Extension of the Velocity-Correction Scheme to General Coordinate Systems

Abstract

The velocity-correction scheme is a time-integration method for the incompressible Navier-Stokes equations, and is a common choice in the context of spectral/hp methods. Although the spectral/hp discretization allows the representation of complex geometries, in some cases the use of a coordinate transformation is desirable, since it may lead to symmetries which allow a more efficient solution of the equations. One example of this occurs when the transformed geometry has a homogeneous direction, in which case a Fourier expansion can be applied in this direction, reducing the computational cost. In this paper, we revisit two recently proposed forms of extending the velocity-correction scheme to general coordinate systems, the first treating the mapping terms explicitly and the second treating them semi-implicitly. We then present some numerical examples illustrating the properties and applicability of these methods, including new tests focusing on the time-accuracy of these schemes.