TY - JOUR

T1 - An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems

AU - Balakrishnan, K.

AU - Sureshkumar, R.

AU - Ramachandran, P. A.

N1 - Funding Information:
P.A.R. and R.S. would like to thank the National Science Foundation for partial financial support under Grants CPE 9527671 and CTS 9874813, respectively. *Author to whom all correspondence should be addressed.

PY - 2002/2

Y1 - 2002/2

N2 - 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.

AB - 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.

KW - Convection-diffusion-reaction equation

KW - Helmholtz equation

KW - Method of fundamental solutions

KW - Nonlinear Poisson problem

KW - Operator splitting

KW - Particular solution method

KW - Radial basis functions

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U2 - 10.1016/S0898-1221(01)00287-5

DO - 10.1016/S0898-1221(01)00287-5

M3 - Article

AN - SCOPUS:0036467777

SN - 0898-1221

VL - 43

SP - 289

EP - 304

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 3-5

ER -