A novel fully compatible and asymptotic preserving semi-implicit scheme on staggered unstructured meshes

We present a novel structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the discretization of first order systems of time-dependent partial differential equations (PDEs). The method is fully compatible and asymptotic preserving due to the staggered discretization of the state variables on unstructured grids that are constituted by primal Delaunay triangles and their dual polygons.

Our method applies to a broad spectrum of governing equations, including the weakly compressible and incompressible Euler and Navier-Stokes equations, the incompressible magnetohydrodynamics (MHD) system, and the incompressible version of the first-order hyperbolic Godunov-Peshkov-Romenski (GPR) model for continuum mechanics.

The computational domain is covered by a primal triangular mesh and a dual tessellation made of so-called star polygons constructed by linking the barycenters of adjacent triangles and the midpoints of the shared edges.

The scalar pressure field, the density, and the viscous stress tensor are defined at the triangles nodes. In particular, the pressure equation is evolved implicitly in a finite element fashion, yielding a symmetric and positive definite system. Instead, the velocity, the momentum, the magnetic and distortion fields are stored at the triangles barycenters and evolved explicitly with a finite volume approach.

Thanks to this semi-implicit conservative treatment, the CFL condition depends only on convective terms, improving computational efficiency and enabling simulations at all Mach numbers.

The fully compatible nature of the method ensures exact divergence-free velocity and magnetic field, exact curl-free distortion field, as well as energy stability and correct asymptotic behavior in the incompressible limit.

Finally, we have performed an extensive validation through classical benchmark test cases, including the Taylor-Green vortex, MHD vortex and the solid rotor problem, which demonstrates the scheme’s accuracy, reliability, and advantages over standard numerical methods.