The Godunov Fluxes And A Relaxation-type Fast Sweeping Method For A Class Of Hamilton-Jacobi Equations

Hamilton-Jacobi equation (H-J equation for simple) is a very important classof nonlinear partial diflerential equations and plays an important role in physicalmechanics field. Because of its closely relation to hyperbolic diflerential equations,as well as to discontinuous solution, the study of numerical solution of H-J equationhas both theoretical value and practical application.In this paper, we construct an algorithm combining Godunov fluxes with arelaxation-type fast sweeping method to compute the numerical solution of a classof H-J equations with strictly convex and homogeneous Hamiltonians, based on thework by Zhao H.-K. in 2003.The Godunov fluxes of H-J equations with strictly convex and homogeneousHamiltonians are analyzed. We give an improvement of a proposition on the Go-dunov fluxes derived by Zhao H.-K. and the exact expression of the Godunov fluxes.Especially, an equal expression of the Godunov fluxes in a compact way is presentedto prepare for the construction of the numerical algorithm.Then, we present a numerical algorithm for model equations with strictly con-vex and homogeneous Hamiltonians by first discretizing the equations with Go-dunov fluxes and then solving the nonlinear equations with relaxation-type fastsweeping method. The algorithm is valid for general H-J equations with strictlyconvex and homogeneous Hamiltonians. Some numerical examples indicate theconvergence of the algorithm. The computing eflciency is eflected by diflerentrelaxation factors.