Abstract: In solving one-dimensional convection-diffusion equations, present numerical methods are prone to suffer from stability and accuracy problems caused by numerical oscillation and pseudo-diffusion. In view of this, an idea of applying Numerical Manifold Method (NMM) based on independent covers (the approximation using polynomial series piecewise-defined) to the numerical solution is proposed. The solution formula of the one-dimensional convection-diffusion equation is derived based on the standard Galerkin method. The posterior error estimation method about the continuity of the first-order derivative of field variable in the narrow overlapping area between independent covers is used for the automatic solving by h-p hybrid self-adaptive analysis with mesh refinement and ascending series order. The results of the steady-state and unsteady-state analysis examples show that the numerical solution of the piecewise-defined series steadily approximates and finally well fits the exact solution. For the convection-dominated problem, the adaptive solution effectively avoids numerical oscillation. In addition, the error index of the residual by substituting the numerical result back to the differential equation is successfully attempted. If the differential equation is solved point by point, the method would the most stringent error judgment for the numerical solution so far.

Key words: convection-diffusion equation, numerical oscillation, self-adaptive analysis, Numerical Manifold Method (NMM), independent covers