On the Representation of Numerical Solutions Using Taylor Series Approximation

Abstract

In this study, the representation of discretization error using Taylor series in finite difference solutions is investigated as well as the behavior of the exact solutions to the finite difference equations as a function of the grid size and grid refinement factor. The results are compared to the classical Richardson Extrapolation method whereby the numerical solution (or the error) is explicitly expressed as a Taylor series expansion. The exact finite difference solutions are used to demonstrate that oscillatory convergence is a common occurrence. The expansion of the numerical solutions in Taylor series is based on the exact finite difference solutions that are obtained using different discretization schemes. It is shown that in some cases the numerical solution exhibited a singular behavior which can not be remedied easily. Some exact finite difference solutions also exhibited oscillatory behavior which was not due to the use of mixed order terms as is usually believed by the Computational Fluid Dynamics community. Moreover, representation of the numerical solution using Taylor series is not always satisfactory even in case of relatively simple one-dimensional problems.