A method for relaxing the Courant-Friedrich-Levy condition in time-explicit schemes

Abstract

We present a method for relaxing the Courant-Friedrich-Levy (CFL) condition, which limits the time step size in explicit numerical methods in computational fluid dynamics. The method is based on re-formulating explicit methods in matrix form. Here an explicit method appears as a special case in which the global matrix of coefficients is replaced by the most simple matrix in algebra: the identity matrix I. This procedure is stable under severe limiting conditions only. Using matrix formulation, one can design various solution methods in arbitrary dimensions that range from explicit to unconditionally stable implicit methods in which the CFL-number may reach arbitrary large values. In addition, we find that adopting a specially varying-time-stepping scheme accelerates convergence toward steady state solutions and improves the efficiency of the solution procedure.