×

Two kinds of modified difference schemes for convective diffusion equations without source term. (Chinese. English summary) Zbl 1463.65345

Summary: In this paper, equidistance difference schemes for the convection diffusion equation without source term are studied. The difference schemes are designed on the three-point template. After expanding the function values at both nodes about the center point by Taylor’s expansion, two Taylor expansions are obtained. While the original differential equation is used repeatedly, the higher derivative terms in two Taylor expansions are transformed into expansions containing only the first-order derivative term by means of the idea of “reduced order”. Then the first-order derivative can be eliminated by combining the two Taylor expansions and a formally accurate difference scheme can be obtained. Since the coefficients of the difference scheme are composed of infinite series, how to preserve finite terms to make the difference scheme suitable for problems with large or small parameters is the focus of this paper. We design two kinds of difference schemes in different situations: when the parameter is large, the power of \(h\) has a greater impact on the difference scheme coefficient, so we design a kind of “horizontal series modified difference schemes (HDS)”, whose accuracy can reach the second order, the fourth order, the sixth order, the eighth order respectively. However, when the parameter \(\varepsilon\) is very small, the power of \(1/\varepsilon\) has a greater impact on the difference scheme coefficient than the step size, therefore we design a kind of “vertical series modified difference schemes (VDS)”. One numerical example is selected to carry on the experiment, and the numerical comparisons are made among the HDS, VDS and the seven difference schemes given in the references. Results show that the HDS designed in this paper are suitable for the case where \(\varepsilon\) is larger, and the VDS are suitable for the case where \(\varepsilon\) is very small. Finally, it is also shown that the accuracy of our method is better than that of the difference schemes in references.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
PDFBibTeX XMLCite