Godunov scheme for the scalar nonlinear conservation laws with flux depending on the space variable

We derive the Godunov scheme for the scalar nonlinear conservation laws with the flux depending on the space variable x and on the unknown function. The Riemann problem for the scalar conservation law with the flux F (x, u) = k (x) f (u) has been solved and the corresponding Godunov scheme is derived. Then we prove that the derived scheme has the TVB property. Finally, the numerical examples involving Burgerʹs equation and the Buckley-Leverett equation are presented.