A physical picture of the time step problem.

Purely for gaining a physical insight into what is happening when the equation solving routine goes unstable, consider what happens when the time step is too large. When Euler's method takes a step, it moves volume from cell 1 to cell 2.

Now if became comparable in size to the difference in level between the two cells, then Euler's method has over compensated for the level difference. If became larger than half the difference, then the cell's would slosh back and forth. If became larger than the difference, then the lake model would "explode", as the difference became larger and larger without bound.

It is important to understand that physically no such thing would happen, no matter what the values of the various constants. This "sloshing" is purely due to our wholesale abuse of Taylor's theorem. Our assumptions in using Euler's method have been ignored, therefore garbage is the result. Again. Physically a pan of water can slosh to and fro. However the physical inertia and energy terms have not been included in our differential equations, therefore these equations, no matter how well or badly solved, cannot predict physical sloshing behaviour. Again. Playing with the constants like relaxation time or the areas represents different physical problems, the only solution really is a smaller time step or better equation solver.