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A proximal-gradient algorithm for crystal surface evolution

Publication ,  Journal Article
Craig, K; Liu, JG; Lu, J; Marzuola, JL; Wang, L
Published in: Numerische Mathematik
November 1, 2022

As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted H- 1 norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.

Duke Scholars

Published In

Numerische Mathematik

DOI

EISSN

0945-3245

ISSN

0029-599X

Publication Date

November 1, 2022

Volume

152

Issue

3

Start / End Page

631 / 662

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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Craig, K., Liu, J. G., Lu, J., Marzuola, J. L., & Wang, L. (2022). A proximal-gradient algorithm for crystal surface evolution. Numerische Mathematik, 152(3), 631–662. https://doi.org/10.1007/s00211-022-01320-0
Craig, K., J. G. Liu, J. Lu, J. L. Marzuola, and L. Wang. “A proximal-gradient algorithm for crystal surface evolution.” Numerische Mathematik 152, no. 3 (November 1, 2022): 631–62. https://doi.org/10.1007/s00211-022-01320-0.
Craig K, Liu JG, Lu J, Marzuola JL, Wang L. A proximal-gradient algorithm for crystal surface evolution. Numerische Mathematik. 2022 Nov 1;152(3):631–62.
Craig, K., et al. “A proximal-gradient algorithm for crystal surface evolution.” Numerische Mathematik, vol. 152, no. 3, Nov. 2022, pp. 631–62. Scopus, doi:10.1007/s00211-022-01320-0.
Craig K, Liu JG, Lu J, Marzuola JL, Wang L. A proximal-gradient algorithm for crystal surface evolution. Numerische Mathematik. 2022 Nov 1;152(3):631–662.
Journal cover image

Published In

Numerische Mathematik

DOI

EISSN

0945-3245

ISSN

0029-599X

Publication Date

November 1, 2022

Volume

152

Issue

3

Start / End Page

631 / 662

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics