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

TLDR: In this paper, a new family of fourth-order compact difference schemes for the three-dimensional semilinear convection-diffusion equation with variable coefficients is presented, combining with the Simpson integral formula and parabolic interpolation, four-order schemes are derived based on two different types of dual partitions.

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.