Local Absorbing Boundary Conditions for the Convection-Reaction-Diffusion Equation in One-Dimensional Unbounded Domain

The exact artificial boundaries of the convection-reaction-diffusion equation present some nonlocal integrals in time when we investigate the numerical solutions of the related models in one dimensional unbounded spatial domain. To avoid solving a nonlocal problem in time, in this work, we derive local artificial boundaries for the considered mathematic model. The main strategy is to approximate the square root function by using Padé approximation in the frequency domain. By introducing auxiliary variables to reduce the order of the high order derivatives, the nonlocal boundaries is approximated by a coupled system of ordinary differential equations with classic first-order derivatives. To examine the effectiveness of local boundaries, we design a fully discrete implicit difference scheme for the reduced problem. The error estimates is established for the present finite difference scheme. Numerical examples are presented to verify our theoretical results.