A Study of Adams-Bashforth Method in the Finite Element Based Model for Nonlinear Water Waves
Authors: Poguluri Sunny Kumar , Parameswaran Krishnankutty , Chiruvai Pattu Vendhan 
Published: February 2017

The problem of nonlinear water waves, which is of great practical importance in ocean engineering, has been studied vigorously for over three decades by adopting a Mixed Eulerian-Lagrangian (MEL) formulation that employs the fully nonlinear potential flow theory (FNPT). In this approach, the free surface equations in the Lagrangian frame are solved using a time marching procedure and the Laplace equation in the fluid domain is solved in the Eulerian frame. While the boundary integral/element method for solving the Laplace equation has been studied for over 4 decades, the finite element (FE) method has been investigated during the last 2 decades.
Time domain (TD) integration of the free surface equations in the MEL model is a crucial step. In the FE based MEL model, FE solution of the Laplace equation, which is the most computationally intensive part, is required at each time step. The fourth order Runge-Kutta (RK4) method, which involves four function evaluations at each time step, has been widely used for solving the free surface equations. In this context, the third and fourth order Adams-Bashforth (AB3 & AB4) methods that involve only one function evaluation at each time step are worth considering. For a chosen time step, the RK4 method is much more accurate than the AB methods, in addition to having much better stability. So, it is essential to study the performance of the AB methods from the view point of accuracy and stability, with a focus on computational economy. In the present paper, such a study has been undertaken employing a MEL computation capability recently developed by the authors. Since the accuracy of MEL solution to sloshing problems is not hindered by radiation boundary condition (r.b.c.), it has been adopted here to carry out simulations over several wave cycles. Since long-time nonlinear simulations using the MEL formulation are generally hampered by instability, the case of small amplitude sloshing has been simulated for about 1000 wave cycles using the AB3 and AB4 algorithms. In the MEL numerical model, errors accrue at every time step because of numerical integration, FE solution of the Laplace equation and estimation of velocities using the FE solution. The errors in the amplitude and phase of free surface waves have been estimated and compared for different simulations. The errors in the FE solution at every time step, which is an input for the next time stepping calculation, appear contained and no solution instability has been noticed.
As a second example, nonlinear sloshing with a moderate steepness of 1/30 has been considered. For this case, simulations could not be carried out beyond about 30 to 50 cycles, because of the well known saw-tooth instability associated with the Lagrangian model for the free surface equations. This instability manifests as Jacobian determinant error in the isoparametric element formulation. Interestingly, this seems to provide a diagnostic to detect saw-tooth instability in the MEL model. It would be useful to develop accurate smoothing techniques to overcome this instability and also extend the computation capability to problems with higher wave steepness.