Cosine methods for second-order hyperbolic equations with time-dependent coefficients

Abstract

We analyze efficient, high-order accurate methods for the approximation of the solutions of linear, second-order hyperbolic equations with time-dependent coefficients. The methods are based on Galerkin-type discretizations in space and on a class of fourth-order accurate, two-step, cosine time-stepping schemes. Preconditioned iterative techniques are used to solve linear systems with the same operator at each time step. The schemes are supplemented by single-step high-order starting procedures and need no evaluations of derivatives of operators. L 2 {L^2} -optimal error estimates are proved throughout.