An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

This paper presents a novel semi-implicit hybrid finite volume / finite element (FV/FE) scheme for the numerical solution of the incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using a direct Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a suitable splitting of the governing partial differential equations into subsystems and employs a staggered grid arrangement, where the pressure is defined on the primal simplex mesh, while the velocity and the remaining flow quantities are defined on an edge-based staggered dual mesh. The key idea of the scheme presented in this paper is to discretize the nonlinear convective and viscous terms at the aid of an explicit finite volume scheme that employs the space-time divergence form of the governing equations on moving space-time control volumes. For the convective terms, an ALE extension of the Ducros flux on moving meshes is introduced, which can be proven to be kinetic energy preserving and stable in the energy norm when adding suitable numerical dissipation terms. The use of closed space-time control volumes inside the finite volume scheme guarantees that the important geometric conservation law (GCL) of Lagrangian schemes is verified by construction. Finally, the pressure equation of the Navier-Stokes system is solved on the new mesh configuration at the aid of a classical continuous finite element method, using traditional P1 Lagrange elements.