A Numerical Study for the Performance of the Runge-kutta Finite Difference Method Based on Different Numerical Hamiltonians

High resolution finite difference methods have been successfully employed in solving Hamilton Jacobi equations, and numerical Hamiltonians are as important as numerical fluxes for solving hyperbolic conservation laws using the finite volume methodology. Though different numerical Hamiltonians have been suggested in the literature, only some simple ones such as the Lax-Friedrichs Hamiltonian are widely used. In this paper, we identify six Hamiltonians and investigate the performance of Runge-Kutta ENO methods introduced by Shu and Osher in [SIAM J. Numer. Anal. (28)(4):907-922, 1991] based on different numerical Hamiltonians, with the objective of obtaining better performance by selecting suitable numerical Hamiltonians. Extensive one dimensional numerical tests indicate that Hamiltonians with the upwinding property perform the best when factors such as accuracy and resolution of kinks are addressed. However, when the order of accuracy of the scheme and the number of grid points are increased, the errors for all Hamiltonians become nearly equal. Numerical tests are also performed for two dimensional problems.