Regularity In, and Multi-scale Discretization of the Solution Construction of Hyperbolic Evolution Equations of Limited Smoothness

We present a multi-scale solution scheme for hyperbolic evolution equations with curvelets. We assume, essentially, that the second-order derivatives of the symbol of the evolution operator are uniformly Lipschitz. The scheme is based on a solution construction introduced by Smith [19] and is composed of generating an approximate solution following a paradifferential decomposition of the mentioned symbol, here, with a second-order correction reminiscent of geometrical asymptotics involving a Hamilton-Jacobi system of equations and, subsequently, solving a particular Volterra equation. We analyze the regularity of the associated Volterra kernel, and then determine the optimal quadrature in the evolution parameter. Moreover, we provide an estimate for the spreading of (finite) sets of curvelets, enabling the multi-scale numerical computation with controlled error.