On the Convergence of Higher Order Upwind Differencing Schemes for Tridiagonal Iterative Solution of the Advection-Diffusion Equation

Abstract

In order to reduce numerical diffusion associated with firstorder upwind differencing schemes UDS, higher order upwind differencing schemes are often employed in computational fluid dynamics CFD calculations. The two most popular higher order UDS are the second-order UDS 1 and the quadratic upwind interpolation for convective kinematics QUICK scheme 2. It is well known that the use of either of the aforementioned two schemes results in a system of equations that may not converge when using an iterative solution method 3. Even for the one-dimensional advection-diffusion equation, since the stencil extends beyond three nodes, a single tridiagonal matrix inversion is not sufficient, and iterations are necessary to solve the resulting system of algebraic equations. Thus, one-dimensional calculations are sufficient for extracting meaningful information pertaining to the stability of these schemes in the context of iterative solution. In this Technical Brief, the convergence characteristics of tridiagonal iterative solution of the equations resulting from use of the second-order UDS and the QUICK scheme are investigated using discrete Fourier analysis. It is found that the chosen iterative method diverges at intermediate Peclet numbers for both schemes. Since these findings are not intuitive, prior to drawing firm conclusions, numerical experiments were performed to verify these findings. The numerical experiments show that the stability and convergence characteristics of both schemes agree perfectly with the results predicted by the discrete Fourier analysis. It is also shown that introduction of a small amount of diagonal dominance through the use of so-called inertial damping factors makes both schemes unconditionally stable. Once again, the convergence characteristics of the modified equations are examined by both methods, and the results are in perfect agreement with each other, lending credibility to the final results and conclusions.