Convergence Analysis for Computation of Coupled Advection-Diffusion-Reaction Problems

TL;DR

This paper analyzes the convergence behavior of numerical methods for coupled advection-diffusion-reaction problems, proposing optimal interface conditions and demonstrating potential for rapid convergence, even in nonlinear cases like viscous Burgers equations.

Contribution

It provides a comprehensive convergence analysis for coupled advection-diffusion-reaction equations, including optimal interface conditions and extension to nonlinear Burgers equations.

Findings

Convergence conditions are derived for explicit schemes.

Optimal interface conditions achieve 'perfect convergence'.

Numerical experiments confirm the theoretical results.

Abstract

A study is presented on the convergence of the computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even types are assigned in two subdomains, and Schwarz iteration is made between the equations when marching from a time level to the next one. The analysis starts with the linear systems resulting from the full discretization of the 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 optimal interface condition for the Schwarz iteration is obtained, 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 the viscous Burgers equations. Numerical experiments indicate that the conclusions, such as the "perfect convergence, " drawn in the linear situations may remain in the Burgers equations' computation.