Article
Mathematics
Jin Wang, Zhengyuan Shi
Summary: The multi-reconstruction algorithm proposed in this study, based on a modified vector-valued Allen-Cahn equation, is able to reconstruct multi-component surfaces without overlapping or self-intersections, producing smooth surfaces and preserving the original data effectively. The algorithm involves one linear equation and two nonlinear equations, with the linear equation discretized using implicit methods and solved using fast Fourier transform. The ability to apply the algorithm directly to graphics processing units allows for faster implementation compared to traditional central processing units.
Article
Computer Science, Interdisciplinary Applications
Rihui Lan, Jingwei Li, Yongyong Cai, Lili Ju
Summary: The paper develops two structure-preserving numerical schemes for solving the mass-conserving convective Allen-Cahn equation. The operator splitting approach is used to split the advancing of the equation into mass-conserving AC equations and transport equations. The discretization is achieved using classic finite volume approximation and stabilized exponential time differencings. The proposed schemes are validated through numerical examples and demonstrate good performance.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics
Jingwen Chen, Pedro Gaspar
Summary: In this article, the eternal solutions to the Allen-Cahn equation in the 3-sphere are studied, focusing on the relationship between the gradient flow of the associated energy functional and the mean curvature flow. Eternal integral Brakke flows connecting Clifford tori to equatorial spheres are constructed and the symmetry properties of such flows are analyzed. The approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen (J Differ Geom 38(2):417-461, 1993) and Tonegawa (Hiroshima Math J 33(3): 323-341, 2003), and it utilizes the classification of ancient gradient flows in spheres by Choi and Mantoulidis (Amer J Math, 2019), as well as the rigidity of stationary solutions with low Morse index proved by Hiesmayr (arXiv:2007.08701 [math.DG], 2020).
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Statistics & Probability
Martin Hairer, Khoa Le, Tommaso Rosati
Summary: In this study, the Allen-Cahn equation is considered with a rapidly mixing Gaussian field as the initial condition. It is shown that if the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.
PROBABILITY THEORY AND RELATED FIELDS
(2023)
Article
Mathematics, Applied
Maxim Olshanskii, Xianmin Xu, Vladimir Yushutin
Summary: The paper studies an Allen-Cahn-type equation defined on a time-dependent surface as a model of phase separation with order-disorder transition in a thin material layer. It shows that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates through formal inner-outer expansion. A geometrically unfitted finite element method, known as trace FEM, is considered for numerical solution with full stability and convergence analysis accounting for interpolation errors and approximate geometry recovery.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics
Raffaele Folino
Summary: This paper investigates some hyperbolic variants of the mass conserving Allen-Cahn equation, focusing on the metastable dynamics of solutions within a bounded interval of the real line. The evolution of profiles with N + 1 transition layers is shown to be very slow in time, with a system of ODEs derived to describe the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and their hyperbolic variations is also conducted.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Andrea Giorgini, Maurizio Grasselli, Hao Wu
Summary: This paper focuses on the global well-posedness of two Diffuse Interface systems that model the motion of an incompressible two-phase fluid mixture in the presence of capillarity effects. The study proves the existence and uniqueness of global weak and strong solutions, as well as their separation from the pure states, using energy and entropy estimates, endpoint estimates, and logarithmic type Gronwall arguments.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Physics, Multidisciplinary
Darae Jeong, Yibao Li, Yongho Choi, Chaeyoung Lee, Junxiang Yang, Junseok Kim
Summary: A simple and practical adaptive finite difference method for the Allen-Cahn equation in two-dimensional space is presented, utilizing a narrow band domain embedded in a discrete rectangular domain. Through various computational experiments, the high performance of the proposed method is demonstrated.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Jingwen Chen
Summary: In this article, the authors generalize their previous results to higher dimensions and prove the existence of eternal weak mean root 1 root-1 curvature flows connecting a Clifford hypersurface to the equatorial spheres.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2024)
Article
Mathematics, Applied
Jiang Yang, Zhaoming Yuan, Zhi Zhou
Summary: This paper presents a class of maximum bound preserving schemes for approximately solving Allen-Cahn equations. The developed schemes include kth-order single-step schemes in time and lumped mass finite element methods in space. By using a cut-off post-processing technique, the numerical solutions satisfy the maximum bound principle, and an optimal error bound is theoretically proved for a certain class of schemes. Furthermore, the combination of the cut-off strategy and the scalar auxiliary value technique leads to the development of energy-stable and arbitrarily high-order schemes in time.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Antoine Laurain, Shawn W. Walker
Summary: This study develops a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. By minimizing an appropriate cost functional, the control of the trajectory of the flow is achieved, demonstrating the efficiency of the approach in two and three dimensions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Xiaowei Chen, Xu Qian, Songhe Song
Summary: We propose a high-order method for the conservative Allen-Cahn equation. The method uses integrating factor Runge-Kutta schemes in the temporal direction and second-order finite difference method in the spatial direction. Theoretical analysis shows that the method conserves mass and preserves the maximum principle under reasonable time step-size restriction, regardless of the space step size.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2023)
Article
Mathematics, Applied
Xiaowei Chen, Xu Qian, Songhe Song
Summary: In this paper, we propose a high-order scheme for solving the Allen-Cahn equation with a non-local Lagrange multiplier. Theoretical analysis shows that under reasonable time step-size restriction, the proposed scheme can preserve the mass and maximum principle of the numerical solution. Numerical experiments are conducted to validate the theoretical analysis.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2022)
Article
Mathematics, Applied
Vesa Julin, Joonas Niinikoski
Summary: We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in Rn+1 is close to a constant in the Ln sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem, and using it we are able to show that in R2 and R3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by a weak solution we mean a flat flow, obtained via the minimizing movements scheme.
Article
Mathematics, Applied
Helmut Abels, Maximilian Moser
Summary: This paper focuses on the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Omega. By studying the limit problem, a local in time convergence result is proven for well-prepared initial data when a smooth solution to the limit problem exists. Through constructing a curvilinear coordinate system and carrying out an asymptotic expansion, the Allen-Cahn equation with the nonlinear Robin boundary condition is rigorously analyzed.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)