Open Access

T. Tiihonen, J.-P. Zolésio

• We propose a new incremental method for solving time dependent free boundary problems. In this work we consider the so-called quasi-steady flows, i.e. flows that are so slow that the inertial effects can be ignored but where the domain of the flow is non-cylindrical in time. Moreover, the shape of the fluid varies according to the flow velocity which leads us to a time dependent free boundary problem. The numerical methods used to solve this kind of problems usually apply a first order explicit Euler schema to time discretization. We propose here an explicit method which is formally of second order in time. Essentially, we take into account the acceleration of the fluid due to the change in the geometry. In other words, we compute the shape derivative of the solution (velocity field) at each time instant to the direction of the change in the geometry. In our case this direction is determined by the flow velocity. Numerical experiments with this ‘acceleration’ technique show quadraticWe propose a new incremental method for solving time dependent free boundary problems. In this work we consider the so-called quasi-steady flows, i.e. flows that are so slow that the inertial effects can be ignored but where the domain of the flow is non-cylindrical in time. Moreover, the shape of the fluid varies according to the flow velocity which leads us to a time dependent free boundary problem. The numerical methods used to solve this kind of problems usually apply a first order explicit Euler schema to time discretization. We propose here an explicit method which is formally of second order in time. Essentially, we take into account the acceleration of the fluid due to the change in the geometry. In other words, we compute the shape derivative of the solution (velocity field) at each time instant to the direction of the change in the geometry. In our case this direction is determined by the flow velocity. Numerical experiments with this ‘acceleration’ technique show quadratic convergence in time.…