Deeper investigation of time discretization for free surface problems is a widely neglected problem. Many existing approaches use an explicit decoupling which is only conditionally stable. Only few unconditionally stable methods are known, and known methods may suffer from too strong numerical dissipativity. They are also usually of first rder only [1, 9]. We are therefore looking for unconditionally stable, minimally dissipative methods of higher order.

Linearly implicit Runge-Kutta (LIRK) methods are a class of one-step methods that require the solution of linear systems in each time step of a nonlinear system. They are well suited for discretized PDEs, e.g. parabolic problems [7]. They have been used successfully to solve the incompressible Navier-Stokes equations [5]. We suggest an adaption of these methods for free surface problems and compare different approximations to the Jacobian matrix needed for such methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)