A new implementation of the finite collocation method for time dependent PDEs

摘要

This paper is concerned with a new implementation of a variant of the finite collocation (FC) method for solving the 2D time dependent partial differential equations (PDEs) of parabolic type. The time variable is eliminated by using an appropriate finite difference (FD) scheme. Then, in the resultant elliptic type PDEs, a combination of the FC and local RBF method is used for spatial discretization of the field variables. Unlike the traditional global RBF collocation method, dividing the collocation of the problem in the global domain into many local regions, the method becomes highly stable. Furthermore, the computational cost of the method is modest due to using strong form equation, collocation approach and that the matrix operations require only inversion of matrices of small size. Different approaches are investigated to impose Neumann's boundary conditions. The test problems consist of three linear convection-diffusion-reaction equations and a 2D nonlinear Burger's equation. An iterative approach is proposed to deal with the nonlinear term of Burger's equation.