Numerical analysis of a continuum model of phase transition
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
On the construction of error estimators for implicit Runge-Kutta methods
Journal of Computational and Applied Mathematics
Spectral methods in MatLab
The Legendre collocation method for the Cahn-Hilliard equation
Journal of Computational and Applied Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Conservative multigrid methods for Cahn-Hilliard fluids
Journal of Computational Physics
Error analysis of a mixed finite element method for the Cahn-Hilliard equation
Numerische Mathematik
A multigrid finite element solver for the Cahn-Hilliard equation
Journal of Computational Physics
A discontinuous Galerkin method for the Cahn-Hilliard equation
Journal of Computational Physics
Journal of Computational Physics
Local discontinuous Galerkin methods for the Cahn-Hilliard type equations
Journal of Computational Physics
On an accurate third order implicit-explicit Runge--Kutta method for stiff problems
Applied Numerical Mathematics
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Numerical Analysis
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A computational framework is presented for materials science models that come from energy gradient flows. The models of interest lead to the evolution of structure involving two or more phases. The framework includes higher order derivative models and vector problems. Solutions are considered in periodic cells and standard Fourier spectral discretization in space is used. Implicit time stepping is used with adaptivity based on local error estimates. The implicit system at every time step is solved iteratively with Newton@?s method. The resulting linear systems are solved in inner iterations with the conjugate gradient method, using a novel preconditioner that is a constant coefficient version of the system, taking values for the coefficients at the pure phase states. Solutions with high spatial and temporal accuracy are obtained. The dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model (that represents the scaled width of transition layers between phases) is investigated numerically and with formal asymptotics in a simple setting. The asymptotic results require a conjecture on the rank of a modified square distance matrix. Results from a fast, graphical processing unit implementation for a three-dimensional model are shown. A comparison to time stepping with operator splitting (into convex and concave parts that guarantees energy decrease in the numerical scheme) is done.