Abstract
In this paper an automatic technique for handling discontinuous IVPs when they are solved by means of adaptive Runge–Kutta codes is proposed. This technique detects, accurately locates and passes the discontinuities in the solution of IVPs by using the information generated by the code along the numerical integration together with a continuous interpolant of the discrete solution. A remarkable feature is that it does not require additional information on the location of the discontinuities. Some numerical experiments are presented to illustrate the reliability and efficiency of the proposed algorithms.
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M. Calvo, J.I. Montijano and L. Rández, A fifth-order interpolant for the Dormand and Prince Runge–Kutta method, J. Comput. Appl. Math. 29 (1990) 91–100.
M. Calvo, J.I. Montijano and L. Rández, On the numerical solution of IVPs with discontinuities by adaptive Runge–Kutta codes, Technical Report, Department of Matemática Aplicada (2002), http://pcmap.unizar.es/∼odes/.
M.B. Carver, Efficient integration over discontinuities in ordinary differential equation simulations, Math. Comput. Simuluation 20(3) (1978) 190–196.
J. Dormand and P. Prince, A family of embedded Runge–Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19–26.
D. Ellison, Efficient automatic integration of ordinary differential with discontinuities, Math. Comput. Simuluation 23(1) (1981) 12–20.
W.H. Enright, K.R. Jackson, S.P. Nørsett and P.G. Thomsen, Effective solution of discontinuous IVPs using a Runge–Kutta formula pair with interpolants, Appl. Math. Comput. 27 (1988) 313–335.
C.W. Gear and O. Østerby, Solving ordinary differential equations with discontinuities, ACM Trans. Math. Software 10(1) (1984) 23–44.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems (Springer, Berlin, 1987).
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential–Algebraic Problems (Springer, Berlin, 1991).
R. Mannshardt, One-step methods of any order for ordinary differential equations with discontinuous right-hand sides, Numer. Math. 31 (1978) 131–152.
P.G. O'Regan, Step size adjustement at discontinuities for fourth order Runge–Kutta methods, Comput. J. 13(4) (1970) 401–404.
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Calvo, M., Montijano, J. & Rández, L. On the Solution of Discontinuous IVPs by Adaptive Runge–Kutta Codes. Numerical Algorithms 33, 163–182 (2003). https://doi.org/10.1023/A:1025507920426
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DOI: https://doi.org/10.1023/A:1025507920426