Threshold dynamics for subdiffusive grain growth
Abstract
We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For α=1, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for 0<α<1 it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.
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BibTeX
@misc{fritz2026threshold,
title = {{Threshold dynamics for subdiffusive grain growth}},
author = {Marvin Fritz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07938},
eprint = {2609.07938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07938},
}
RIS
TY - UNPB
TI - Threshold dynamics for subdiffusive grain growth
AU - Marvin Fritz
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07938
UR - https://arxiv.org/abs/2609.07938
AB - We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For α=1, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for 0<α<1 it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.
ER -