AI 中文总结
该研究构建了四类几何流的高阶Runge–Kutta扩展BGN参数有限元方法，证明其能量稳定特性，实验验证了能量衰减、网格重分布及高阶收敛行为。
AI 中文摘要
我们构建了Barrett–Garcke–Nürnberg（BGN）参数有限元方法的高阶Runge–Kutta扩展，用于平面曲线的曲线缩短流和曲线扩散，以及闭亏格0曲面的平均曲率流和曲面扩散。在每个时间片$I_m=(t_m,t_{m+1}]$上，连续方程通过映射$\boldsymbol{X}^{m,t}:Γ^m\toΓ^t$定义在左端点曲面$Γ^m=Γ^{t_m}$上，该映射的目标是时刻$t$的演化曲面。对于曲线，该公式通过从$Γ^0$到$Γ^m$的调和拉回得到。对于曲面，每个时间片上单独定义保定向调和微分同胚，共形性给出权重$\frac12|\nabla_{Γ^m}\boldsymbol{X}^{m,t}|^2$。在Runge–Kutta内部时刻计算时间片方程，并应用质量集中参数有限元，得到公共定义域$Γ^m$上的系统，其中间目标几何为$Γ^{t_m+c_iτ_m}$。这种内部时刻离散为四类流动构建了高阶BGN结构扩展。对于具有非负权重的代数稳定表式，每个具有非退化中间构型的精确阶段解，在解存在的每个正时间步长下，都满足离散曲线长度或曲面面积的单调衰减。Radau IIA实验显示四类流动均存在能量衰减，且曲线测试中出现BGN型网格重分布。Hausdorff自收敛结果表现出与对应设计阶数一致的高阶特性。
英文摘要
We construct high-order Runge--Kutta extensions of Barrett--Garcke--Nürnberg (BGN) parametric finite element methods for curve-shortening flow and curve diffusion of planar curves and for mean curvature flow and surface diffusion of closed genus-$0$ surfaces. On each time slab $I_m=(t_m,t_{m+1}]$, the continuous equations are posed on the left-endpoint surface $Γ^m=Γ^{t_m}$ through the map $\X^{m,t}:Γ^m\toΓ^t$, whose target is the evolving surface at time $t$. For curves, this formulation follows by harmonic pullback from $Γ^0$ to $Γ^m$. For surfaces, an orientation-preserving harmonic diffeomorphism is posed separately on each time slab, and conformality yields the weight $\frac12|\nabla_{Γ^m}\X^{m,t}|^2$. Evaluating the time-slab equations at the Runge--Kutta internal times and applying mass-lumped parametric finite elements yields systems on the common domain $Γ^m$, with $Γ^{t_m+c_iτ_m}$ as the intermediate target geometry. This internal-time discretization constructs high-order BGN-structured extensions for the four flows. For an algebraically stable tableau with nonnegative weights, every exact stage solution with nondegenerate intermediate configurations satisfies monotone decay of the discrete curve length or surface area for every positive time step for which that solution exists. Radau IIA experiments exhibit energy decay for all four flows and BGN-type mesh redistribution in the curve tests. The Hausdorff self-convergence results exhibit high-order behavior consistent with the corresponding design orders.
