Fourier spectral methods for solving the Korteweg-de Vries equation

This work is concerned with studying spectral collocation or pseudospectral (PS) methods, (Fourier transform methods), combined with temporal discretization techniques to numerically compute solutions of partial differential equations (PDEs) of the form ut = F( u), where F(u) := Lu+Nfu . Here, L and N are linear differential operators and f( u) is a nonlinear function. In particular, numerical solutions of the Korteweg-de Vries (KdV) equation will be computed.;We use a Fourier spectral collocation method to discretize the space variable and a leap-frog (LF) or an Xth order Runge-Kutta (RK X ) scheme for time dependence. Our implementation employs the Fast Fourier Transform (FFT) algorithm. It costs only O (Nlog2N) provided that N is chosen in an appropriate manner, i.e. N is a power of 2. Numerical schemes such as Fourier leap-frog, Fourier based RK4, and Fourier Split-step for the KdV equation will be presented.