An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems. (December 2018) - Lin, Ji; Reutskiy, S.Y.
Highlights: A new accurate meshless method is proposed for time-dependent fully nonlinear problems. The obtained solutions are remarkable better than the ones obtained using other techniques. The proposed method can be easily extended to general high order nonlinear problems. Abstract: In this paper, a new meshless semi-analytical collocation technique is presented for the simulation of time-dependent linear and fully nonlinear advection diffusion reaction equations (ADREs). The time-dependent nonlinear problem is reduced to the system of linear problems in each time layer by using the quasilinearization technique in combination with the Crank–Nicolson scheme for the temporal approximation. Then it may be solved by the meshless backward substitution method. In the backward substitution method the approximation is given to satisfy the boundary conditions as a prior. Then the approximation is substituted back to the governing equations where the unknown weighted parameters are obtained using the collocation approach. The developed method has been tested on seven concrete problems for a range of free parameters. The comparison of the obtained numerical results with those available in literatures reveals that the proposed method can accurately approximate linear and nonlinear advection diffusion reaction problems. This method can also be easily extended onto other linear and nonlinear transient problems in 1D, 2D, and 3D due to its general idea and easy implementation.