一种用于曲面上扩散和反应-扩散方程的高阶核方法
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
AI总结：
提出一种基于径向基函数的高阶核方法，通过半离散配点逼近曲面导数算子，用于数值求解光滑闭曲面上的扩散和反应-扩散偏微分方程，避免了坐标畸变和奇异性。
摘要：
本文提出了一种高阶核方法，用于数值求解嵌入在 $\mathbb{R}^d$ 中的光滑闭曲面上的扩散和反应-扩散偏微分方程。对于嵌入在 $\mathbb{R}^3$ 中的二维曲面，这类问题在生物学、化学和计算机图形学中日益受到关注，用于模拟诸如生物细胞或膜上的化学物质扩散、生物学中的模式形成、可激发介质中的非线性化学振荡器以及纹理映射等。我们的核方法基于径向基函数，采用半离散方法（或线方法），其中 PDE 中出现的曲面导数算子通过配点进行逼近。该方法仅需要曲面上的“散乱”位置节点以及相应的曲面法向量。此外，它不依赖于任何基于曲面的度量，并避免了任何内在坐标系，因此不会受到坐标畸变或奇异性的影响。我们提供了基于核的近似曲面导数算子的误差估计，并数值研究了该方法的精度和稳定性。还展示了在生物学和化学中出现的不同非线性 PDE 系统的应用。
英文摘要：
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.
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