Abstract
We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation,u tt =c 2 u xx , given by
It has already been proved that the maximal order of accuracyp of such schemes is given byp ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Padé schemes, withS = 0,s > 0, ands = 0,S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.
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The work of the second author was supported by an NSF US-Switzerland cooperative research grant INT9123314 and funding from the Forschungsinstitut für Mathematik, ETH, Zürich. The work of the first and third authors was supported under project Nr. 21-33551.92 of the Schweizerische Nationalfonds. Travel funds for the third author were provided by the University of Stellenbosch.
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Jeltsch, R., Renaut, R.A. & Smit, K.J.H. The maximal accuracy of stable difference schemes for the wave equation. Bit Numer Math 35, 83–115 (1995). https://doi.org/10.1007/BF01732980
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DOI: https://doi.org/10.1007/BF01732980