Runge-Kutta methods for the incompressible Navier-Stokes equations

Time integration of the incompressible Navier-Stokes equations with Runge-Kutta methods is not straightforward due to the differential-algebraic nature of the equations. In this work we investigate the temporal order of accuracy of velocity and pressure for both explicit and implicit methods. This is done by applying existing theory on Runge-Kutta methods for differential-algebraic equations to the incompressible Navier-Stokes equations. We focus on a specific class of Runge-Kutta methods, namely symplectic Runge-Kutta methods, which in the case of the incompressible Navier-Stokes equations are energy-conserving.