发表机构
Wuhan University; Tsinghua University（武汉大学; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种单步隐式-显式龙格-库塔框架，用于几何流的高阶时间积分，通过降系数结构在每个阶段仅需线性求解，无需额外预测步骤，并扩展到平均曲率流和表面扩散，数值实验验证了收敛阶数和网格质量。
AI 中文摘要
我们提出了一种用于几何流高阶时间积分的单步隐式-显式（IMEX）龙格-库塔框架。从曲线缩短流的Dziuk和Deckelnick-Dziuk参数有限元公式出发，我们将空间半离散方程重新表述为常微分方程组，并构造了二阶和三阶格式。每个阶段仅需一次线性求解。关键要素是一种降系数结构，它允许每个阶段的变分问题在前一阶段生成的几何上提出。因此，龙格-库塔阶段提供了所需的预测几何，而无需单独的预测过程。所得格式不需要额外的起始值。该框架还直接扩展到曲线和曲面的平均曲率流及表面扩散的Barrett-Garcke-Nürnberg公式。数值实验支持所设计的时间收敛阶数，并表明合适的系数选择可产生与相应一阶方法相当的网格质量。
英文摘要
We propose a one-step implicit-explicit (IMEX) Runge-Kutta framework for high-order time integration of geometric flows. Starting from the Dziuk and Deckelnick-Dziuk parametric finite element formulations of curve shortening flow, we recast the spatially semidiscrete equations as systems of ordinary differential equations and construct second- and third-order schemes. Each stage requires only a linear solve. The key ingredient is a reduced-coefficient structure that allows the variational problem at each stage to be posed on the geometry generated by the preceding stage. The Runge-Kutta stages thus supply the required prediction geometry without a separate prediction procedure. The resulting schemes require no additional starting values. The framework also extends directly to the Barrett-Garcke-Nürnberg formulations of mean curvature flow and surface diffusion for curves and surfaces. Numerical experiments support the designed orders of temporal convergence and indicate that suitable coefficient choices yield mesh quality comparable to that of the corresponding first-order methods.
Comments27 pages, 11 figures
