A Novel Method to Deduce a High-Order Compact Difference Scheme for the Three-Dimensional Semilinear Convection-Diffusion Equation with Variable Coefficients

Abstract

In this article, a new family of fourth-order compact difference schemes for the three-dimensional semilinear convection-diffusion equation with variable coefficients is presented. Like the finite-volume method, a dual partition is introduced. Combining with the Simpson integral formula and parabolic interpolation, fourth-order schemes are derived based on two different types of dual partitions. Moreover, a sixth-order finite-difference discretization strategy is developed, which is based on the fourth-order compact discretization and Richardson extrapolation technique. This extrapolation technique can achieve a sixth-order-accurate solution on fine grids directly, without the need for interpolation. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order schemes and extrapolation formulas.

Acknowledgments

The authors are very thankful to the editor and referee who meticulously read through the article, and made many helpful comments and corrections of the English and typesetting mistakes. This work is in part supported by the National Science Foundation of China (nos. 61163027 and 11271313), the China Postdoctoral Science Foundation (nos. 201104702 and 2012M512056), the Key Project of Chinese Ministry of Education (no. 212197), the Doctoral Foundation of Xinjiang University (no. BS110101), and the Excellent Doctor Innovation Program of Xinjiang University (no. XJUBSCX-2012003).