High order accurate two-step approximations for hyperbolic equations

Cosine Methods for Second-Order Hyperbolic Equations With Time-Dependent Coefficients

“…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.…”

“…[12], [3], [4]. The rational functions that we shall consider are of the form 1 + pxx2 + p2x4 '(*) = --¡--J.…”

“…For example, the point B represents the approximation to the cosine obtained as the real part of the (2,2) Padé rational approximation to e'x. The special approximations r(x) with denominators (1 + ßx2)2, ß > 0, considered in [3], [4], that are especially effective for problems with time-independent coefficients, correspond to qx = 2ß, q2 = ß2 (with px, p2 given by (1.15), (1.16)), i.e., lying on the parabola q2 = q\/4 which crosses d0tx and lies inside^for qx = 2ß > 1/2 -I-1/ ¿6 .…”

“…(4.3) now follows from the above and (1.5) and (1.10). D Now with Vo, Vx given elements of Sh, let V"+l, 1 < n < J -1, be the approximation to the exact solution V"+x of çn + lyn+l _ 2P"V" -Q"~lV"~l (4)(5)(6) +**(ft-¿){Lp-K-+ 2Ljf>-^(^" -^"1)-|4^" obtained by applying pn+1 steps of an iterative method, such as the ones described above, with a preconditioner PQ that satisfies (4.4), so that, for 1 < n < 7 -1, and a preassigned tolerance ß"+x, we have…”

“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Now, by (4,8), (1.5), (1.10) it follows that for 1 < n < 7 -1, (4.14) \{Q"+l)1/2{Z"+1 -TiUZ))l< C £ \\(QJ)1/2(ZJ j-n-i+2…”