AI 中文总结
该研究提出一种高阶无网格拉格朗日-欧拉RBF-FD方法，用于求解三维移动流形上的平流-扩散-反应偏微分方程，通过多种策略提升稳定性与效率，数值实验验证了其收敛性、守恒性与长期积分稳定性。
AI 中文摘要
我们提出了一种用于处理三维空间中余维数为1的移动流形$\boldsymbol{\textit{M}}(t)$上偏微分方程的高阶径向基函数生成有限差分（RBF-FD）方法。该方法基于曲面RBF-FD的切平面公式，结合了拉格朗日与欧拉处理方式：以拉格朗日方式演化流形和物质导数，同时在瞬时点云上重构其余曲面微分算子，并在必要时通过拟解析超粘性公式进行稳定化处理。采用移动曲面的紧凑全局参数模型进行自适应重排，以处理拉格朗日标记点漂移，该模型还能提供精确的法向量和基于几何的求积。重排后，通过反向半拉格朗日追踪和插值重构多步历史，随后恢复拉格朗日时间离散化。我们通过三种更新策略利用时间相干性：局部RBF-FD权重的缺陷校正、谱重计算间基于曲率的超粘性系数更新，以及在预处理广义最小残差（GMRES）迭代前重用不完全LU（ILU）分解的全局缺陷校正迭代。最后，基于演化的曲面求积通过标量投影强制规定的全局质量守恒。数值实验表明，该方法具有高阶收敛性、无源问题下的舍入误差守恒性、稳定的长时间积分，且所提更新策略可大幅节省计算量。
英文摘要
We present a high-order radial basis function-generated finite difference (RBF-FD) method for partial differential equations on moving manifolds $\mathcal M(t)\subset\mathbb R^3$ of co-dimension one. Our method builds on the tangent-plane formulation of surface RBF-FD and combines Lagrangian and Eulerian treatments: the manifold and material derivative are evolved in a Lagrangian fashion, while the remaining surface differential operators are reconstructed on the instantaneous point cloud and stabilized, when necessary, by a quasi-analytical hyperviscosity formulation. Lagrangian marker drift is handled by adaptive rearrangement using a compact global parametric model of the moving surface, which also supplies accurate normals and geometry-based quadrature. After rearrangement, we reconstruct the multistep history by backward semi-Lagrangian tracing and interpolation before resuming the Lagrangian time discretization. We exploit temporal coherence through three update strategies: defect correction for the local RBF-FD weights, a curvature-based update of the hyperviscosity coefficients between spectral recomputations, and a global defect-correction iteration that reuses an incomplete LU (ILU) factorization before preconditioned generalized minimal residual (GMRES) iterations. Finally, we enforce the prescribed global mass balance through a scalar projection based on the evolving surface quadrature. Numerical experiments demonstrate high-order convergence, conservation to roundoff in source-free problems, stable long-time integration, and substantial savings from the proposed update strategies.
Comments27 pages + 12 page appendix, 21 figures
