A ROBUST HYPERVISCOSITY FORMULATION FOR STABLE RBF-FD DISCRETIZATIONS OF ADVECTION-DIFFUSION-REACTION EQUATIONS ON MANIFOLDS
A ROBUST HYPERVISCOSITY FORMULATION FOR STABLE RBF-FD DISCRETIZATIONS OF ADVECTION-DIFFUSION-REACTION EQUATIONS ON MANIFOLDS
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DOI：
10.1137/19m1288747
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发表时间：
2020-01-01
影响因子：
3.1
通讯作者：
Narayan, Akil
中科院分区：
文献类型：
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作者：
Shankar, Varun;Wright, Grady B.;Narayan, Akil
我们提出了一种新的高效率公式，用于稳定径向基础功能 - 限制差异（RBF-FD）对流 - 扩散 - 反应方程的离散量，consimimension 1的R-3子集M子集合1。我们的技术涉及自动添加人工超胶合性以降低人造超胶合性，以降低潮湿的潮湿。在克服最近开发的欧几里得的技术局限性的过程中，分化矩阵中的伪模式与表面梯度相对应配方。像欧几里得配方一样，歧管配方依赖于对辅助差分算子进行的von Neumann稳定性分析，这些辅助差异算子模仿了由RBF-FD分化矩阵引起的杂种溶液生长。我们在涉及表面对流和表面对流扩散的问题上证明了高阶收敛速率。最后，我们证明了我们的配方对纯粹描述为点云的流形方程的适用性。我们的表面离散化使用了最近开发的RBF最高正交插值方法，并且通过添加了高粘度，现在在经验上是高阶的准确，稳定且没有停滞误差。
We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds M subset of 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 advectiondiffusion-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.