Studies in the Stability of a Numerical Wave Tank Model for Generation and Propagation of Steep Nonlinear Long-Crested Surface Waves

Abstract

We are studying numerically the problem of generation and propagation of gravity long-crested waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite depth. A non-orthogonal curvilinear coordinate system, which follows the free surface is constructed which gives a realistic “continuity condition”, since it tracks the entire fluid domain at all times. A depth profile of the potential is assumed, and employed to perform a waveform relaxation algorithm to decouple the discrete Laplacian along dimensional lines, thereby reducing it’s computation over this total fluid domain. In addition, the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm, and a suitably tuned numerical beach is used to avoid reflections. It is well known that instability, in the form of generated spurious “sawtooth waves”, plagues this problem, leading to numerical overflow. This makes it very difficult to generate steep waves for sufficiently long simulation times. The authors have struggled with this problem for some time, with significant success, by employing an “aliasing filter”. This paper outlines our ongoing study of the stability of the model, including an analysis of the possible nature of the underlying causes including compatibility conditions. We conclude by giving a simple practical technique for greatly improving the stability.