Some explicit three-level finite-difference simulations of advection

Finding an accurate solution to the one-dimensional constant-coefficient advection equation is basic to the problem of modelling, for example, the time-dependent spread of contaminants in fluids. A number of explicit finite-difference methods involving two levels in time have been used for this purpose in the past. Unfortunately, the more accurate ones have very wide computational stencils, making it difficult to obtain approximations near the upstream boundary and at the outflow boundary. For explicit methods, more compact stencils are possible only if they involve three levels in time. Until recently, the second-order leapfrog method has been the only available three-level technique. Using a weighted discretisation on a three-level computational stencil, explicit finite-difference equations of third and fourth order have now been developed. Both methods are more accurate at simulating advection in the absence of shocks, than the explicit two-level methods of corresponding order.