RBF-based meshless local Petrov Galerkin method for the multi-dimensional convection-diffusion-reaction equation. (English) Zbl 1404.65176
Summary: In this paper, the meshless local Petrov Galerkin (MLPG) method is employed to analyze convection-diffusion-reaction equation based on radial basis function (RBF) collocation method. Compared with traditional Petrov Galerkin method, MLPG method is not limited to the particular interpolation domain and integration domain. In the MLPG method, interpolation domain is chosen through a special method which ensures that the number of neighboring nodes around the interpolation point (or collocation point) is constant. And all integrations are carried out locally over small quadrature domains of regular shapes such as two-dimensional (2D) squares and three-dimensional (3D) cubes. Crank-Nicolson scheme is applied to the time discretization to guarantee the unconditional stability of present method. Thanks to the Kronecker delta function property of the shape functions for RBF interpolation, we can easily obtain the error estimates that MLPG method has second order convergent rate in time and space simultaneously. Finally, numerical examples are presented to show the accuracy and efficiency of the MLPG method.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
convection-diffusion-reaction equation; radial basis function collocation method; meshless local Petrov Galerkin method; Kronecker delta function; unconditional stabilityReferences:
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