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Spectral methods in time for hyperbolic equations. (English) Zbl 0613.65091
It is well known that pseudospectral space discretizations of linear hyperbolic problems with periodic boundary conditions possess an infinite order of accuracy. These semidiscretizations are usually integrated in time by means of standard finite-difference procedures (which clearly are of finite order of accuracy), thus resulting in an unbalanced overall scheme. In this paper an explicit time-integration technique is employed which achieves infinite order of accuracy. The underlying idea, first suggested by W. J. Cody, G. Meinardus and R. S. Varga [J. Approx. Theory 2, 50-65 (1969; Zbl 0187.116)] in the field of parabolic problems, is to approximate the exponential exp(tA) by a high degree minimax polynomial P(tA) rather than by a power $Q{\left({\Delta }tA\right)}^{t/{\Delta }t}$, with Q(${\Delta }$ tA) a Taylor polynomial of exp(${\Delta }$ tA). Resolution, stability and accuracy are discussed. The paper includes numerical results for several scalar model problems.
Reviewer: J.M.Sanz-Serna

##### MSC:
 65M06 Finite difference methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 35L45 First order hyperbolic systems, initial value problems