Multidimensional pseudo-spectral methods on lattice grids

作者：Munthe Kaas Hans; Sorevik Tor^*

来源：Applied Numerical Mathematics, 2012, 62(3): 155-165.

DOI：10.1016/j.apnum.2011.11.002

• Multidimensional Fourier expansion
• Lattice rules
• Pseudo-spectral methods

When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L-2-integrable, the contributions from many of the N-s terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.

• 出版日期2012-3