Some three-level finite difference methods for simulating advection in fluids

The one-dimensional advection equation forms the basis for modeling transient forced convection in fluids. A weighted differencing on a (1, 3, 2) computational stencil is used to produce a number of finite difference equations for finding approximate solutions to this equation supplemented by smooth initial-boundary conditions. Among these are several well-known two-level versions, as well as the leapfrog equation, the only one presently used which involves three levels in time. In addition, two recently developed and five new three-level equations are obtained. Their accuracy has been estimated theoretically by comparing their amplitude response and relative wave speed obtained in series form using the coefficients of their modified equivalent partial differential equations. These theoretical estimates are checked by numerical tests. The results show that several of the new three-level methods have a number of advantages when compared to those in common use, in particular having much greater accuracy when the tests involve smooth initial-boundary data. Superior accuracy is retained, but to a smaller degree, when the initial-boundary conditions are discontinuous.