L∞-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

Abstract

a b s t r a c tThis work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.

$$L^\infty $$ -Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

“…This section is devoted to the design of a numerical scheme to approximate (2), using the vertex-based MUSCL finite volume methods introduced in [13] and used in [2] for a second-order accuracy in space, and an implicit-explicit (IMEX) linear multistep methods [10] for a second-order in time.…”

“…Here, we describe the finite volume scheme for solving (2). In order to obtain the density reconstruction on the interfaces Γ (T ) i j l , we use the MUSCL technique with a multislope gradient reconstruction.…”

“…In [2], G i j l is the upstream flux, but many other numerical fluxes can be considered, as for instance Lax-Friedrichs or Engquist-Osher fluxes. We underline that for multi-physics coupled models, a particular attention must be paid in the approximation of the continuous velocity associated to any point of ∂ C i (see [3]).…”

“…where Lim is a so-called "τ-limiter" (for details see [2]). In particular, they have the following result:…”

“…This work is based on a implicit-explicit (IMEX) second order method and it is issued from the seminal paper [2]. In the framework of MUSCL methods, we will prove that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition and the classical hypothesis for the triangulation of the domain.…”