Numerical stabilisation of motion integration

Abstract

Explicit time stepping of initial value problems may experience instability due to feedback when the highest derivative can not be expressed explicitly, such as in the equation of motion $\ddot{x}=F(x,\dot{x},\ddot{x} )/m$. The particular example of ship motion computations in the time domain is studied, in which the instability arises from feedback from implied acceleration terms in the hydrodynamic force as a result of the `added mass' effect of the water surrounding the hull. By combining the acceleration computed from the hydrodynamic forces with one obtained by extrapolating the motion history from the last few time steps a stable and accurate solution may be obtained. The problem is studied using a simpler approximation after demonstrating its equivalence. The proposed stabilising technique may be applied to any problem exhibiting the same type of instability.