An Order-N Complexity Meshless Algorithm Based on Local Hermitian Interpolation

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 11)

This work describes a new numerical method utilising RBF interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time-advancement formulation, the method is of order N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size.

The performance of the method is examined for a variety of linear convection-diffusion-reaction problems, featuring both steady and unsteady solutions. The technique is named the Local Hermitian Interpolation (LHI) method.


Local Hermitian Interpolation Radial Basis Functions Meshless Methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Schaback, R., Convergence of unsymmetric kernal-based meshless collocation methods. SIAM Journal of Numerical Analysis, 2007. 45(1): p. 333–351MATHMathSciNetGoogle Scholar
  2. 2.
    Kansa, E.J., Multiquadrics — A scattered data approximation scheme with applications to computational fluid-dynamics-I: Surface approximations and partial derivatives estimates. Computers & Mathematics with Applications, 1990. 19: p. 127–145MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kansa, E.J., Multiquadrics — A scattered data approximation scheme with applications to computational fluid dynamics-II: Solution to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 1990. 19: p. 147–161MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ling, L., R. Opfer, and R. Schaback, Results on meshless collocation techniques. Engineering Analysis with Boundary Elements, 2006. 30(4): p. 247–253CrossRefGoogle Scholar
  5. 5.
    Brown, D., On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Engineering Analysis with Boundary Elements, 2005. 29(4): p. 343–353CrossRefGoogle Scholar
  6. 6.
    Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions. Surface fitting and Multiresolution Methods, 1997Google Scholar
  7. 7.
    Wu, Z., Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approximation Theory, 1992. 8(2): p. 1–11MATHMathSciNetGoogle Scholar
  8. 8.
    Wu, Z., Solving PDEs with radial basis functions and the error estimation;, in Advances in Computational Mathematics, Z. Chen, et al., Editors. 1998Google Scholar
  9. 9.
    Schaback, R., Multivariate interpolation and approximation by translates of a basis function, in Approximation Theory VIII, C. Chui and L. Schumaker, Editors. 1995Google Scholar
  10. 10.
    Munoz-Gomez, J., P. Gonzalez-Casanova, and G. Rodriguez-Gomez. Domain decomposition by radial basis functions for time dependent partial differential equations, in Proceedings of the 2nd IASTED international conference on Advances in computer science and technology. 2006. Puerto Vallarta, MexicoGoogle Scholar
  11. 11.
    Ling, L. and E.J. Kansa, Preconditioning for Radial Basis Functions with Domain Decomposition Methods. Mathematical and Computer modelling, 2004. 38(5): p. 320–327MathSciNetGoogle Scholar
  12. 12.
    Hernandez Rosales, A. and H. Power, Non-overlapping domain decomposition algorithm for the Hermite radial basis function meshless collocation approach: applications to convection diffusion problems. Journal of Algorithms and Technology, 2007 (preprint)Google Scholar
  13. 13.
    Ingber, M., C. Chen, and J. Tanski, A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. International Journal for Numerical Methods in Engineering, 2004. 60: p. 2183–2201MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Vertnik, R., M. Zaloznik, and B. Sarler, Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method. Engineering Analysis with Boundary Elements, 2006. 30: p. 847–855CrossRefGoogle Scholar
  15. 15.
    Wright, G. and B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions. Journal of Computational Physics, 2006. 212(1): p. 99–123MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Saad, Y. SPARSKIT: A basic tool-kit for sparse matrix computations. 1988–2000 [cited; Available from:]
  17. 17.
    Power, H. and V. Barraco, A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations. Computers and Mathematics with Application, 2002. 43: p. 551–583MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Portapila, M. and H. Power, Iterative schemes for the solution of a system of equations arising from the DRM in multi domain approach, and a comparative analysis of the performance of two different radial basis functions used in the interpolation. Engineering Analysis with Boundary Elements, 2005. 29: p. 107–125CrossRefGoogle Scholar
  19. 19.
    Fornberg, B. and J. Zuev, The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers and Mathematics with Application, 2007. 54(3): p. 379–398MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Van Genuchten, M. and W. Alves, Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation. United States Department of Agriculture, Agricultural research service, 1982. Technical bulletin number 1661 Google Scholar
  21. 21.
    Cecil, T., J. Qian, and S. Osher, Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. Journal of Computational Physics, 2004. 196: p. 327– 347MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    LaRocca, A. and H. Power, Comparison between the single and the double collocation methods, 2007 (preprint)Google Scholar
  23. 23.
    Orsini, P., H. Power, and H. Morvan, The use of local radial basis function meshless scheme to improve the performance of unstructured volume element method, in ICCES Special Symposium on MESHLESS METHODS 2007: Patras, GreeceGoogle Scholar

Copyright information

© Springer Science + Business Media B.V 2009

Authors and Affiliations

  1. 1.School of Mechanical, Materials and Manufacturing Engineering, Faculty of Engineering RoomUniversity of NottinghamNottinghamUK

Personalised recommendations