Solving differential-algebraic equations (DAEs) eﬃciently by means of appropriate numerical schemes for time-integration is an ongoing topic in applied mathematics. In this context, especially when considering large systems that occur with respect to many ﬁelds of practical application eﬀective computation becomes relevant. In particular, corresponding examples are given when having to simulate network structures that consider transport of ﬂuid and gas or electrical circuits.
Due to the stiﬀness properties of DAEs, time-integration of such problems generally demands for implicit strategies. Among the schemes that prove to be an adequate choice are linearly implicit Rung-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes. Compared to fully implicit methods, they are easy to implement and avoid the solution of non-linear equations by including Jacobian information within their formulation. However, Jacobian calculations are a costly operation. Hence, necessity of having to compute the exact Jacobian with every successful time-step proves to be a considerable drawback.
To overcome this drawback, a ROW-type method is introduced that allows for non-exact Jacobian entries when solving semi-explicit DAEs of index one. The resulting scheme thus enables to exploit several strategies for saving computational eﬀort. Examples include using partial explicit integration of non-stiﬀ components, utilizing more advantageous sparse Jacobian structures or making use of time-lagged Jacobian information. In fact, due to the property of allowing for non-exact Jacobian expressions, the given scheme can be interpreted as a generalized ROW-type method for DAEs. This is because it covers many diﬀerent ROW-type schemes known from literature. To derive the order conditions of the ROW-type method introduced, a theory is developed that allows to identify occurring diﬀerentials and coeﬃcients graphically by means of rooted trees. Rooted trees for describing numerical methods were originally introduced by J.C. Butcher. They signiﬁcantly simplify the determination and deﬁnition of relevant characteristics because they allow for applying straightforward procedures. In fact, the theory presented combines strategies used to represent ROW-type methods with exact Jacobian for DAEs and ROW-type methods with non-exact Jacobian for ODEs. For this purpose, new types of vertices are considered in order to describe occurring non-exact elementary diﬀerentials completely. The resulting theory thus automatically comprises relevant approaches known from literature. As a consequence, it allows to recognize order conditions of familiar methods covered and to identify new conditions.
With the theory developed, new sets of coeﬃcients are derived that allow to realize the ROW-type method introduced up to orders two and three. Some of them are constructed based on methods known from literature that satisfy additional conditions for the purpose of avoiding eﬀects of order reduction. It is shown that these methods can be improved by means of the new order conditions derived without having to increase the number of internal stages. Convergence of the resulting methods is analyzed with respect to several academic test problems. Results verify the theory determined and the order conditions found as only schemes satisfying the order conditions predicted preserve their order when using non-exact Jacobian expressions.