A hybrid Fourier-Chebyshev method for partial differential equations

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability.