A Comparison of Convergence Rates for Godunov’s Method and Glimm’s Method in Resonant Nonlinear Systems of Conservation Laws

We obtain time independent bounds on derivatives, prove convergence, and establish a rate of convergence for Godunov’s numerical method as applied to the initial value problem for a resonant inhomogeneous conservation law which is treated as a $2 \times 2$ nonstrictly hyperbolic system. We compare the results with a corresponding analysis of Glimm’s method and see that our analysis gives equivalent (sharp) convergence rates in the strictly hyperbolic setting, but an improvement is seen in Godunov’s method over Glimm’s method in the nonstrictly hyperbolic resonant regime. The $2 \times 2$ Glimm and Godunov methods are the only methods for which we can obtain time independent bounds on derivatives; these bounds represent a purely nonlinear phenomenon because there are no corresponding time independent bounds for the linearized equations which blow up at a linear rate in time.