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

## TL;DR

This paper introduces a spatially coupled hybrid time integration scheme combining explicit and implicit methods, extended to adaptive time stepping on unstructured grids, ensuring stability, accuracy, and conservation.

## Contribution

The paper develops an extended hybrid scheme that adaptively couples explicit and implicit time integration methods on unstructured grids with local mesh refinement.

## Key findings

- Linearly stable and second-order accurate scheme.
- Validated with wave packet, Sod's tube, and vortex transport tests.
- Maintains conservation across interfaces of different cell ranks.

## Abstract

The 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 the implicit scheme of Crank and Nicolson using a dedicated transition function. This scheme is linearly stable and second-order accurate. In this paper, 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 with local mesh refinement. Depending on its characteristic size, each mesh cell is assigned 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 paper recalls briefly the coupling procedure, and details the extension to the temporal adaptive procedure. The new time integration scheme is validated with the propagation of 1D wave packet, the Sod's tube, and the transport of a bi-dimensional vortex in an uniform flow.