An arbitrary Lagrangian-Eulerian semi-implicit hybrid method for continuum mechanics with GLM cleaning

Abstract

This paper proposes a semi-implicit arbitrary Lagrangian-Eulerian (ALE) method for the solution of the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics. To handle the curl free involutions arising in the solid limit of the model, the original system is augmented by adopting a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach. Next, an operator splitting strategy decouples the computation of fast pressure waves from the bulk velocity of the medium yielding a transport subsystem, containing convective terms and non-conservative products, and a Poisson-type subsystem, for the pressure. A second splitting yields an ODE subsystem comprising only the potentially stiff source terms, responsible for the relaxation of the model between its fluid and solid limits. The mesh motion can be driven by two sources: the local fluid velocity and a prescribed boundary displacement. For the spatial discretization, we employ unstructured staggered grids, with the pressure defined on the primal mesh and all remaining variables on the dual grid. The transport subsystem is advanced via an explicit finite volume method, in which integration over closed space-time control volumes ensures verification of the geometric conservation law (GCL). On the other hand, implicit continuous finite elements are used for the discretization of the pressure subsystem and an implicit DIRK scheme is employed to solve the ODE subsystem. Consequently, the proposed approach is well suited to address all Mach number flows. A comprehensive set of benchmarks is employed to assess the accuracy and robustness of the proposed methodology in tackling both fluid and solid mechanics problems.