The authors discuss the stability of the Godunov method for the \(2 \times 2\) system of resonant nonlinear conservation laws \[ a_t = 0, \quad u_t + f(a,u)_x = 0 \] with initial conditions \(a(x,0) = a_0 (x) = a(x)\), \(u(x,0) = u_0(x)\). They show that total variation of \(u\) can grow at most linearly with \(t\) if the variation of \(a(x)\) and \(a'(x)\) are bounded. This is the first such stability result for a numerical method based on the solution of the Riemann problem for a resonant nonstrictly hyperbolic system.
Moreover the averaging step in the Godunov method wipes out the numerical oscillations that can occur in the Riemann problem solution step.