Reducing Round-Off Error in Chebyshev Pseudospectral Computations
Ernest E. Rothman · 1991
Spectral methods have become indispensable tools for scientists involved in solving partial differential equations, because they can achieve infinite-order accuracy for problems with smooth data. In order to realize this accuracy in real large-scale applications, one must be able to compute with pseudo-spectral operators without excessive damaging effects of round-off error. There are essentially three distinct approaches to computing derivatives with Chebyshev pseudospectral operators: One approach is to explicitly construct the matrix representation of the pseudospectral derivative operator and perform a matrix-vector multiplication. For small sizes of the matrix (i.e., order of the approximation) this straightforward approach can give good results and may even yield performance superior to that of the other approaches. Current codes, however, can require up to 500 Chebyshev modes. This number is likely to rise as the solutions of more complex problems are attempted. The other two approaches involve performing transformations between physical space and Chebyshev space. One transform approach is to apply an FFT and the other is to apply the Clenshaw recursion algorithm. In this paper, all three approaches are compared in terms of accuracy and are found to have different properties with respect to round-off error. In fact, the second-derivative approximation via matrix-vector multiplication may be accomplished in several ways. All these approaches are found to have different properties with respect to round-off error. However, in each case, the error is worst neat the boundaries. A preconditioning technique that attempts to damp the (worst) errors near the boundaries, first suggested by Breuer and Everson (1990) to be applied to the FFT approach, is extended for application to the other approaches and is found to substantially improve accuracy.