Surface PDE: a minimizing movement approach

表面 PDE：最小化运动方法

• 批准号: 22K03440
• 负责人: Ginder Elliott
• 金额: $ 2.66万
• 依托单位: Meiji University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2022
• 资助国家: 日本
• 起止时间: 2022-04-01 至 2025-03-31
• 项目状态: 未结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22K03440/
• 关键词: surface PDE; minimizing movements; interfacial dynamics; level set method; approximation methods; Minimizing Movements; Surface PDE; Interfacial dynamics; Curvature flow; Approximation methods

Our research focused on developing minimizing movments for surface-constrained partial differential equations. The corresponding approximation methods were successfully realized by incorporating the closest point method into a functional minimization scheme.Using our surface-type minimizing movements, we were able to effectively simulate mean curvature flow and hyperbolic mean curvature flow of interfaces on surfaces. These methods represent generalizations of the MBO (Merriman, Bence, and Osher) and HMBO (Hyperbolic Merriman, Bence, and Osher) algorithms, specifically tailored for the surface-constrained setting.In addition, we designed a surface-constrained signed distance vector field (SDVF) for describing phase geometries on surfaces in multiphase settings. We further implemented the numerical algorithms that enable the application of the SDVF to computational problems.Regarding our approximation method that combines the closest point method and minimizing movements, numerical error analyses were conducted for the heat and wave equations on surfaces, under various conditions. Convergence of our surface-type minimizing movement, with respect to the spatial discretization, was also examined. Our the results revealed that the numerical solution converges to the exact solution.

我们的研究重点是开发最小化表面约束偏微分方程的运动。通过将最接近点的方法纳入功能最小化方案中，成功实现了相应的近似方法。使用表面型最小化运动，我们能够有效地模拟表面上界面的平均曲率流量和双曲线平均曲率流量。 These methods represent generalizations of the MBO (Merriman, Bence, and Osher) and HMBO (Hyperbolic Merriman, Bence, and Osher) algorithms, specifically tailored for the surface-constrained setting.In addition, we designed a surface-constrained signed distance vector field (SDVF) for describing phase geometries on surfaces in multiphase settings.我们进一步实施了使SDVF应用于计算问题的数值算法。在各种条件下，对表面上的热量和波动方程进行了数值误差分析，以将最接近点方法和最小化的运动结合起来。还检查了我们的表面类型的收敛性最大程度地减少有关空间离散化的运动。我们的结果表明，数值解决方案会收敛到精确解决方案。

项目成果

On the construction of minimizing movements for surface PDE

表面偏微分方程最小化运动的构造

• 发表时间: 2023
• 作者: T. Muramatsu;E. Ginder
• 通讯作者: E. Ginder

Applications of the closest point method for surface PDE

最近点法在曲面偏微分方程中的应用

• 发表时间: 2023
• 作者: Y. Ote;E. Ginder
• 通讯作者: E. Ginder

Numerical analysis and application of the signed distance vector field

带符号距离矢量场的数值分析及应用

• 发表时间: 2023
• 作者: I. Aoki;E. Ginder
• 通讯作者: E. Ginder

制約条件付き曲面上界面運動に対する近似解法

约束条件下曲面界面运动近似求解方法

• 发表时间: 2022
• 作者: 村松拓真;E. Ginder
• 通讯作者: E. Ginder

Numerical analysis and application of minimizing movements for hyperbolic problems

双曲线问题最小化运动的数值分析及应用

• 发表时间: 2023
• 作者: R. Sakai;E. Ginder
• 通讯作者: E. Ginder