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

*(English)*Zbl 0999.65111Summary: 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 gridfree, 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.

Radial basis functions are used to construct approximate particular solutions, and a gridfree, 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.

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

35K55 | Nonlinear parabolic equations |

35K57 | Reaction-diffusion equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

##### Keywords:

numerical examples; method of fundamental solutions; operator splitting; nonlinear Poisson problem; particular solution method; radial basis functions; convection-diffusion-reaction equation; Helmholtz equation; Adams-Bashforth time marching; heat and mass transfer
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\textit{K. Balakrishnan} et al., Comput. Math. Appl. 43, No. 3--5, 289--304 (2002; Zbl 0999.65111)

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