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In this article we consider wave equations with constant and linear time dependent diffusion-coefficients which are solved numerically by iterative operator splitting with interval overlapping algorithms. The benefits of overlappling for time dependent equations are discussed. A splitting analysis and the assemblation are presented. Numerical examples for 2D wave equations are discussed at the end of this paper.

In this paper we present computational schemes for solving transport and chemical reaction processes. The model is based on a convection-diffusion-reaction equation which models our fluid dynamics of the deposition process. For the spatial discretization we combine characteristics methods for the advection equations and use finite volume methods for the diffusion equation. The reaction parts are solved analytically and the solutions are embedded into the spatial discretized advection parts. We present decomposition methods in time to couple the different scales of the multi-component transport equations. The different time-scales are coupled by transfer-operators and we discuss a multi-scale solver. In the numerical experiments we present a microscopic model for pratical transport. The accuracy of the methods are discussed.