Fractional Time-Stepping Methods for Unsteady Flow Problems

We study accuracy of fractional time-stepping (FTS) methods such as the alternating direction implicit method for the computation of heat ows and the velocity-pressure splitting methods for Navier-Stokes ows. The FTS techniques can be viewed as a perturbation of conventional temporal-spatial discretization schemes; they in theory invoke a perturbation error that is on the same order as the discretization error. However, it has been observed in simulation that the perturbation error can be much (one-order) larger than the discretization error unless the timestep size is suÆciently small, which often degrades bene ts of the FTS method. The article suggests variants of FTS methods in which the splitting error is on one-order higher than the discretization error and therefore negligible for moderate timestep sizes. The new FTS methods require a small fraction of additional computation cost over the original ones. Numerical veri cations and applications for the numerical solution of the heat equation and unsteady Navier-Stokes equations are presented. Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027 USA Email: [email protected] Department of Mathematics, Purdue University, W. Lafayette, IN 47097 USA Email: [email protected]