In this paper, a conserved domain decomposition method for solving
convection-diffusion equations with variable coefficients is analyzed.
The interface fluxes over the sub-domains are firstly obtained by the
explicit fluxes scheme. Secondly, the interior solutions and fluxes over
each sub-domains are computed by the modified upwind implicit scheme.
Then, the interface fluxes are corrected by the obtained solutions. We
prove rigorously that our scheme is mass conservative, unconditionally
stable and of second-order convergence in spatial step. Numerical
examples test the theoretical analysis and efficiencies. Lastly, we
extend our scheme to the nonlinear convection-diffusion equations and
give the error estimate.