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"# Cahn-Hilliard with Primitive and Legendre Bases\n",
"\n",
"This example uses a Cahn-Hilliard model to compare two different bases representations to discretize the microstructure. One basis representation uses the primitive (or hat) basis and the other uses Legendre polynomials. The example includes the background theory about using Legendre polynomials as a basis in MKS. The MKS with two different bases are compared with the standard spectral solution for the Cahn-Hilliard solution at both the calibration domain size and a scaled domain size. "
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"### Cahn-Hilliard Equation\n",
"\n",
"The Cahn-Hilliard equation is used to simulate microstructure evolution during spinodial decomposition and has the following form,\n",
"\n",
"$$ \\dot{\\phi} = \\nabla^2 \\left( \\phi^3 - \\phi \\right) - \\gamma \\nabla^4 \\phi $$\n",
"\n",
"where $\\phi$ is a conserved ordered parameter and $\\sqrt{\\gamma}$ represents the width of the interface. In this example, the Cahn-Hilliard equation is solved using a semi-implicit spectral scheme with periodic boundary conditions, see [Chang and Rutenberg](http://dx.doi.org/10.1103/PhysRevE.72.055701) for more details."
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"### Basis Functions for the Microstructure Function and Influence Function\n",
"\n",
"In this example, we will explore the differences when using the\n",
"Legendre polynomials as the basis function compared to the primitive\n",
"(or hat) basis for the microstructure function and the influence coefficients.\n",
"\n",
"For more information about both of these basis please see the [theory section](../../THEORY.html)."
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