Explicit finite difference methods for the EW and RLW equations

An extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal-width (EW) and regularized long-wave (RLW) equations is reported. Such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing. Two of the methods presented here make use of three-point, fourth-order accurate, finite difference formulae for the first- and second-order spatial derivatives. Two methods are based on the analytical solution of second-order ordinary differential equations which have locally exponential solutions, and the fourth technique is a standard finite difference scheme. A linear stability analysis of the four methods is presented. It is shown that, for the EW and RLW equations, a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations. The latter are reported to be more accurate than exponential techniques that employ second-order accurate approximations, and, these, in turn, are more accurate than the standard explicit method.