A three-stage, VSVO, Hermite-Birkhoff-Taylor, ODE solver |
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Authors: | Vladan Bozic Rémi Vaillancourt |
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Institution: | Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 |
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Abstract: | The new variable-step, variable-order, ODE solver, HBT(p) of order p, presented in this paper, combines a three-stage Runge-Kutta method of order 3 with a Taylor series method of order p-2 to solve initial value problems , where y:R→Rd and f:R×Rd→Rd. The order conditions satisfied by HBT(p) are formulated and they lead to Vandermonde-type linear algebraic systems whose solutions are the coefficients in the formulae for HBT(p). A detailed formulation of variable-step HBT(p) in both fixed-order and variable-order modes is presented. The new method and the Taylor series method have similar regions of absolute stability. To obtain high-accuracy results at high order, this method has been implemented in multiple precision. |
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Keywords: | Hermite-Birkhoff-Taylor method Vandermonde-type systems VSVO method High-order solver Multiple precision Comparing ODE solvers |
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