Spectral Methods in Nonlinear Optics Equations for Non-Uniform Grids Using an Accelerated NFFT Scheme

Abstract

In this work, we propose the use of non-homogeneous grids in 1D and 2D for the study of various nonlinear physical equations using spectral methods. As is well known, the use of spectral methods allow a faster resolution of the problem via the application of the ubiquitous Fast Fourier Transform (FFT) algorithm. We will center our investigation on the search of fast and accurate schemes to solve the spectral operators in the Fourier space. In particular, we will use the Conjugate Gradient (CG) iterative method, with a preconditioning matrix to accelerate the inversion process of the non-uniform Fast Fourier Transform (NFFT). As it will be shown, the results obtained are in good agreement with the expected values.

Keywords

• spectral methods
• nonlinear optics
• non-uniform fast Fourier transform
• Julia language
• conjugate gradient algorithm