Assessment of high-order IMEX methods for incompressible flow

TL;DR

This paper evaluates high-order IMEX methods, specifically semi-implicit Runge-Kutta and spectral deferred correction schemes, for incompressible flow simulations, highlighting their accuracy, efficiency, and suitability for complex boundary conditions.

Contribution

It introduces a partitioned approach combining RK and SDC methods for Navier-Stokes problems and identifies effective third-order RK schemes that outperform second-order methods.

Findings

Third-order RK methods outperform second-order schemes.

SDC methods are more accurate but only competitive at very low error levels.

Two third-order RK methods perform well across various test cases.

Abstract

This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two third-order RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca $10^{-5}$.