Hello, I want to know how to solve exactly Riemann problems of nonstrictly hyperbolic conservation laws, such as the modified Burgers' equation, u_t+(a(x)u^2)_x=0. Can you give me some advices?Many thanks.

I know the iterate method can solve this problem. Are there any references?

You need to solve the characteristic equations

dt/ds = 1, dx/ds = 2au, du/ds = - a'u^2

with, at s=0, t=0, x=X, u = f(X).

See the following: Toro, E. F. 1997. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer New York.

or

Godlewski, E. and P.-A. Raviart. 1996. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer New York.

These will tell you everything you need to know. And then some more.

-- Jarmo