Can somebody suggest me that what exactly is required in order to show that a non-linear scheme for non-linear hyperbolic consevation laws, converges to right weak solution. Someone suggested me that scheme should satisfy the entropy stability condition. You people out there, please throw some light on it.

regards Ritesh

For a scalar conservation law, satisfaction of entropy condition will ensure that you get the correct weak solution. See for example

Godlewski and Raviart: Hyperbolic Systems of Conservation Laws, Paris, Ellipses

However for systems of equations, even in 1-D, it is not known in general whether entropy condition is sufficient to ensure uniqueness.

Also see

Godlewski and Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer.