The fundamentals of Fourier spectrum methods are covered in this book, along with how to use them in code to solve partial differential equations. Several Fourier spectrum implementations' performance considerations are explored, and techniques for efficient scaling on parallel computers are described.

It begins by quickly reviewing finite precision arithmetic. The topic of employing the separation of variables to solve ordinary differential equations (ODE) and partial differential equations (PDE) follows. Following that, numerical time-stepping techniques are presented that can be utilized to resolve PDEs and ODEs. Following this, a description of the discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) algorithm, which is used to compute the DFT quickly, is given before an introduction to pseudo spectral approaches. In order to solve a few different PDEs, it will finally merge all of these, solving them in both a serial and parallel mode.

Fortran and Matlab1 will be used in the programs. There is also a Python2 implementation of a few of the Matlab programs.