Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Highlights: • High-order accurate temporal integration methods for the incompressible Navier–Stokes equations are discussed. • Comparison between primary variable, gauge, and auxiliary variable formulations. • Design of a single-step, spectral deferred pressure correction scheme with arbitrary formal order of accuracy. • Numerical tests demonstrating schemes of up to eighth-order accuracy in square and curved domains. • Discussion of order reduction owing to time-dependent boundary conditions and in gauge formulations. This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier–Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.