Local Error Control in the RK5GL3 Method

Authors: J.S.C. Prentice

Abstract:

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe an effective local error control algorithm for RK5GL3, which uses local extrapolation with an eighth-order Runge-Kutta method in tandem with RK5GL3, and a Hermite interpolating polynomial for solution estimation at the Gauss-Legendre quadrature nodes.

Keywords: RK5GL3, RKrGLm, Runge-Kutta, Gauss-Legendre, Hermite interpolating polynomial, initial value problem, local error.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070711

References:

[1] J.S.C. Prentice, "The RKGL method for the numerical solution of initialvalue problems", Journal of Computational and Applied Mathematics, 213, 2 (2008) 477.
[2] D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd ed., Pacific Grove: Brooks/Cole, 2002, pp492−498.
[3] E. Hairer, S.P. Norsett, and G. Wanner, Solving ordinary differential equations I: Nonstiff problems, Berlin: Springer-Verlag, 2000, p177.
[4] E. Hairer, S.P. Norsett, and G. Wanner, Solving ordinary differential equations I: Nonstiff problems, Berlin: Springer-Verlag, 2000, p180.
[5] J.C. Butcher, Numerical methods for ordinary differential equations, Chichester: Wiley, 2003, p192.
[6] R.L. Burden and J.D. Faires, Numerical analysis, 7th ed., Pacific Grove: Brooks/Cole, 2001, pp133 − 135.
[7] E. Hairer, S.P. Norsett, and G. Wanner, Solving ordinary differential equations I: Nonstiff problems, Berlin: Springer-Verlag, 2000, pp165− 185.
[8] J.C. Butcher, Numerical methods for ordinary differential equations, Chichester: Wiley, 2003, pp181 − 196.
[9] J.R. Dormand and P.J. Prince, "A family of embedded Runge-Kutta formulae", Journal of Computational and Applied Mathematics, 6 (1980) 19. Note that DOPRI853 is actually an embedded triple (8th-order, 5thorder and 3rd-order) but it is only the fifth- and eighth-order components that interest us here.
[10] T.E. Hull, W.H. Enright, B.M Fellen, and A.E. Sedgwick, "Comparing numerical methods for ordinary differential equations", SIAM Journal of Numerical Analysis, 9, 4 (1972) 603.