On computation of coupled advection-diffusion-reaction problems by Schwarz waveform relaxation methods
share
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.
READ FULL TEXT