An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems

K. Balakrishnan, R. Sureshkumar, P. A. Ramachandran

Research output: Contribution to journalArticlepeer-review

31 Scopus citations


This paper presents an operator splitting-radial basis function (OS-RBF) method as a generic solution procedure for transient nonlinear Poisson problems by combining the concepts of operator splitting, radial basis function interpolation, particular solutions, and the method of fundamental solutions. The application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit Adams-Bashforth time marching for half the time step. This leaves a nonhomogeneous, modified Helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step. The resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions. Radial basis functions are used to construct approximate particular solutions, and a grid-free, dimension-independent method with high computational efficiency is obtained. This method is demonstrated for some prototypical nonlinear Poisson problems in heat and mass transfer and for a problem of transient convection with diffusion. The results obtained by the OS-RBF method compare very well with those obtained by other traditional techniques that are computationally more expensive. The new OS-RBF method is useful for both general (irregular) two- and three-dimensional geometry and provides a mesh-free technique with many mathematical flexibilities, and can be used in a variety of engineering applications.

Original languageEnglish (US)
Pages (from-to)289-304
Number of pages16
JournalComputers and Mathematics with Applications
Issue number3-5
StatePublished - Feb 2002
Externally publishedYes


  • Convection-diffusion-reaction equation
  • Helmholtz equation
  • Method of fundamental solutions
  • Nonlinear Poisson problem
  • Operator splitting
  • Particular solution method
  • Radial basis functions

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics


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