A generalized Lagrangian method for solving the steady shallow water equations

A generalized Lagrangian formulation of the shallow water equations in conservation law form using the Lagrangian distance and stream function as independent variables is described. The resulting system is fully hyperbolic and renders the flux vector continuous at tangential discontinuities. A Godunov-type shock capturing method is developed through the solution of the steady Riemann problem. A second-order nonoscillatory extension is also devised to enhance the resolution. Numerical experiments indicate that the generalized Lagrangian method attains good resolution of oblique standing waves and is excellent in resolving tangential discontinuities in high-speed steady shallow water flows.