An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u 1 + au x1 − vδu = 0 in ℝ n × ℝ + with initial data u 0 ; the Support of u 0 is contained in ℝ   − n (x 1 < 0) and a: ℝ n → ℝ is positive. In order to approximate the full space solution by the solution of a problem in ℝ   − n× ℝ + , we propose the artificial boundary condition: u 1 + au x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O ( v 2 ) for small values of the viscosity v .