Simulating the 2D wave equation with varying speed

[rewolf]

As a computer scientist and applied math guy, I've recently taken interest in learning to simulate fluids. I've done simple wave equation and KDV simulations before, but after reading the paper
"Rapid, Stable Fluid Dynamics for Computer Graphics" by Kass and Miller 1990.
I thought I'd like to try varying the depth of the water and see its effects.

The paper imposes assumptions on the shallow water equations such that the wave equation is derived:
[tex]h_{tt}=dg\nabla^2h[/tex]
where g is the gravity and [tex]d=d(x,z)[/tex] is the depth of the water and [tex]h(x,z,t)=ground+d[/tex] is the water surface height as the sum of ground height and depth.

I have a single initial gaussian disturbance, and the boundaries are reflective.

As a first attempt, I tried a simple explicit finite difference scheme. I calculated stability, and am pretty sure that my parameters lead to stable behaviour. if i set d(x,z) to a constant, the simulation appears as expected. If however i let d(x,z) be a continous,smooth surface the results are incorrect. The gaussian propogates outwards as a circular wave BUT the region inside the circle just sinks and sinks and sinks, instead of returning to the equilibrium waterlevel.
I am struggling to see why this is so. The paper seems to think this works (although they use an implicit discretization).

This surely cannot be a stability issue as all possible values of d for my given ocean floor, yield stable coefficients.

It has nothing to do with the shape of the ocean floor either, because the same happens when the floor is flat.

Each iteration, for each mesh point, [tex]d = h - ground[/tex] is calculated, and it seems that once depth starts decreasing, it doesn't want to stop. :(

If anyone can shed any light on this matter i'd appreciate it a lot !

Here are two images of the equation working (when d(x,z)=const - ground(x,z) )
http://img191.imageshack.us/img191/2551/wavesim2.th.jpg
http://img59.imageshack.us/img59/3576/wavesim1.th.jpg


The following two, are examples of when it doesn't work, ie. d(x,z,t) = h(x,z,t) - ground(x,z)
http://img641.imageshack.us/img641/6181/wavesim4.th.jpg
http://img199.imageshack.us/img199/2434/wavesim3.th.jpg

 

[jvicens]

Hello,

I am also a computer scientist and applied math expert, so I understand your interest in simulating fluids. After reading the paper you mentioned and looking at your simulations, I have a few suggestions that may help you solve the issue you are facing.

Firstly, it is important to note that the paper you referenced is from 1990 and may not take into account all aspects of fluid dynamics. It would be beneficial to look at more recent research and papers in this field to see if there are any new developments or techniques that could improve your simulation.

Secondly, it seems like the issue may lie in how you are calculating the depth of the water (d) at each time step. In the paper, they mention using an implicit discretization, which may be more accurate and stable in this case. It would be worth trying an implicit discretization to see if it improves the results.

Additionally, it may be helpful to check your boundary conditions and make sure they are properly implemented. Reflective boundaries should not cause the sinking behavior you are seeing, so there may be an error in how you are handling them.

Lastly, it may be useful to try different initial conditions and see if the issue persists. It is possible that the initial gaussian disturbance is causing an unstable behavior in the simulation.

I hope these suggestions help you in solving the issue you are facing. Good luck with your simulation!