Efficient integration of ordinary differential equations by transformation.

*(English)*Zbl 0686.65035The paper deals with the numerical solution of initial value problems of ordinary differential equations with a parameter \(\epsilon\) such as \(y'=\epsilon f(x,y,\epsilon).\) Under certain assumptions this problem becomes easier for the typical numerical method as \(\epsilon \to 0,\) still the problem might be difficult for non-vanishing \(\epsilon\). However, if the limit problem with \(\epsilon =0\) may be solved analytically, it is possible to transform the original problem to a form which is easier for numerical methods. For practical purposes this transformation must be well-conditioned.

The paper gives as beautiful example for this method a stiff ordinary differential equation with slowly varying Jacobian and refers to similar techniques by J. D. Lawson [SIAM J. Numer. Anal. 4, 372-380 (1967; Zbl 0223.65030)] and P. Deuflhard [Z. Angew. Math. Phys. 30, 177- 189 (1979; Zbl 0406.70012); Celestial Mech. 21, 213-223 (1980; Zbl 0422.70019)]. The displacement of a nonlinear spring and Bessel’s equation illustrate the numerical behaviour of this technique and the improvements on the computation time. Compared to other approaches one of the main advantages of this method is the possibility to transform the equation and to use existing codes, it is not necessary to develop new software, whereas the main problems are the analytic solution and the condition of the transformation.

The paper gives as beautiful example for this method a stiff ordinary differential equation with slowly varying Jacobian and refers to similar techniques by J. D. Lawson [SIAM J. Numer. Anal. 4, 372-380 (1967; Zbl 0223.65030)] and P. Deuflhard [Z. Angew. Math. Phys. 30, 177- 189 (1979; Zbl 0406.70012); Celestial Mech. 21, 213-223 (1980; Zbl 0422.70019)]. The displacement of a nonlinear spring and Bessel’s equation illustrate the numerical behaviour of this technique and the improvements on the computation time. Compared to other approaches one of the main advantages of this method is the possibility to transform the equation and to use existing codes, it is not necessary to develop new software, whereas the main problems are the analytic solution and the condition of the transformation.

Reviewer: C.H.Cap

##### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

##### Keywords:

stiff system; parameter; transformation; slowly varying Jacobian; nonlinear spring; Bessel’s equation
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\textit{L. F. Shampine} and \textit{W. Zhang}, Comput. Math. Appl. 15, No. 3, 213--220 (1988; Zbl 0686.65035)

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##### References:

[1] | Shampine, L.F., Efficiency of phase function methods for Sturm-Liouville eigenvalues, J. inst. maths. applic., 23, 413-420, (1979) · Zbl 0419.65053 |

[2] | Deuflhard, P., A study of extrapolation methods based on multistep schemes without parasitic solutions, Zamp, 30, 177-189, (1979) · Zbl 0406.70012 |

[3] | Deuflhard, P., Kepler discretization in regular celestial mechanics, Celest. mech., 21, 213-223, (1980) · Zbl 0422.70019 |

[4] | Gautschi, W., Numerical integration of odes based on trigonometric polynomials, Numer. math., 3, 381-397, (1961) · Zbl 0163.39002 |

[5] | Stiefel, E.; Bettis, D.G., Stabilization of cowells’ method, Numer. math., 13, 154-175, (1969) · Zbl 0219.65062 |

[6] | Hull, T.E.; Enright, W.H.; Fellen, B.M.; Sedgwick, A.E., Comparing numerical methods for ordinary differential equations, SIAM J. numer. analysis, 9, 603-637, (1972) · Zbl 0221.65115 |

[7] | Krogh, F.T., On testing a subroutine for the numerical integration of ordinary differential equations, J. ACM, 20, 545-562, (1973) · Zbl 0292.65039 |

[8] | Lawson, J.D., Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. numer. analysis, 4, 372-380, (1967) · Zbl 0223.65030 |

[9] | Gragg, W.B., On extrapolation algorithms for ordinary differential equations, SIAM J. numer. analysis, 2, 384-403, (1965) · Zbl 0135.37803 |

[10] | Bauer, H.-J., Entwicklung leistungsfähiger extrapolationscodes, () |

[11] | Hairer, E.; Norsett, S.P.; Wanner, G., Solving ordinary differential, equations I nonstiff problems, (1987), Springer Berlin · Zbl 0638.65058 |

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