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A hybrid Fourier-Chebyshev pseudospectral method for nonperiodic time-dependent problems
Rodrigo Platte
ASU
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A hybrid Fourier-Chebyshev pseudospectral method for nonperiodic time-dependent problems Rodrigo Platte ASU |
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ABSTRACT Fourier pseudospectral methods produce outstanding results for smooth problems with periodic boundary conditions. For nonperiodic problems, Chebyshev pseudospectral methods are commonly used to obtain exponential rates of convergence. However, while Fourier methods require 2 points per wavelength, Chebyshev methods require pi. Moreover, the clustering of nodes near the boundaries required by polynomial methods also impose an O(1/N^2) restriction on the time-step size when these methods are used for hyperbolic problems. In order to circumvent these difficulties in the solution of nonperiodic problems, we propose a hybrid Fourier Chebyshev pseudospectral method. The method requires a window function w, where w and its derivatives are approximately zero at the boundary. The target function $u$ is then replaced by the product uw which is periodic and can be accurately approximated by a trigonometric series. A polynomial approximation is needed only in the region where w is close to zero (near the boundaries), since u cannot be accurately recovered from uw if w is small. Numerical results confirm the spectral accuracy and stability of the method. |