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Regularity and multi-scale discretization of the solution construction of hyperbolic evolution equations with limited smoothness. (English) Zbl 1252.35287

A multi-scale solution scheme for hyperbolic evolution equations with curvelets is presented. The authors assume 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 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. They analyze the regularity of the associated Volterra kernel, and then determine the optimal quadrature in the evolution parameter. Moreover, an estimate for the spreading of (finite) sets of curvelets, enabling the multi-scale numerical computation with controlled error is obtained.

MSC:

35S10 Initial value problems for PDEs with pseudodifferential operators
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35S30 Fourier integral operators applied to PDEs
35L45 Initial value problems for first-order hyperbolic systems
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
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