This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in $\mathbb{R}^3$. The method is built upon the principles of the closest point method, without the use of a grid or a closest point mapping. We show that the combination of local embedded stencils with these principles can be employed to approximate surface derivatives using polyharmonic spline kernels and polynomials (PHS+Poly) RBF-FD. Specifically, we show that it is enough to consider a constant extension along the normal direction only at a single node to overcome the rank deficiency of the polynomial basis. An extensive parameter analysis is presented to test the dependence of the approach. We demonstrate high-order convergence rates on problems involving surface advection and surface diffusion, and solve Turing pattern formations on surfaces defined either implicitly or by point clouds. Moreover, a simple coupling approach with a particle tracking method demonstrates the potential of the proposed method in solving PDEs on evolving surfaces in the normal direction. Our numerical results confirm the stability, flexibility, and high-order algebraic convergence of the approach.

翻译：本文提出了一种基于径向基函数-有限差分（RBF-FD）近似的新型无网格方法，用于数值求解嵌入在 $\mathbb{R}^3$ 中的余维1曲面上的偏微分方程（PDEs）。该方法建立在最近点法的原理之上，但无需使用网格或最近点映射。我们证明了将局部嵌入模板与这些原理相结合，可以利用多谐样条核与多项式（PHS+Poly）RBF-FD 来近似曲面导数。具体而言，我们表明仅需在单个节点处考虑沿法向的常值延拓，即可克服多项式基的秩亏问题。本文进行了广泛的参数分析以检验该方法的依赖性。我们在涉及曲面平流和曲面扩散的问题上展示了高阶收敛率，并求解了在隐式定义或点云定义的曲面上的图灵模式形成问题。此外，通过与粒子追踪方法的简单耦合策略，证明了所提方法在求解沿法向演化的曲面上的偏微分方程方面的潜力。我们的数值结果验证了该方法的稳定性、灵活性以及高阶代数收敛性。