A Monotonicity Preserving Scheme I : A Finite Variable Difference Method Based on A Locally Exact Solution of Steady Advection-Diffusion Equation

Abstract

We discuss a numerical scheme with monotonicity preserving properties without additionally introducing artificial diffusions for advection-diffusion equations. This paper proposes "Finite Variable Difference Method (FVDM)", in which the convection term is discretized by using locally optimized numerical fluxes so that the resulting difference equation may satisfy a locally exact solution of steady advection-diffusion equations. The present scheme ensures the monotnicity up to the cell Reynolds number Rm ≒ 3.4 in keeping the second-order accuracy, while the conventional central scheme and the QUICK scheme up to Rm = 2 and Rm = 8/3, respectively. For Rm > 3.4, though the present scheme has the first-order accuracy, the lowest order of its truncation error can be finite but arbitrary small. Numerical experiments show solutions with good quality.