This chapter analyzes the computation of advection–diffusion–reaction equations via the domain decomposition method. The equations in subdomains may differ, e.g., a diffusion equation in a subdomain and an advection–diffusion in another, and they are coupled at a subdomain interface. It deals with two computation methods: the conventional Schwarz method and the Schwarz waveform relaxation method. For the conventional Schwarz method, the chapter discusses the computation by explicit and implicit schemes at the full discretization level. Expressions are derived for the convergence speed associated with the classical interface algorithm based on the Dirichlet condition, and then with its optimized interface algorithm to accelerate the convergence. The optimized interface algorithm leads to “perfect convergence,” that is, convergence within two iterations. The classical and optimized algorithms also extend to nonlinear equations, and numerical examples illustrate their validity. For the Schwarz waveform relaxation method, this chapter discusses the computation at the semi-discretization level, starting from linear equations and then proceeding to nonlinear equations. It presents the expressions of convergence speeds with the classical interface algorithm and also the optimized interface algorithm that effectively speeds up the convergence of computation. Finally, the chapter compares the computational loads of the conventional method and the waveform relaxation method, showing that the former method has a considerably lower load.