A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media

Abstract

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 for approximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semianalytical technique based on the RBFs. The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters. The free parameters are determined by substitution of the expansion in the equation and collocation in the solution 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 the linear ones on each time layer. The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

Reliability Investigation of BiCGStab and IDR Solvers for the Advection-Diffusion-Reaction Equation

“…Advection-diffusion-reaction (ADR) equations and their discrete approximations are ubiquitous in the modeling of physical systems [1][2][3][4][5]. A wide variety of discretization schemes for the ADR equation, such as finite difference, (pseudo)spectral, finite element and finite volume methods exist.…”