spectral-methods

Using Chebyshev spectral methods to solve the Allen-Cahn and Cahn-Hilliard equations. Instead of an equispaced grid we consider the Chebyshev points $x_j = \cos(j \pi/N)$. We implement the Chebyshev method as given in Trefethen. [1] We use it to first solve the Allen-Cahn equation[2] in 1D and 2D: $$\frac{\partial\phi}{\partial t} = \Gamma(a\phi - b\phi^3 + c\nabla^2\phi) \label{ac}$$ with the no flux boundary condition, $\frac{\partial \phi}{\partial \textbf{n}} = 0$ and initial condition, $\phi(x,0) = \phi(x)$. We then proceed to try and solve the Cahn-Hilliard equation[3] $$\frac{\partial\phi}{\partial t} = D\nabla^2(b\phi^3 - a\phi - c\nabla^2\phi) \label{ch}$$ with the same boundary and initial conditions. We only solved it in 1D and could not go to higher dimensions due to unavailibility of resources

[1] Lloyd N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, 2000
[2] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallalurgica, 27 (1979), 1085–1095
[3] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.