Asynchronous domain decomposition methods for continuous casting problem

Two asynchronous domain decomposition methods (which appear to be a two-stage Schwarz alternating algorithms) to solve the finite difference schemes approximating dynamic continuous casting problem are theoretically and numerically studied. Fully implicit and semi-implicit (implicit for the diffusion operator while explicit for the nonlinear convective term) finite difference schemes are considered. Unique solvability of the finite difference schemes as well as a monotone dependence of the solution on the right-hand side (the so-called comparison theorem) are proved. Geometric rate of convergence for the iterative methods is investigated, the comparison theorem being the main tool of this study. Numerical results are included and analyzed.