A study is presented on the  computation of coupled advection-diffusion-reaction equations by Schwarz waveform relaxation methods. Unlike in most earlier work, which primarily focuses on linear, homogeneous scenarios at the continuous/semi-continuous levels, we deal with the equations that may not be identical in different subdomains and analyze their computation at the full discretization level, plus a discussion on nonlinear equations .  The analysis starts with the linear systems resulting from the  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 discretization by an implicit scheme, and a recursive expression for convergence speed is derived. An optimal interface condition for Schwarz waveform relaxation is also obtained, which 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 mostly in the computation of the Burgers equations .
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 .
A
convergence of the
D
by Schwarz waveform relaxation methods. Unlike in most earlier work, which primarily focuses on linear, homogeneous scenarios at the continuous/semi-continuous levels, we deal with the equations that may not be identical in different subdomains and analyze their computation at the full discretization level, plus a discussion on nonlinear equations
A
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.
A
full
D
discretization by
R
Schwarz waveform relaxation is also obtained, which
the Schwarz iteration is obtained, and it
R
",
, "
R
mostly in the computation of the Burgers equations
in the Burgers equations' computation