  We study the computation of coupled advection-diffusion-reaction equations by
the Schwarz waveform relaxation method. The study starts with linear equations,
and it analyzes the convergence of the computation with a Dirichlet condition,
a Robin condition, and a combination of them as the transmission conditions.
Then, an optimized algorithm for the Dirichlet condition is presented to
accelerate the convergence, and numerical examples show a substantial speedup
in the convergence. Furthermore, the optimized algorithm is extended to the
computation of nonlinear equations, including the viscous Burgers equation, and
numerical experiments indicate the algorithm may largely remain effective in
the speedup of convergence.
