Stability of Radial Basis Function Methods for Convection Problems on the Circle and Sphere

Abstract

This paper investigates the stability of the radial basis function (RBF) collocation method for convective problems on the circle and sphere. We prove that the RBF method is Lax-stable for problems on the circle when the collocation points are equispaced and the transport speed is constant. We also show that the eigenvalues of discretization matrices are purely imaginary in the case of variable coefficients and equispaced nodes. By studying the $\epsilon$-pseudospectra of these matrices we argue that approximations are also Lax-stable in the latter case. Based on these results, we conjecture that the discretization of transport operators on the sphere present a similar behavior. We provide strong evidence that the method is Lax-stable on the sphere when the collocation points come from certain polyhedra. In both geometries, we demonstrate that eigenvalues of the differentiation matrix deviate from the imaginary axis linearly with perturbations off the set of ideal collocation points. When the ideal set is impractical or unavailable, we propose a least-squares method and present numerical evidence suggesting that it can substantially improve stability without any increase to computational cost and with only a minor cost to accuracy.

Appendix

1.1 Proof that A and B commute

Proof

Fix $1\le i,j\le N$ and let $\psi _{ij}:\{1,\ldots ,N\}\rightarrow \{1,\ldots ,N\}$ be given by

$$\begin{aligned} \psi _{ij}(k)=i+j-k+N(\mathbf {1}_{i+j-k\,<\,1}-\mathbf {1}_{N\,<\,i+j-k}) \end{aligned}$$

(72)

where $\mathbf {1}$ denotes the indicator function. It can be shown that $\psi _{ij}$ is an involution (i.e. $\psi _{ij}$=$\psi _{ij}^{-1}$) and hence a bijection. Next, fix $1\le k\le N$ and observe that

$$\begin{aligned} \cos (2\pi (\psi _{ij}(k)-j)/N)&=\cos (2\pi (i-k)/N-2\pi (\mathbf {1}_{i+j-k\,<\,1}-\mathbf {1}_{N\,<\,i+j-k})) \nonumber \\&=\cos (2\pi (i-k)/N), \end{aligned}$$

(73)

$$\begin{aligned} \cos (2\pi (i-\psi _{ij}(k))/N)&=\cos (2\pi (k-j)/N-2\pi (\mathbf {1}_{i+j-k\,<\,1}-\mathbf {1}_{N\,<\,i+j-k}))\nonumber \\&=\cos (2\pi (k-j)/N) \end{aligned}$$

(74)

and similarly for sine. Comparing (73)–(74) to (21)–(22), we see that

$$\begin{aligned} a_{ik}b_{kj}=b_{i\psi _{ij}(k)}a_{\psi _{ij}(k)j}. \end{aligned}$$

(75)

Since $\psi _{ij}$ is a bijection,

$$\begin{aligned} AB\equiv X=[x_{ij}]=\left[ \sum _{k=1}^{N}a_{ik}b_{kj}\right] =\left[ \sum _{k=1}^{N}b_{i\psi _{ij}(k)}a_{\psi _{ij}(k)j}\right] =[y_{ij}]=Y\equiv BA \end{aligned}$$

(76)

and so A and B commute.