.

Contribution

It presents a novel flux splitting scheme combining FV and VEM, allowing larger time steps and high accuracy on polygonal grids for compressible flows.

Findings

Method is energy conserving and captures shocks effectively.

Suitable for low Mach and high Mach flows, validated on classical benchmarks.

Allows high order accuracy in space and time with complex geometries.

Abstract

We present a novel high order semi-implicit hybrid finite volume/virtual element numerical scheme for the solution of compressible flows on Voronoi tessellations. The method relies on the flux splitting of the compressible Navier-Stokes equations into three sub-systems: a convective sub-system solved explicitly using a finite volume (FV) scheme, and the viscous and pressure sub-systems which are discretized implicitly at the aid of a virtual element method (VEM). Consequently, the time step restriction of the overall algorithm depends only on the mean flow velocity and not on the fast pressure waves nor on the viscous eigenvalues. As such, the proposed methodology is well suited for the solution of low Mach number flows at all Reynolds numbers. Moreover, the scheme is proven to be globally energy conserving so that shock capturing properties are retrieved in high Mach number flows. To reach high order of accuracy in time and space, an IMEX Runge-Kutta time stepping strategy is employed together with high order spatial reconstructions in terms of CWENO polynomials and virtual element space basis functions. The chosen discretization techniques allow the use of general polygonal grids, a useful tool when dealing with complex domain configurations. The new scheme is carefully validated in both the incompressible limit and the high Mach number regime through a large set of classical benchmarks for fluid dynamics, assessing robustness and accuracy.