arXiv:2604.21906v2 Announce Type: replace Abstract: We present a structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the compatible discretization of first-order systems of time-dependent partial differential equations (PDEs). The method preserves divergence-free and curl-free vector fields exactly thanks to the compatible vertex-staggered discretization of the state variables on unstructured grids consisting of primal Delaunay triangles and their dual polygons. For the weakly compressible Euler equations, the scheme is asymptotic preserving, yielding a consistent discretization of the incompressible limit as the Mach number goes to zero. The scheme applies to a broad spectrum of PDEs, 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 star polygons. Scalar quantities (pressure, density, viscous stress) are defined at nodes, with pressure updated implicitly in a continuous finite element fashion, yielding a symmetric and positive definite pressure system. Instead, vector fields (velocity, momentum, magnetic and distortion fields) are stored at triangle barycenters and evolved explicitly using a compatible finite volume scheme. With the semi-implicit discretization, the CFL restriction is independent of the sound speed, allowing simulations at low Mach numbers. The compatible formulation ensures an exactly divergence-free velocity field in the incompressible limit, an exactly divergence-free magnetic field for MHD, and an exactly curl-free inverse deformation gradient in solid mechanics. The scheme has been extended to second-order accuracy. The method is validated through a set of test cases.