Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

A boundary value problem for a singularly perturbed parabolic convection–diffusion equation is considered in a rectangular domain in $x$ and $t$; the perturbation parameter $\varepsilon$ multiplying the highest derivative takes arbitrary values in the half-open interval $(0,1]$. For the boundary value problem, we construct a scheme based on the method of lines in $x$ passing through $N_0+1$ points of the mesh with respect to $t$. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width $\delta$), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter $\varepsilon$ (as well as the values of $N_0$, $\delta$) influence the convergence rate of the decomposition scheme (as $N_0\to\infty$), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection–diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.