Second order accuracy of Brenier's time-discrete method for nonlinear systems of conservation laws

An approximation $\tilde u(x,t)$ to the smooth solution of a hyperbolic system of conservation laws is shown to be second order accurate in time, i.e., $\tilde u(x,t) = u(x,t) + O(t^3 )$. This approximation is based on linearized characteristics and is related to various other perturbation techniques for hyperbolic systems. Proving second order accuracy is of interest because certain fully discrete numerical methods can be viewed as spatial discretization of the time-discrete method.