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14 pages, 1751 KiB

Open AccessArticle

An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation

by Chaeyoung Lee, Sungha Yoon, Jintae Park and Junseok Kim

Symmetry 2020, 12(8), 1218; https://doi.org/10.3390/sym12081218 - 25 Jul 2020

Cited by 3 | Viewed by 2272

We extend the explicit hybrid numerical method for solving the Allen–Cahn (AC) equation to the scheme for the nonlocal AC equation with isotropically symmetric interfacial energy. The proposed method combines the previous explicit hybrid method with a space-time dependent Lagrange multiplier which enforces conservation of mass. We perform numerical tests for the area-preserving mean curvature flow, which is the basic property of the nonlocal AC equation. The numerical results show good agreement with the theoretical solutions. Furthermore, to demonstrate the usefulness of the proposed method, we perform a cell growth simulation in a complex domain. Because the proposed numerical scheme is explicit, it is remarkably simple to implement the numerical solution algorithm on complex discrete domains. Full article

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52 pages, 2846 KiB

Open AccessArticle

R-Adaptive Multisymplectic and Variational Integrators

by Tomasz M. Tyranowski and Mathieu Desbrun

Mathematics 2019, 7(7), 642; https://doi.org/10.3390/math7070642 - 18 Jul 2019

Cited by 7 | Viewed by 3262

Abstract

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper, we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations, and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. Numerical results for the Sine–Gordon equation are also presented. Full article

(This article belongs to the Special Issue Geometric Numerical Integration)

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