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

作     者：John P.Boyd Lei Wang 

作者机构：Department of AtmosphericOceanic and Space ScienceUniversity of MichiganAnn Arbor MI 48109USA Department of Mathematics and Program in Applied and Interdisciplinary MathematicsUniversity of MichiganAnn Arbor MI 48109USA 

出 版 物：《Communications in Computational Physics》 (计算物理通讯（英文）)

年 卷 期：2009年第5卷第1期

页      面：42-60页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Pseudospectral radial basis function high order finite difference nonstandard finite differences spectral differences 

摘      要：Radial basis functions(RBFs)can be used to approximate derivatives and solve differential equations in several ***,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 differencesare 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 *** 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(O(N2))unless N is ***,for Gaussian RBFs,which are exponentially localized,there is another option,which is to 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 *** a grid of spacing h with the RBF asφ(x)=exp(−α^(2)(x/h)^(2)),d f dx(0)≈∑^(∞)_(m)=1 wm{f(mh)−f(−mh)},where without approximation wm=(−1)m+12α^(2)/sinh(mα^(2)).We derive explicit formula for the differentiation of the linear function,f(X)≡X,and the errors *** 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 *** error of the infinite series(M=∞)decreases exponentially asα→***,truncated GARBF series have a second error(truncation error)that grows exponentially asα→*** forα∼O(1)where the sum of these two errors is minimized,it is