Second Order Implicit Schemes for Scalar Conservation Laws.

The today's demands for simulation and optimization tools for water supply networks are permanently increasing. Practical computations of large water supply networks showthat rather small time steps are needed to get sufficiently good approximation results - a typical disadvantage of low order methods. Having this application in mind we use higher order time discretizations to overcome this problem. Such discretizations can be achieved using so-called strong stability preserving Runge-Kutta methods which are especially designed for hyperbolic problems. We aim at approximating entropy solutions and are interested in weak solutions and variational formulations. Therefore our intention is to compare different space discretizations mostly based on variational formulations, and combine them with a second-order two-stage SDIRK method. In this paper, we will report on first numerical results considering scalar hyperbolic conservation laws.