Improving the accuracy of convexity splitting methods for gradient flow equations

This paper introduces numerical time discretization methods which significantly improve the accuracy of the convexity-splitting approach of Eyre (1998) [7], while retaining the same numerical cost and stability properties. A first order method is constructed by iteration of a semi-implicit method ba...

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Bibliographic Details
Published in:Journal of computational physics 2016-06, Vol.315, p.52-64
Main Authors: Glasner, Karl, Orizaga, Saulo
Format: Article
Language:English
Subjects:
Accuracy
Algorithms
Cahn–Hilliard
Computation
Discretization
Eyre splitting
Gradient flows
Iterative methods
Materials science
Mathematical analysis
Mathematical models
Phase-field crystal
Spectral methods
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Summary:This paper introduces numerical time discretization methods which significantly improve the accuracy of the convexity-splitting approach of Eyre (1998) [7], while retaining the same numerical cost and stability properties. A first order method is constructed by iteration of a semi-implicit method based upon decomposing the energy into convex and concave parts. A second order method is also presented based on backwards differentiation formulas. Several extrapolation procedures for iteration initialization are proposed. We show that, under broad circumstances, these methods have an energy decreasing property, leading to good numerical stability. The new schemes are tested using two evolution equations commonly used in materials science: the Cahn–Hilliard equation and the phase field crystal equation. We find that our methods can increase accuracy by many orders of magnitude in comparison to the original convexity-splitting algorithm. In addition, the optimal methods require little or no iteration, making their computation cost similar to the original algorithm.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2016.03.042