Journal of Scientific Computing

, Volume 51, Issue 3, pp 683–702

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

Article

## Abstract

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.

## Keywords

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

## References

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)
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)
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)
4. 4.
Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003)
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)
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)
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)
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)
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)
12. 12.
Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley Interscience, New York (1995)
13. 13.
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge UP, Cambridge (1990)
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)
15. 15.
Jung, J.-H.: A note on the Gibbs phenomenon with multiquadric radial basis functions. Appl. Numer. Math. 57, 213–229 (2007)
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)
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)
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)
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)
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)
21. 21.
Sarra, S.A.: Adaptive radial basis function methods for time dependent partial differential equations. Appl. Numer. Math. 54, 79–94 (2005)
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)
23. 23.
Shaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995)
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)
25. 25.
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
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)
27. 27.
Zhou, X., Hon, Y.C., Li, J.: Overlapping domain decomposition method by radial basis functions. Appl. Numer. Math. 44, 241–255 (2003)
28. 28.
Zingg, D.W.: Aspects of linear stability analysis for higher-order finite-difference methods. AIAA Paper 97-1939 (1997) Google Scholar