An Explicit Fourth-Order Compact Finite Difference Scheme
for Three Dimensional Convection-Diffusion Equation

Jun Zhang
Department of Computer Science and Engineering
University of Minnesota
Minneapolis, MN 55455, USA


We present an explicit fourth-order compact finite difference scheme for approximating the three dimensional convection-diffusion equation with variable coefficients. This $19$-point formula is defined on a uniform cubic grid. We compare advantages and implementation cost of the new scheme with the standard $7$-point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourth-order convergence rate of the scheme and to show that the Gauss-Seidel iterative method converges for large values of the convection coefficients. Some algebraic properties of the coefficient matrices arising from different discretization schemes are compared. We also comment on potential use of the fourth-order compact scheme with multi-level iterative methods.

Key words: Three dimensional convection-diffusion equation, fourth-order compact scheme, iterative methods.

This paper has been published in Communications in Numerical Methods in Engineering, Vol. 14, 263--280 (1998).