Hybrid finite-difference scheme for solving the dispersion equation

An efficient hybrid finite-difference scheme capable of solving the dispersion equation with general Peclet conditions is proposed. In other words, the scheme can simultaneously deal with pure advection, pure diffusion, and/or dispersion. The proposed scheme linearly combines the Crank-Nicholson second-order central difference scheme and the Crank-Nicholson Galerkin finite-element method with linear basis functions. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multidimensional problems without much difficulty. It is found that the proposed scheme produces the best results, in terms of numerical damping and oscillation, among several non-split-operator schemes. In addition, the accuracy of the proposed scheme is comparable with a well-known and accurate split-operator approach in which the Holly-Preissmann scheme is used to solve the pure advection process while the Crank-Nicholson second-order central difference scheme is applied to the pure diffusion process. Since the proposed scheme is a non-split-operator approach, it does not compute the two processes separately. Therefore, it is simpler and more efficient than the split-operator approach.