Limitations of Richardson Extrapolation and Possible Remedies for Estimation of Discretization Error

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady convection diffusion equation with variable coefficients and possible scenarios are shown that exhibit the oscillatory convergence. Also a manufactured solution to difference equations is formulated that exhibit desired oscillatory behavior in gird convergence with varying formal order of accuracy. This model-error equation is used to assess statistically the performance of several methods of extrapolation. Alternative extrapolation schemes such as the deferred extrapolation to limit technique to calculate the coefficients in the Taylor series expansion of the error function are also considered. A new method is proposed that is based on the extrapolation of approximate error and shown to be a viable alternative to the other methods. This paper elucidates the problem of oscillatory convergence and brings a new look into the problem of estimating discretization error by optimizing the information from a minimum number of calculations.