Review on A Third-order Optimized Numerical Scheme with Positive Difference Coefficients for Advection-diffusion Equations.

This article overviews our studies aiming at absolutely stable numerical schemes for any transporting velocities and any gradient of transported quantities in advection-diffusion equations. According to the Godunov theory, there exists only the first-order polynomial scheme with positive difference coefficients in numerical calculations of advection equations. We show that a third-order polynomial scheme with positive difference coefficients exists in case of advection-diffusion equations. We construct a stable polynomial scheme with third-order accuracy under an allowance condition among the Courant numbers and the diffusion numbers for advection-diffusion equations. We extend the present method into two-dimensional and three-dimensional equations. Numerical experiments show numerical solutions of good quality with the present scheme. [ABSTRACT FROM AUTHOR]