Extension of the Velocity-Correction Scheme to General Coordinate Systems
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.
D.S. and J.R.M. are grateful for the support received from CNPq (grants 231787/2013-8 and 312755/2014-7) and FAPESP (grants 2012/23493-0 and 2014/50279-4). S.J.S. would like to acknowledge support under the Royal Academy of Engineering Research Chair Scheme and support under EPSRC grant EP/K037536/1.
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