Solving Phase-Field Models with Fourier Spectral Methods

Abstract

As described in Chap. 2, finite difference and finite element methods have local character and the unknown functions are interpreted by usually low-order polynomials over small sub-domains. In contrast, spectral methods make use of global representation, usually with high-order polynomials or Fourier series. The rate of convergence of spectral approximations depends only on the smoothness of the solution. They achieve much higher accuracy with much smaller number of sampling points in comparison to other two methods. This fact is known in the literature as “spectral accuracy.” The spectral methods most often are successful with domains in periodic nature, which is the case in most of the phase-field modeling simulations. Again, the application of the Fourier spectral method will be demonstrated to the solution of one-dimensional transient heat conduction in this section. This source code, solving this simple problem given below, forms the foundation of the algorithms that will be developed in this chapter.