A New Explicit Formulation by the Finite Element Method and the Finite Difference Method for the Two-Dimensional Convection-Diffusion Equation. Approach by the Error Analysis Techique.

Abstract

In our previous papers, the Modified Galerkin Method (MGM) was proposed for the one-dimensional and the two-dimensional convection-diffusion equations as one of the most efficient methods using the error analysis technique. In this paper, this approach was applied for two explicit finite element methods and eight explicit finite difference methods. The corresponding correction coefficients were obtained for these methods in order to improve their numerical accuracies. All of these newly formulated schemes were evaluated from the standpoints of phase error and dissipation error. From the error analysis results, we confirmed the higher accuracy of our proposed finite difference method for a convection dominated problem and our proposed finite element method for a diffusion dominated problem. Through numerical experiments using the explicit finite element schemes, we confirmed the effectiveness of our proposed method and the general correspondence between the error analysis and the numerical simulation.