# A coupled implicit-explicit time integration method for compressible unsteady flows

## TL;DR

This paper develops a stable coupled implicit-explicit time integration method for compressible unsteady flows, enabling efficient large eddy simulation and RANS computations in industrial applications.

## Contribution

It introduces a novel coupling procedure for explicit and implicit schemes that maintains stability and spectral properties suitable for industrial flow simulations.

## Key findings

- The alternative coupling procedure is stable and spectrally consistent.
- Validation with standard test cases confirms effectiveness.
- The method enables efficient LES/RANS coupled simulations.

## Abstract

This paper addresses how two time integration schemes, the Heun's scheme for explicit time integration and the second-order Crank-Nicolson scheme for implicit time integration, can be coupled spatially. This coupling is the prerequisite to perform a coupled Large Eddy Simulation / Reynolds Averaged Navier-Stokes computation in an industrial context, using the implicit time procedure for the boundary layer (RANS) and the explicit time integration procedure in the LES region. The coupling procedure is designed in order to switch from explicit to implicit time integrations as fast as possible, while maintaining stability. After introducing the different schemes, the paper presents the initial coupling procedure adapted from a published reference and shows that it can amplify some numerical waves. An alternative procedure, studied in a coupled time/space framework, is shown to be stable and with spectral properties in agreement with the requirements of industrial applications. The coupling technique is validated with standard test cases, ranging from one-dimensional to three-dimensional flows.