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

Abstract

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. L2-optimal error estimates are proved throughout.1. Introduction. In this paper we shall study efficient, high-order accurate methods for the approximation of the solutions of linear, second-order hyperbolic equations with time-dependent coefficients. We shall use Galerkin-type discretizations in the space variables and base the time-stepping scheme on a class of fourth-order accurate, two-step methods generated by rational approximations to the cosine; cf., e.g., [3], [4] for the case of time-independent coefficients. The implementation of these "base" schemes requires solving linear systems of equations with operators that vary from time step to time step. Following Douglas, Dupont and Ewing, [9], and Bramble and Sammon, [7], we shall modify the schemes by using preconditioned iterative methods for the approximate solution of the linear systems and thereby only solve linear systems with the same, time-independent operator at every step. If k is the time step, we show that solving 0(ln(k~x)) systems at each time step suffices to preserve the overall accuracy and stability of the base schemes.Preconditioned iterative techniques have been used for second-order hyperbolic problems already by Ewing in [10], where a nonlinear equation is solved by a second-order accurate, two-step time discretization. In addition, one of us, [5], [6], has used such techniques coupled with up to fourth-order accurate single-step discretizations for the problem (1.1) below, written in first-order system form, cf. [2]. These single-step schemes are based on rational approximations to e'x and give rise to quite different time-stepping methods from the ones that we study here. In this paper we take a different approach, discretizing the second-order equation without reducing it first to a first-order system. Our two-step schemes require then, as

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