(Springer, 2004) Portero Egea, Laura; Bujanda Cirauqui, Blanca; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
In this paper we develop parallel numerical algorithms to
solve linear time dependent coefficient parabolic problems. Such methods
are obtained by means of two consecutive discretization procedures.
Firstly, we realize a time integration of the original problem using a
Fractional Step Runge Kutta method which provides a family of elliptic
boundary value problems on certain subdomains of the original domain.
Next, we discretize those elliptic problems by means of standard techniques.
Using this framework, the numerical solution is obtained by solving,
at each stage, a set of uncoupled linear systems of low dimension.
Comparing these algorithms with the classical domain decomposition
methods for parabolic problems, we obtain a reduction of computational
cost because of, in this case, no Schwarz iterations are required. We give
an unconditional convergence result for the totally discrete scheme and
we include two numerical examples that show the behaviour of the proposed
method.