SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2371-A2401, January 2020.
We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds $\mathbb{M} \subset \mathbb{R}^3$ of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and surface advection-diffusion. Finally, we demonstrate the applicability of our formulation to advection-diffusion-reaction equations on manifolds described purely as point clouds. Our surface discretizations use the recently developed RBF-least orthogonal interpolation method and, with the addition of hyperviscosity, are now empirically high-order accurate, stable, and free of stagnation errors.

中文翻译：

---

流形上平流-扩散-反应方程稳定RBF-FD离散化的鲁棒高粘度公式

SIAM科学计算杂志，第42卷，第4期，第A2371-A2401页，2020年1月。
我们提出了一种新的高粘度公式，用于稳定流形1的流元$ \ mathbb {M} \ subset \ mathbb {R} ^ 3 $上的对流扩散反应方程式的径向基函数有限差分（RBF-FD）离散化。我们的这项技术涉及自动添加人工高粘度，以消除与表面梯度对应的微分矩阵中的虚假模式，这一过程克服了最近开发的欧几里得配方的技术限制。与欧几里得公式一样，流形公式也依赖于在辅助微分算子上进行的冯·诺依曼稳定性分析，该算术子模仿了RBF-FD微分矩阵引起的杂散生长。我们证明在涉及表面对流和表面对流扩散的问题上的高阶收敛速度。最后，我们证明了我们的公式对纯粹描述为点云的流形上的对流扩散反应方程的适用性。我们的表面离散化使用最近开发的RBF-最小正交插值方法，并且在添加了高粘度的情况下，从经验上讲是高阶准确，稳定的，并且没有停滞误差。