Dam break problem is also a Riemann problem type. It does not have b.c. but initial values only. It is similar to a shock tube in gas dynamics. For further reference see Leveque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.

In my own practice, I have assumed that both ends of the 1D domain were bounded with vertical walls and made sure that the domain is long enough so the flow does not affect the ends. Along the vertical walls, a simple fully reflective condition can be applied.

Furthermore, based on my experience, finite difference scheme may not work well for dam-break problems as it lacks shock-capturing capability. So, I would like to follow Eugenio to recommend finite volume method. You may try "Clawpack", which is an open source package (http://depts.washington.edu/clawpack/).

I suggest you take a look on this paper: 

http://www.sciencedirect.com/science/article/pii/S0022169499000074

Dear Hamed Farhadi,

Do you mean dam break cases like in a laboratory scale? Like there is a gate between the reservoir and the channel, don't you? If so, I will try to share my experience with you. I have some researches before about dam-break flow modeling, but in 2-Dimensional (2D) using finite volume. (Of course here it is same when you want to use finite difference, I will explain it later).

My domain is a reservoir which surrounded with wall, and a gate is between the reservoir and the channel, so the flow opening is only governed by the gate. That means at upstream part of the reservoir there is no flow boundary which could be applied, meanwhile at downstream part of the channel, the outflow boundary condition of course could be applied, since the water flows out of the system. So, our trick is to apply the "no slip boundary condition" at the upstream boundary of the reservoir which mathematically explains that there is no flux allowed through the wall. By this condition, we should apply the "ghost cell technique" which will ensure this "no slip boundary condition".

So, in 1D I assume this trick could also be applied particularly at the upstream boundary of the reservoir. We should apply this technique at the upstream boundary instead of determining the "zero velocity" or even knowing the "head-velocity relationship". Knowing the head-velocity relationship is the best but determining the zero velocity seems not appropriate to explain the true condition. 

So, in my opinion this no slip technique of course could explain the head-velocity relationship particularly at the upstream boundary. To prove it, we could check the "emptying curve" of the reservoir. Without the "head-velocity relationship" this emptying curve will still be computed accurately compared to the measured one by the "no slip boundary condition".

Regarding to the instability numerical procedure, in my opinion we "could not" say that finite difference will be not stable for the dam-break problem. The main point is because we use the shallow water equations with naturally there is no viscous term in these equations. (As we know that the viscous term could damp the oscillations naturally). So in my case, even I used finite volume method, I should also developed the artificial viscosity technique as the viscous term to deal with all discontinuities and oscillations in the computation and the results were amazing.

It should be noted that without this artificial viscosity, my scheme maybe also not stable in simulating the dam-break flows even I used finite volume.

So, regarding to your case my conclusion is:

No matter if you want to use the 1D finite difference or 1D finite volume, you could apply the "no slip boundary condition" at the upstream boundary of the reservoir and to ensure your numerical model runs stably just put the dissipation technique like 1D TVD or 1D artificial viscosity. I believe the 2D artificial viscosity technique which I developed before could be used also for your 1D problem.

NB:

I developed the artificial viscosity based on Jameson, et. al (1981) and Jameson, et. al (1992) which is well-known as JST scheme. If you are also interested you could search Mavriplis (1987). They are the pioneers of it.

Hopefully this will help you.

Best wishes 

Dear Hamed, I know that there were many models of dam destruction. The main interest was how the water wave will propagate along the river below the point of destruction. I found this recent paper in English with equations and boundary conditions: http://www.sv-jme.eu/data/upload/2012/04/04_2010_216_Krzyk_03.pdf . Normally you have vector of water speed always tangent to hard surface (normal component of speed is zero). There is also special condition on free surface (which is unknown a priori), but people who do not study waves use a simplified one.

Typical problem for the evolution of water surface over time contains both initial and boundary conditions. See, for example, my article with initial problem formulation, model derivation and solution: https://www.researchgate.net/publication/29629917_Tsunami_wave_generation_by_the_eruption_of_underwater_volcano

You can also read about the case of real destruction causing huge catastrophe in China in 1975: https://en.wikipedia.org/wiki/Banqiao_Dam

Article Tsunami wave generation by the eruption of underwater volcano

Dan break problem is a initial value problem, like a shock tube. As long your solution does not reach the end of your numerical domain you do not need B.C. In case you are interested you must consider the wall reflections.