Journal of Scientific Computing

, Volume 51, Issue 3, pp 683–702 | Cite as

Matrix Stability of Multiquadric Radial Basis Function Methods for Hyperbolic Equations with Uniform Centers

  • Xinjuan Chen
  • Jae-Hun Jung


The fully discretized multiquadric radial basis function methods for hyperbolic equations are considered. We use the matrix stability analysis for various methods, including the single and multi-domain method and the local radial basis function method, to find the stability condition. The CFL condition for each method is obtained numerically. It is explained that the obtained CFL condition is only a necessary condition. That is, the numerical solution may grow for a finite time. It is also explained that the boundary condition is crucial for stability; however, it may degrade accuracy if it is imposed.


Multiquadric radial basis functions Numerical stability Matrix stability Eigenvalue stability CFL condition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bernal, F., Kindelan, M.: An RBF meshless method for injection molding modelling. In: Meshfree Methods for Partial Differential Equations III. Lecture Notes in Computational Science and Engineering, vol. 57, pp. 41–56 (2007) CrossRefGoogle Scholar
  2. 2.
    Boyd, J.P., Wang, L.: Truncated Gaussian RBF differences are always inferior to finite differences of the same stencil width. Commun. Comput. Phys. 5, 42–60 (2009) MathSciNetGoogle Scholar
  3. 3.
    Boyd, J.P., Wang, L.: An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice. Appl. Math. Comput. 215, 2215–2223 (2009) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003) CrossRefMATHGoogle Scholar
  5. 5.
    Driscoll, T.A., Heryudono, A.R.H.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53, 927–939 (2007) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Flyer, N., Wright, G.B.: A radial basis function method for the shallow water equations on a sphere. Proc. R. Soc. A 465, 1949–1976 (2009) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fornberg, B., Flyer, N.: The Gibbs phenomenon for radial basis functions. In: Jerri, A. (ed.) The Gibbs Phenomenon in Various Representations and Applications, pp. 201–224. Sampling, Potsdam (2008) Google Scholar
  8. 8.
    Fornberg, B., Dirscoll, T., Wright, G., Charles, R.: Observations on the behavior of radial basis function approximations near boundaries. Comput. Math. Appl. 43, 473–490 (2003) CrossRefGoogle Scholar
  9. 9.
    Fasshauer, G.E.: RBF collocation methods as pseudospectral methods. In: Kassab, A., Brebbia, C.A., Divo, E., Poljak, D. (eds.) Boundary Elements XXVII, pp. 47–56. WIT Press, Southampton (2005) Google Scholar
  10. 10.
    Fornberg, B., Zuev, J.: The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput. Appl. Math. 54, 379–398 (2007) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fornberg, B., Driscoll, T.A., Wright, G., Charles, R.: Observations on the behavior of radial basis function approximations near boundaries. Comput. Math. Appl. 43, 473 (2002) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley Interscience, New York (1995) MATHGoogle Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge UP, Cambridge (1990) MATHGoogle Scholar
  14. 14.
    Hon, Y.C., Kansa, E.J.: Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appl. 39, 123–137 (1998) MathSciNetGoogle Scholar
  15. 15.
    Jung, J.-H.: A note on the Gibbs phenomenon with multiquadric radial basis functions. Appl. Numer. Math. 57, 213–229 (2007) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kansa, E.J.: Muliquadrics scattered data approximation scheme with applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic, and elliptic partial differential equations. Comput. Math. Appl. 19, 147–161 (1990) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lai, S.J., Wang, B.Z., Duan, Y.: Meshless radial basis function method for transient electromagnetic computations. IEEE Trans. Magn. 44, 2288–2295 (2008) CrossRefGoogle Scholar
  18. 18.
    Platte, R.B., Driscoll, T.A.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43, 750–766 (2005) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Platte, R.B., Driscoll, T.A.: Eigenvalue stability of radial basis function discretizations for time-dependent problems. Comput. Math. Appl. 51, 1251–1268 (2006) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Platte, R.B., Trefethen, L.N., Kuijlaars, A.B.J.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev. 53(2), 308–318 (2011) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sarra, S.A.: Adaptive radial basis function methods for time dependent partial differential equations. Appl. Numer. Math. 54, 79–94 (2005) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sarra, S.A.: A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs. Numer. Methods Partial Differ. Equ. 24, 670–686 (2008) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shokri, A., Dehghan, M.: A Not-a-Knot meshless method using radial basis functions and predictor corrector scheme to the numerical solution of improved Boussinesq equation. Comput. Phys. Commun. 181, 1990–2000 (2010) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005) MATHGoogle Scholar
  26. 26.
    Wright, G.B., Flyer, N., Yuen, D.A.: A hybrid radial basis function—pseudospectral method for thermal convection in a 3D spherical shell. Geochem. Geophys. Geosyst. 11, Q07003 (2010) CrossRefGoogle Scholar
  27. 27.
    Zhou, X., Hon, Y.C., Li, J.: Overlapping domain decomposition method by radial basis functions. Appl. Numer. Math. 44, 241–255 (2003) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zingg, D.W.: Aspects of linear stability analysis for higher-order finite-difference methods. AIAA Paper 97-1939 (1997) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsState University of New York at BuffaloBuffaloUSA

Personalised recommendations