Convergence analysis on computation of coupled advection-diffusion-reaction problems

We present a study on the convergence of computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even dissimilar types are assigned in two subdomains, and Schwarz iteration is made between the equations when marching from a time step to the next one. The analysis starts with the algebraic systems resulting from the full discretization of the linear advection-diffusion-reaction equations by explicit schemes. Conditions for convergence are derived, and its speedup and the effects of difference in the equations are discussed. Then, it proceeds to an implicit scheme, and a recursive expression for convergence speed is derived. An optimized interface condition for the Schwarz iteration is presented, and it leads to “perfect convergence”, that is, convergence within two times of iteration. Furthermore, the methods and analyses are extended to the coupling of advection-diffusion-reaction equations with nonlinear advection or/and reaction terms. Numerical experiments indicate that the conclusions, such as the “perfect convergence”, drawn in the linear situations may remain in the nonlinear equations’ computation. It is observed that the slowdown of convergence associated with the optimized interface condition occurs when the diffusion coefficients in subdomains become different, even slightly.