Interaction of time stepping and convection schemes for unsteady flow simulation

For the time dependent flow simulation, despite the existence of extensive literature dealing with either the convection or the unsteady terms, their interaction has not been adequately investigated. In order to shed more light on this important issue, three time stepping methods, including the first-order backward Euler scheme, the second-order Crank-Nicolson scheme, and the third-order Adams-Bashforth/Adams-Moulton predictor-corrector scheme are studied along with four convection schemes, including the first-order upwind, the second-order upwind, the second-order central differencing, and QUICK schemes. The Burgers equation of both linear and nonlinear forms is used as the test problem, aided by the von Neumann stability analysis and the FFT spectral analysis. The results indicate that a second-or higher-order accuracy for both time and space discretizations can produce satisfactory results for smooth solution profiles. Overall, among the schemes tested, either a combination of first-order upwind for convection and Crank-Nicolson for time, or a combination of second-order upwind for convection and backward Euler for time performs better. It appears that by selectively utilizing the dispersive and diffusive characteristics of the time stepping and convection schemes in complementary manners, overall accuracy can be improved.