On High Order MIRK Schemes and Hermite-Birkhoff Interpolants

Date of Online Publication: 26/08/2006
Authors: S. D. Capper and D. R. Moore
Pages: 27-47
Mono-Implicit Runge-Kutta (MIRK) formulae present an effective means for solving general non-linear two-point boundary value problems. High order finite difference schemes provide significant savings in both computational time and memory when the problem exhibits the required smoothness. In this paper we introduce MIRK methods of orders 10 & 12. The local truncation error of the tenth order scheme is presented and verified by numerical experiments. A piecewise Hermite-Birkhoff interpolant of order 10 is also introduced, which allows for event locations (such as roots or extrema) of the solution to be found. The corresponding error analysis and numerical data are provided for the interpolant as well. We perform numerical experiments on the set of 32 test problems from Cash & Wright [9], and find that the order 12 scheme provides a signi_cantly greater accuracy than observed with the lower order schemes.


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