Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913908072120320 |
|---|---|
| author | Dondl, Patrick Onwunta, Akwum Striet, Ludwig Wojtowytsch, Stephan |
| author_facet | Dondl, Patrick Onwunta, Akwum Striet, Ludwig Wojtowytsch, Stephan |
| contents | The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces Dondl, Patrick Onwunta, Akwum Striet, Ludwig Wojtowytsch, Stephan Numerical Analysis Analysis of PDEs 65M12, 35A35, 49Q05 The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow. |
| title | Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces |
| topic | Numerical Analysis Analysis of PDEs 65M12, 35A35, 49Q05 |
| url | https://arxiv.org/abs/2506.18869 |