Limitations of Richardson extrapolation and some possible remedies

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady one-dimensional 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 exhibits desired oscillatory behavior in grid convergence, with a varying formal order of accuracy. This model-error equation is used to statistically assess 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 is shown to be a viable alternative to other methods. This paper elucidates the problem of oscillatory convergence, and brings a new perspective to the problem of estimating discretization error by optimizing the information from a minimum number of calculations.