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Solving differential-algebraic equations (DAEs) efficiently is an ongoing topic in applied mathematics. Applications are given with respect to many fields of practical interest, such as multiphysics problems or network simulations. Due to the stiffness properties of DAEs, linearly implicit Runge-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes are an appropriate choice for effecitive numerical time-integration. Compared to fully implicit schemes, they are easy to implement and avoid having to solve non-linear equations by including Jacobian information in their formulation explicity. But, especially when having to solve large coupled systems, computing the Jacobian is costly and proves to be a considerable drawback. Inspired by the works of Steihaug and Wolfbrandt [4], we introduce concepts to realize linearly-implicit Runge-Kutta methods for DAEs in the form of so-called W-methods. These schemes allow for arbitrary approximations to given Jacobian entries and, thus, for versatile strategies to reduce computational effort significantly when solving semi-explicit DAE problems of index-1. An approach extending Roche’s procedure [3] will be presented that enables to derive order conditions of the resulting methods by an algebraic theory using rooted trees, a strategy originally introduced by Butcher regarding Runge-Kutta schemes [1,2]. Besides, suitable sets of coefficients for implementing embedded schemes and their potential of increasing efficincy when solving DAEs will be demonstrated.

Process simulation tools for sewer systems are built-up by modules for the simulation of flow, transport and chemical or biological reactions. Moreover, the coupling of the models for a single channel to a network is important. In this paper a splitting approach is presented for the suitable numerical treatment of the governing equations. The flow simulation is based on the one-dimensional Saint-Venant equations. These are combined with advection-diffusion-reaction equations for the transport of substances and their chemical or biological reactions. The Saint-Venant equations are splitted into quasi-linear and non-linear components which are semi-discretised in space separately. This approach is adapted to the equations for channels or rivers with strongly varying bottom elevation or cross sections and partial dry channels. The advection-diffusion-reaction equations are treated by an appropriate discretisation of the different parts. WENO-schemes are proposed for the space-discretisation of the advection terms. Numerical results of a case-study with eight chemical reactions are presented.