Iterative Operator-Splitting Methods and Continuous and Discrete Case

Abstract

In this paper, we contribute waveform relaxation and iterative splitting methods for systems of parabolic differential equations. We could present an analysis comparing both methods and see advantages in the iterative splitting method. Here the benefits are combination of large and small time-scales, which one the large time-scale the computational effort is less and on the small time-scale the computational work is tremendous. We discuss the convergence analysis in the finite and infinite time-interval, see [Vandewalle 1993]. The applications can be done for parabolic equations with nonlinear parts. Such problems can be decoupled in two problems, where on the one side the less investigated operator is solved with cheap methods, e.g. implicit Euler methods and the other part with high accurate methods, e.g. Runge-Kutta methods of higher order. We present the method with comparison to standard Fractional-Stepping methods. The benefit will be the individual handling of each operators with adapted standard higher order time-integrators. The methods are applied to convection-diffusion-reaction equations as used to model financial options. Finally we discuss the modified methods for multi-dimensional and multi-physical problems.