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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.

A new type of Rosenbrock-Wanner (ROW) methods for solving semi-explicit DAEs of index-1 is introduced. The scheme considers arbitrary approximations to Jacobian entries resulting for the differential part and thus corresponds to a first attempt of applying W methods to DAEs. Besides, it is a generalized class covering many ROW-type methods known from literature. Order conditions are derived by a consistent approach that combines theories of ROW methods with exact Jacobian for DAEs (Roche, 1988) and W methods with arbitrary Jacobian for ODEs (Steihaug and Wolfbrandt, 1979). In this context, rooted trees based on Butcher’s theory that include a new type of vertices are used to describe non-exact differentials of the numerical solution. Resulting conditions up to order four are given explicitly, including new conditions for realizing schemes of higher order. Numerical tests emphasize the relevance of satisfying these conditions when solving DAEs together with approximations to Jacobian entries of the differential part.