Strange oscillating ripples in my shallow water implementation

I've been trying to implement the shallow water equations in Unity, but I've run in a weird bug. I get these strange oscillating ripples in my water. I made some screenshot:

And a video you can find here: https://www.youtube.com/watch?v=crXLrvETdjA

I based my code on the paper Fast Hydraulic Erosion Simulation and Visualization on GPU by Xing Mei. And you can find the entire solver code here: http://pastebin.com/JktpizHW (or see below.) Each time I use a formula from the paper, I added its number as a comment.

I tried different timesteps, for the video I used 0.02, lowering it just made it oscillate slower. I also tried a bigger grid (video uses 100, I tried 200 but then the ripples just were smaller.) I checked all formulas several times, and can't find any error.

Anyone here who can figure out what's going wrong?

Extra info:

As you can see from the pastebin, I programmed it in c#. I used Unity as my engine for visualisation, and I'm just using a grid mesh to visualise the water. I alter the mesh's vertex y component to match the height I calculate.

The DoUpdate method gets a float[][] lowerLayersHeight parameter, that's basically the height of the terrain under the water. In the video it's just all 0.

public override void DoUpdate(float dt, float dx, float[][] lowerLayersHeight) {
        int x, y;
        float totalHeight, dhL, dhR, dhT, dhB;
        float dt_A_g_l = dt * _A * g / dx; //all constants for equation 2
        float K; // scaling factor for the outﬂow ﬂux
        float dV;

        for (x=1 ; x <= N ; x++ ) {
                for (y=1 ; y <= N ; y++ ) {
                        //
                        // 3.2.1 Outﬂow Flux Computation
                        // --------------------------------------------------------------
                        totalHeight = lowerLayersHeight[x][y] + _height[x][y];
                        dhL = totalHeight - lowerLayersHeight[x-1][y] - _height[x-1][y]; //(3)
                        dhR = totalHeight - lowerLayersHeight[x+1][y] - _height[x+1][y];
                        dhT = totalHeight - lowerLayersHeight[x][y+1] - _height[x][y+1];
                        dhB = totalHeight - lowerLayersHeight[x][y-1] - _height[x][y-1];

                        _tempFlux[x][y].left =   Mathf.Max(0.0f, _flux[x][y].left        + dt_A_g_l * dhL ); //(2)
                        _tempFlux[x][y].right =  Mathf.Max(0.0f, _flux[x][y].right       + dt_A_g_l * dhR );
                        _tempFlux[x][y].top =    Mathf.Max(0.0f, _flux[x][y].top         + dt_A_g_l * dhT );
                        _tempFlux[x][y].bottom = Mathf.Max(0.0f, _flux[x][y].bottom      + dt_A_g_l * dhB );

                        float totalFlux = _tempFlux[x][y].left + _tempFlux[x][y].right + _tempFlux[x][y].top + _tempFlux[x][y].bottom;
                        if (totalFlux > 0) {
                                K = Mathf.Min(1.0f, _height[x][y] * dx * dx / totalFlux / dt);  //(4)

                                _tempFlux[x][y].left =   K * _tempFlux[x][y].left;  //(5)
                                _tempFlux[x][y].right =  K * _tempFlux[x][y].right;
                                _tempFlux[x][y].top =    K * _tempFlux[x][y].top;
                                _tempFlux[x][y].bottom = K * _tempFlux[x][y].bottom;
                        }
                        //swap temp and the real one after the for-loops

                        //
                        // 3.2.2 Water Surface
                        // ----------------------------------------------------------------------------------------
                        dV = dt * (
                                //sum in
                                _tempFlux[x-1][y].right + _tempFlux[x][y-1].top + _tempFlux[x+1][y].left + _tempFlux[x][y+1].bottom
                                //minus sum out
                                - _tempFlux[x][y].right - _tempFlux[x][y].top - _tempFlux[x][y].left - _tempFlux[x][y].bottom
                                ); //(6)
                        _tempHeight[x][y] = _height[x][y] + dV / (dx*dx); //(7)
                        //swap temp and the real one after the for-loops
                }
        }
        Helpers.Swap(ref _tempFlux, ref _flux);
        Helpers.Swap(ref _tempHeight, ref _height);
}

I fixed it myself! Though of it while driving to a friend. The problem is quite simple, what I do in the bugged code is for each cell (or grid-point) calculate the flux, then the height and then I go to the next cell. What I should do is first calculate the flux for all cells, then iterate a second time over all the cells and calculate their height. So the code becomes:

for (x=1 ; x <= N ; x++ ) {
    for (y=1 ; y <= N ; y++ ) {
        //
        // 3.2.1 Outﬂow Flux Computation
        // --------------------------------------------------------------
        ***
    }
}

for (x=1 ; x <= N ; x++ ) {
    for (y=1 ; y <= N ; y++ ) {
        //
        // 3.2.2 Water Surface
        // ---------------------------------------------------------------------------
        ***
    }
}
Helpers.Swap(ref _tempFlux, ref _flux);
Helpers.Swap(ref _tempHeight, ref _height);

(Of course *** becomes the corresponding code from the question above.)

Now I have a working water simulation.

来源：https://stackoverflow.com/questions/24533294/strange-oscillating-ripples-in-my-shallow-water-implementation

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fluid-dynamics