An Essentially Non-oscillatory Crank–Nicolson Procedure for the Simulation of Convection-Dominated Flows

Abstract

The Crank–Nicolson (CN) time-stepping procedure incorporating the second-order central spatial scheme is unconditionally stable and strictly non-dissipative for linear convection flows; however, its numerical solution in practice can be oscillatory for nonsmooth solutions. This article studies variants of the CN method for the simulation of linear convection-dominated diffusion flows, in which the explicit convection part is approximated by an upwind scheme, to effectively suppress nonphysical oscillations. The second-order essentially non-oscillatory scheme incorporated in the CN procedure (ENO-CN) has been found effective for a non-oscillatory numerical solution of minimum numerical dissipation. A stability analysis is provided for ENO-CN, which turns out to be unconditionally stable for problems of nonzero diffusion. However, for purely convective flows, it is stable only when the CFL condition is satisfied. Numerical results are presented to demonstrate its stability and accuracy.