A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media (Communications in Computational Physics, 2019)
This paper presents a new numerical technique for solving initial and boundary value problems with unsteady strongly nonlinear advection diffusion reaction(ADR) equations. The method is based on the use of the radial basis functions (RBF)for the approximation space of the solution. The Crank-Nicolson scheme is used forapproximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs. The approximate solution is sought in the formof the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundaryconditions of the problem for any choice of the free parameters. The free parametersare determined by substitution of the expansion in the equation and collocation in thesolution domain. In the case of a nonlinear equation, we use the well-known procedure of quasilinearization. This transforms the original equation into a sequence of thelinear ones on each time layer. The numerical examples confirm the high accuracy androbustness of the proposed numerical scheme.