A COMPARATIVE-STUDY OF FINITE-DIFFERENCE METHODS FOR SOLVING THE ONE-DIMENSIONAL TRANSPORT-EQUATION WITH AN INITIAL BOUNDARY-VALUE DISCONTINUITY

Six explicit and six implicit finite difference methods are used to solve the transport (convective-diffusion) equation where the intersection of the boundary and initial conditions is discontinuous. Each of the methods are classified according to their theoretical order of accuracy for continuous initial-boundary value data. The transport equation with the given conditions has an exact solution, so the root mean squared error can be obtained for each method with changes in time and space. It is found that the theoretical order of accuracy has little effect on the results and that unless the diffusion number r = Δt/(Δx)2 is very small then the second and third order implicit methods are more accurate. It is also found that the implicit methods have an advantage in terms of lower CPU times over the explicit methods if a desired level of accuracy is required because they are able to use larger time steps. The third order implicit methods are found to reduce to second order as a result of the discontinuity in the initial-boundary conditions. However, this reduction in accuracy can be overcome by the use of a fine grid adjacent to the discontinuity. © 1990.