2016/11/13 by Matthew F. Causley, Causley, Matthew, Hana Cho +3
Computer Science · Materials Science · Physics and Astronomy · #35L05 #65N12 #65N40 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Magnetic properties of thin films #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1611.04214
openalex publication_date 2016/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we develop an \O(N) implicit real space method in 1D\nand 2D for the Cahn Hilliard (CH) and vector Cahn Hilliard (VCH) equations,\nbased on the Method Of Lines Transpose (MOL^\T) formulation. This\nformulation results in a semi-discrete time stepping algorithm, which we prove\nis gradient stable in the H-1 norm.\n The spatial discretization follows from dimensional splitting, and an\n\O(N) matrix-free solver, which applies fast convolution to the\nmodified Helmholtz equation. We propose a novel factorization technique, in\nwhich fourth order spatial derivatives are incorporated into the solver. The\nsplitting error is included in the nonlinear fixed point iteration, resulting\nin a high order, logically Cartesian (line-by-line) update. Our method is fast,\nbut not restricted to periodic boundaries like the fast Fourier transform\n(FFT).\n The basic solver is implemented using the Backward Euler formulation, and we\nextend this to both backward difference (BDF) stencils, implicit Runge Kutta\n(SDIRK) and spectral deferred correction (SDC) frameworks to achieve high\norders of temporal accuracy. We demonstrate with numerical results that the CH,\nand VCH equations maintain gradient stability in one and two spatial\ndimensions. We also explore time-adaptivity, so that meta-stable states and\nripening events can be simulated both quickly and efficiently.\n