Commun. Comput. Phys., 5 (2009), pp. 42-60.


Truncated Gaussian RBF Differences are Always Inferior to Finite Differences of the Same Stencil Width

John P. Boyd 1*, Lei Wang 2

1 Department of Atmospheric, Oceanic and Space Science, University of Michigan, Ann Arbor MI 48109, USA.
2 Department of Mathematics and Program in Applied and Interdisciplinary Mathematics, University of Michigan, Ann Arbor MI 48109, USA.

Received 21 December 2007; Accepted (in revised version) 1 May 2008
Available online 15 July 2008

Abstract

Radial basis functions (RBFs) can be used to approximate derivatives and solve differential equations in several ways. Here, we compare one important scheme to ordinary finite differences by a mixture of numerical experiments and theoretical Fourier analysis, that is, by deriving and discussing analytical formulas for the error in differentiating $\exp(ikx)$ for arbitrary $k$.`Truncated RBF differences'' are derived from the same strategy as Fourier and Chebyshev pseudospectral methods: Differentiation of the Fourier, Chebyshev or RBF interpolant generates a differentiation matrix that maps the grid point values or samples of a function $u(x)$ into the values of its derivative on the grid. For Fourier and Chebyshev interpolants, the action of the differentiation matrix can be computed indirectly but efficiently by the Fast Fourier Transform (FFT). For RBF functions, alas, the FFT is inapplicable and direct use of the dense differentiation matrix on a grid of $N$ points is prohibitively expensive ($\mathcal{O}(N^{2})$) unless $N$ is tiny. However, for Gaussian RBFs, which are exponentially localized, there is another option, which is to \emph{truncate} the dense matrix to a banded matrix, yielding ``truncated RBF differences''. The resulting formulas are identical in form to finite differences except for the difference weights. On a grid of spacing $h$ with the RBF as $\phi(x)=\exp(-\alpha^{2} (x/h)^{2})$, $$\frac{df}{dx}(0)\approx \sum_{m=1}^{\infty} w_{m} \{f(mh) - f(-mh) \},$$ where without approximation $w_{m}=(-1)^{m+1} 2 \alpha^{2}/\mbox{sinh}(m \alpha^{2})$. We derive explicit formula for the differentiation of the linear function, $f(X) \equiv X$, and the errors therein. We show that Gaussian radial basis functions (GARBF), when truncated to give differentiation formulas of stencil width $(2M+1)$, are significantly less accurate than $(2M)$-th order finite differences of the same stencil width. The error of the infinite series $(M=\infty)$ decreases exponentially as $\alpha \rightarrow 0$. However, truncated GARBF series have a second error (truncation error) that grows exponentially as $\alpha \rightarrow 0$. Even for $\alpha \sim \mathcal{O}(1)$ where the sum of these two errors is minimized, it is shown that the finite difference formulas are always superior. We explain, less rigorously, why these arguments extend to more general species of RBFs and to an irregular grid. There are, however, a variety of alternative differentiation strategies which will be analyzed in future work, so it is far too soon to dismiss RBFs as a tool for solving differential equations.

AMS subject classifications: 65D05

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Key words: Pseudospectral, radial basis function, high order finite difference, nonstandard finite differences, spectral differences.

*Corresponding author.
Email: jpboyd@umich.edu (J. P. Boyd), olivewl@umich.edu (L. Wang)
 

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