Dear Sadegh and Junping,

In this case there is an alternative. This is a linear hyperbolic system. you can write is as U_t =A U_x (LaTeX notation) with U being the vector (h, V) and A a matrix containing the coefficients in the equation.  Then you can write the Schur decomposition of A (A = QMQ^{-1}) to write the system as U_t = Q M Q^{-1} U, redefine W = Q^{-1} U the new vector of unknowns (note that U = Q Q^{-1} U), and work with the resulting system, which is decoupled. You end up with linear, scalar wave equations. 

I'm sure, books on hyperbolic problems (like the one already suggested) do give more details. 

Good luck,

Sorin

Use chain rule and some substitutions and then rearrange the terms, it will be easier then!

What's T? h? Are they known function or unknowns too?

T is known,but h is unknown.

You have 2 equations yet three unknown. It cannot solved

First of all, what are the unknown functions? I guess A and Q, right?

There are both temporal and spatial partial derivatives in your equations. I assume that the unknown functions actually depend on time and space, so that your equations are not ODEs, but rather PDEs. Usually one cannot derive closed form solutions to PDEs, only for special cases analytical solutions are known. Instead the solution might be approximated by means of numerical methods for PDEs, e.g. finite differences, finite elements, finite volume, spectral methods, or others. It depends on the PDEs what particular method would be suitable. Here are some book I found helpful for learning about finite difference and finite element method:

[Evans 2010] L.C. Evans: Partial Differential Equations, 2nd Edition. Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 2010

[LeVeque 2007] R.J. LeVeque: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007

[Ern & Guermond 2010] A. Ern, J.-L Guermond: Theory and Practice of Finite Elements. Springer, 2010

Is there a relationship between the three unknowns?

Dear Martin Wlotzka, thank you very much.

i solved it by finite difference but i want to solve that by finite volume method. do you have any example of solving by finite volume method?

Dears,

i modified attached file,please see this file...

Dear Sadegh Moodi,

I recommend the following books:

Eleuterio F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, third edition, Springer, 2009.

Charles Hirsch: Numerical Computation of internal and external flows, Volume 1, John Wiley & Sons, 1994.

Weiming Wu: Computational River Dynamics, Taylor & Francis, 2007.

I presume that You are trying to solve a one-dimensional open channel flow (by the looks of the attached equations). In my experience the Preissmann scheme of the finite difference methods works just fine in these cases.

FDM ＆ FVM are the same just different way of discretization. FDM comes from Taylor series expansion. FVM comes from the physical sense of conservation law. 

This looks like the Saint Vernon equations for channel flow. I assume the S term is related to Mannings. This is not so easy to solve numerically. The book by Toro is a good reference. You need to have a numerical scheme that can deal with different flow type: critical, supercritical, subcritical etc. A good local Riemann solver usually helps a lot. Toro had a paper to deal with this problem, and the method works well. 

Dear Sadegh and Junping,

In this case there is an alternative. This is a linear hyperbolic system. you can write is as U_t =A U_x (LaTeX notation) with U being the vector (h, V) and A a matrix containing the coefficients in the equation.  Then you can write the Schur decomposition of A (A = QMQ^{-1}) to write the system as U_t = Q M Q^{-1} U, redefine W = Q^{-1} U the new vector of unknowns (note that U = Q Q^{-1} U), and work with the resulting system, which is decoupled. You end up with linear, scalar wave equations. 

I'm sure, books on hyperbolic problems (like the one already suggested) do give more details. 

Good luck,

Sorin