Analytical and numerical approximations to highly oscillatory solutions of nonlinear Friedrichs systems

We consider semilinear Friedrichs systems which model high-frequency wave propagation in dispersive media. Typical solutions oscillate in time and space with frequency of $\mathcal{O}({\epsilon }^{-1})$ and have to be computed on time intervals of length of $\mathcal{O}({\epsilon }^{-1})$, where $\epsilon \ll 1$ is a small positive parameter. For such problems, we present an approach which combines analytical approximations with tailor-made time integration. First, we replace the original problem by a fine-tuned modification of the classical slowly varying envelope approximation and prove that the corresponding error is only of $\mathcal{O}({\epsilon }^2)$. The resulting system of partial differential equations has the advantage that solutions do not oscillate in space, but still in time. For this system, we devise a novel time integrator and prove first-order convergence uniformly in $\epsilon$. Essential to this is the careful analysis of interactions between oscillatory and non-oscillatory parts of the solution, which are identified by suitable projections.