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The Reynolds-Averaged Navier-Stokes equations and the Large-Eddy Simulationequations can be coupled using a transition function to switch from a set ofequations applied in some areas of a domain to the other set in the other partof the domain. Following this idea, different time integration schemes can becoupled. In this context, we developed a hybrid time integration scheme thatspatially couples the explicit scheme of Heun and the implicit scheme of Crankand Nicolson using a dedicated transition function. This scheme is linearlystable and second-order accurate. In this paper, an extension of this hybridscheme is introduced to deal with a temporal adaptive procedure. The idea is totreat the time integration procedure with unstructured grids as it is performedwith Cartesian grids with local mesh refinement. Depending on itscharacteristic size, each mesh cell is assigned a rank. And for two cells fromtwo consecutive ranks, the ratio of the associated time steps for time marchingthe solutions is $2$. As a consequence, the cells with the lowest rank iteratemore than the other ones to reach the same physical time. In a finite-volumecontext, a key ingredient is to keep the conservation property for theinterfaces that separate two cells of different ranks. After introducing thedifferent schemes, the paper recalls briefly the coupling procedure, anddetails the extension to the temporal adaptive procedure. The new timeintegration scheme is validated with the propagation of 1D wave packet, theSod's tube, and the transport of a bi-dimensional vortex in an uniform flow.

Spatial coupling of an explicit temporal adaptive integration scheme with an implicit time integration scheme