It is known that Harten's uniformly non-oscillatory scheme is a second-order accurate scheme for discretizing conservation laws. In this paper a multigrid technique and Runge-Kutta time stepping with frozen dissipation is applied to Harten's scheme in order to obtain the steady-state solution. It is shown that these techniques applied to Harten's scheme lead to a better convergence to the steady-state solution of a first-order conservation law than applied to Jameson's scheme.
Burg, van der, J. W., Kuerten, J. G. M., & Zandbergen, P. J. (1991). Multigrid and Runge-Kutta time-stepping applied to the uniform non-oscillatory scheme. Journal of Engineering Mathematics, 25(3), 243-263. https://doi.org/10.1007/BF00044333