We show here some more details about the simulator for weakly dispersive and nonlinear water waves. The purpose of the simulator is to predict the wave motion in ocean basins and in the vicinity of the coastline. The mathematical model consists of a system of two nonlinear partial differential equations. These equations are solved by a finite element method in space and a finite difference scheme in time. Click here to see a description of the model and the numerical methods. The simulator is based on the Diffpack libraries which makes the code quite compact even for a rather complicated problem like the present one.
We present here two numerical examples: One involving a rapidly varying bottom topography with an associated adaptive grid, and one involving a curved coastline. In the first example, we consider wave propagation over a sea mountain.
Figure 1: An underwater mountain. The scale is not the same in the horizontal and vertical direction: The height of the mountain is about 5 km while the width of the bottom is about 80 km.
The water depth at the top of the mountain is 50 meter in the simulation while the surrounding depth is 5000 m. The rapid change in depth requires a very fine mesh at the top of the mountain. This is easily accomplished by using a preprocessor that enables adaptive grid refinement. When the wave propagates over the sea moutain one expects a significant amplification of the wave amplitude. Below you can see a plot of the surface elevation. The top of sea mountain is located at the point (30,0). The line y=0 is a symmetry line.
Figure 2: The water surface elevation.
The other example that we show here concerns the run-up of waves on a curve coastline.
Figure 3: The bottom topography outside a curved beach, seen from above. The line y=10 is a symmetry line.
For simplification of the run-up problem, the coastline is modelled as a wall. Initially, a plain wave is propagating in negative x-direction towards the coast. Below is a figure displaying the surface elevation when the wave has reached the coast.
Figure 4: The water surface elevation.
Diffpack has the ability to automatically run through a large number of test cases and produce LaTeX, ASCII and HTML reports containing selected results. If you want to know more about this facility click here