How can I solve system of nonlinear hyperbolic equations

I have to solve 1-D nonlinear hyperbolic pde system but I can't understand how to use extension of lax friedrich scheme for the nonlinear system .

But i solved system of linear hyperbolic pde using characteristic variable but in this problem a dependent variable is included in the jacobian matrix, so i cannot connect it with the linear case.

Answer 1

[Jesús Zambrano]

Are you using the pdepe command?

Solve System of PDEs - MATLAB & Simulink (mathworks.com)

If so, would it make sense to define f matrix like this?:

f = [-1/u(2); -u(1)] .* dudx;

Another detail, the initial conditions are given but the boundary conditions are missing.

[Sumanta Shagolshem]

I didn't used pdepe command but used fdm and the boundary condition is not given, it should be constructed using some numerical scheme.