Geodesic
A simple and efficient scheme for phase field crystal simulation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1413-1432

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We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error analysis as well as numerical experiments to validate the method's efficiency.

DOI : 10.1051/m2an/2013074
Classification : 65M12, 74S25, 74N05, 74N20, 82C26
Keywords: phase field crystal, semi-implicit time discretization, convex-concave splitting 
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     author = {Elsey, Matt and Wirth, Benedikt},
     title = {A simple and efficient scheme for phase field crystal simulation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1413--1432},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     doi = {10.1051/m2an/2013074},
     mrnumber = {3100769},
     zbl = {1286.74118},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013074/}
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Elsey, Matt; Wirth, Benedikt. A simple and efficient scheme for phase field crystal simulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1413-1432. doi: 10.1051/m2an/2013074

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