Third-order-accurate semi-implicit Runge-Kutta scheme for incompressible Navier-Stokes equations

A semi-implicit three-step Runge-Kutta scheme for the unsteady incompressible Navier-Stokes equations with third-order accuracy in time is presented. The higher order of accuracy as compared to the existing semi-implicit Runge-Kutta schemes is achieved due to one additional inversion of the implicit operator I - tau gamma L, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure-Poisson equation or evaluation of Navier-Stokes operator is needed. The scheme is Supplied with a local error estimation and time-step control algorithm. The temporal third-order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright (c) 2005 John Wiley & Sons, Ltd.