Abstract
Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.
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References
Bateman, H.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)
Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003)
Boyd, J.P.: Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations. J. Comput. Phys. 204, 253–264 (2005)
Carr, J., Beatson, R., Cherrie, J., Mitchell, T., Fright, W., McCallum, B., Evans, T.: Reconstruction and representation of 3D objects with radial basis functions. SIGGRAPH pp. 67–76 (2001)
Driscoll, T.A., Heryudono, A.R.H.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53(6), 927–939 (2007)
Fornberg, B., Flyer, N.: The Gibbs phenomenon for radial basis functions. In: Jerri, A.J. (ed.): Advances in the Gibbs Phenomenon with Detailed Introduction. Sampling Publishing, Potsdam (2007)
Flyer, N., Wright, G.B.: Transport schemes on a sphere using radial basis functions. J. Comput. Phys. 226, 1059–1084 (2007)
Gelb, A.: Parameter optimization and reduction of round off error for the Gegenbauer reconstruction method. J. Sci. Comput. 20(3), 433–459 (2004)
Gelb, A., Tanner, J.: Robust reprojection methods for the resolution of Gibbs phenomenon. Appl. Comput. Harmon. Anal. 20, 3–25 (2006)
Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon IV: Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function. Math. Comput. 64, 1081–1095 (1995)
Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon V: Recovering exponential accuracy from collocation point values of a piecewise analytic function. Numer. Math. 71, 511–526 (1995)
Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–668 (1997)
Gottlieb, D., Shu, C.-W., Solomonoff, A., Vandeven, H.: On the Gibbs phenomenon I: Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math. 43, 81–92 (1992)
Gottlieb, S., Gottlieb, D., Shu, C.-W.: Recovering high order accuracy in WENO computations of steady state hyperbolic systems. J. Sci. Comput. 28, 307–318 (2006)
Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge University Press, Cambridge (2006)
Jung, J.-H.: A note on the Gibbs phenomenon with multi-quadric radial basis functions. Appl. Numer. Math. 57, 213–229 (2007)
Jung, J.-H., Durante, V.: An iterative adaptive multi-quadric radial basis function method for the detection of local jump discontinuities. Appl. Numer. Math. 59, 1449–1466 (2009)
Jung, J.-H., Gottlieb, S., Kim, S.O.: Iterative adaptive RBF methods for detection of edges in two dimensional functions. Appl. Numer. Math. (2010, submitted)
Kansa, E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic, and elliptic partial differential equations. Comput. Math. Appl. 19(6-8), 147–161 (1990)
Kansa, E.J., Carlson, R.E.: Improved accuracy of multi-quadric interpolation using variable shape parameters. Comput. Math. Appl. 24(12), 99–120 (1992)
Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)
Ohtake, Y., Belyaev, A., Seidel, H.-P.: 3D scattered data approximation with adaptive compactly supported radial basis functions. In: Shape Modeling International 2004, Genova, Italy, pp. 31–39. IEEE Comp. Soc., Los Alamitos (2004)
Platte, R., Driscoll, T.A.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43, 750–766 (2005)
Sarra, S.A.: Adaptive radial basis function methods for time dependent partial differential equations. Appl. Numer. Math. 54(1), 79–94 (2005)
Sarra, S.A.: A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs. Numer. Meth. Partial Differ. Equ. 24, 670–686 (2008)
Shu, C.-W., Wong, P.: A note on the accuracy of spectral method applied to nonlinear conservation laws. J. Sci. Comput. 10, 357–369 (1995)
Yee, P.V., Haykin, S.: Regularized Radial Basis Function Networks: Theory and Applications. Wiley, New York (2001)
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In memory of David Gottlieb.
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Jung, JH., Gottlieb, S., Kim, S.O. et al. Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems. J Sci Comput 45, 359–381 (2010). https://doi.org/10.1007/s10915-010-9360-7
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DOI: https://doi.org/10.1007/s10915-010-9360-7