Mathematical analysis of the spatial coupling of an explicit temporal adaptive integration scheme with an implicit time integration scheme

Abstract

AbstractThe Reynolds‐averaged Navier–Stokes equations and the large eddy simulation equations can be coupled using a transition function to switch from a set of equations applied in some areas of a domain to the other set in the other part of the domain. Following this idea, different time integration schemes can be coupled. In this context, we developed a hybrid time integration scheme that spatially couples the explicit scheme of Heun and Crank–Nicolson's implicit scheme using a dedicated transition function. This scheme is linearly stable and second‐order accurate. In this article, an extension of this hybrid scheme is introduced to deal with a temporal adaptive procedure. The idea is to treat the time integration procedure with unstructured grids as it is performed with Cartesian grids and local mesh refinement. Depending on its characteristic size, each mesh cell is assigned to a rank. And for two cells from two consecutive ranks, the ratio of the associated time steps for time marching the solutions is 2. As a consequence, the cells with the lowest rank iterate more than the other ones to reach the same physical time. In a finite‐volume context, a key ingredient is to keep the conservation property for the interfaces that separate two cells of different ranks. After introducing the different schemes, the article recalls briefly the coupling procedure, and details the extension to the temporal adaptive procedure. The new time integrator is validated with the propagation of 1D wave packet, the Sod's tube, and the advection of 2D vortex.