Solving Phase-Field Models with Fourier Spectral Methods
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
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Solving Phase-Field Models with Fourier Spectral Methods
5.1
Introduction
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. There are numerous textbooks and lecture notes available for spectral methods, of which some of them are listed in the reference section, [1–6].
5.2
One-Dimensional Transient Heat Conduction: A Solution with Fourier Spectral Algorithm
Recall Eq. 4.9, for constant thermal conductivity, density, and heat capacity, the one-dimensional heat conduction equation without heatgenerating sources was expressed as: ∂T ∂2 T ¼μ 2, ∂t ∂x
and
μ¼
λ ρcp
ð5:1Þ
where ρ is the density, cp is the heat capacity, λ is the thermal conductivity, T is the temperature, t is the time, and x is the distance. The solution of this equation with Fourier spectral method starts with forward Fourier transform of both sides of the equation 2 ∂ fT gk ∂ T ¼μ ð5:2Þ ∂x2 k ∂t where fgk is the Fourier transform of the quantity inside the bracket and k is the coefficient of the kth Fourier mode. The general relationship for spatial derivatives in Fourier space is given as n pffiffiffiffiffiffiffi n ∂ u ¼ 1 k fugk ð5:3Þ ∂xn k
# Springer International Publishing Switzerland 2017 S.B. Biner, Programming Phase-Field Modeling, DOI 10.1007/978-3-319-41196-5_5
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Solving Phase-Field Models with Fourier Spectral Methods
Then, in Fourier space, Eq. 5.2 becomes ∂fT gk ¼ μk2 fT gk ∂t
ð5:4Þ
By taking forward difference for time derivative fT gknþ1 fT gkn ¼ μk2 fT gkn Δt
ð5:5Þ
in which Δt is the time between the time steps n þ 1 and n. With forward Euler time marching, for the time step n + 1 results in: fT gknþ1 ¼ fT gkn μk2 ΔtfT gkn
ð5:6Þ
Note that Eq. 5.6 is evaluated at Fourier space, and the results are converted back to real space with an inverse Fourier transformation. The program heat_1d_fft.m given below utilizes the above steps to solve the one-dimensional heat conductivity problem. The program uses the identical parameters as given in heat_1d_fd.m. However, in this case the domain is assumed to be periodic without any imposed boundary conditions. Th
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