A new formulation by the finite element method and finite difference method for the one-dimensional convection-diffusion equation. Approach by the error analysis technique.

Abstract

The unstationary convection-diffusion equation is one of the most important equations in heat and fluid analysis. The new effective numerical nethod was proposed for this equation by an error analysis technique based on the Fourier series. First, the conventional numerical methods of Finite-element method (FEM) and Finite-difference method (FDM), were revised by an error analysis technique combined with an analytical solution. Then, all of these numerical methods were evaluated from the standpoints of phase error and dissipation error. Twelve conventional methods were investigated (3 by FEM and 9 by FDM) and five methods were newly developed (1 by FEM and 4 by FDM). There were eight implicit methods and nine explicit methods. The moditied Galerkin method was proven to be the most efficient numerical method among these 17 methods. We confirmed this result through several numerical experiments.