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An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems. (English) Zbl 0999.65111
Summary: 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.

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
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