Strong contractivity properties of numerical methods for ordinary and delay differential equations.

*(English)*Zbl 0749.65042Some new stability and contractivity concepts are introduced for ordinary differential equations for linear and nonlinear test problems. The linear problem is the standard constant coefficient test problem with the addition of a forcing term. The motivation for these test problems is to study stability properties of methods for delay differential equations.

Characterizations of these concepts are provided for the case of Runge- Kutta methods and some relationships are established among various concepts of stability for ordinary and delay differential equations. Runge-Kutta methods with upto two stages are studied in detail and the Lobatto III-C method is shown to be the only two-stage formula of order 2 satisfying certain stability properties.

Characterizations of these concepts are provided for the case of Runge- Kutta methods and some relationships are established among various concepts of stability for ordinary and delay differential equations. Runge-Kutta methods with upto two stages are studied in detail and the Lobatto III-C method is shown to be the only two-stage formula of order 2 satisfying certain stability properties.

Reviewer: G.Hall (Manchester)

##### MSC:

65L05 | Numerical methods for initial value problems |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems, general theory |

34K05 | General theory of functional-differential equations |

##### Keywords:

systems; stability; contractivity; test problems; linear problem; delay differential equations; Runge-Kutta methods; Lobatto III-C method; two- stage formula
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\textit{A. Bellen} and \textit{M. Zennaro}, Appl. Numer. Math. 9, No. 3--5, 321--346 (1992; Zbl 0749.65042)

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##### References:

[1] | Al-Mutib, A.N., Stability properties of numerical methods for solving delay differential equations, J. comput. appl. math., 10, 71-79, (1984) · Zbl 0542.65040 |

[2] | Barwell, V.K., Special stability problems for functional differential equations, Bit, 15, 130-135, (1975) · Zbl 0306.65044 |

[3] | Bellen, A., Constrained mesh methods for functional differential equations, (), 42-70 · Zbl 0577.65125 |

[4] | Bickart, T.A., P-stable and P[α,β]-stable integration/interpolation methods in the solution of retarded differential-difference equations, Bit, 22, 464-476, (1982) · Zbl 0531.65044 |

[5] | Butcher, J.C., The numerical analysis of ordinary differential equations, (1987), Wiley Chichester · Zbl 0616.65072 |

[6] | Crouzeix, M.; Hundsdorfer, W.H.; Spijker, M.N., On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, Bit, 23, 84-91, (1983) · Zbl 0506.65030 |

[7] | Dekker, K.; Verwer, J.G., Stability of Runge-Kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057 |

[8] | Feldstein, A., Discretization methods for retarded ordinary differential equations, () |

[9] | in’t Hout, K.J., The stability of a class of Runge-Kutta methods for delay differential equations, Appl. numer. math., 9, 347-355, (1992), (this issue) · Zbl 0749.65048 |

[10] | in’t Hout, K.J.; Spijker, M.N.; in’t Hout, K.J.; Spijker, M.N., Stability analysis of numerical methods for delay differential equations, (), 807-814 · Zbl 0724.65084 |

[11] | Jackiewicz, Z., Asymptotic stability analysis of θ-methods for functional differential equations, Numer. math., 43, 389-396, (1984) · Zbl 0557.65047 |

[12] | Jackiewicz, Z., One step methods of any order for neutral functional differential equations, SIAM J. numer. anal., 21, 486-511, (1984) · Zbl 0562.65056 |

[13] | Jackiewicz, Z., Quasilinear multistep methods and variable step predictor-corrector methods for neutral functional differential equations, SIAM J. numer. anal., 23, 423-456, (1986) · Zbl 0602.65056 |

[14] | Kraaijevanger, J.F.B.M.; Schneid, J., On the unique solvability of the Runge-Kutta equations, () · Zbl 0703.65038 |

[15] | Liu, M.Z.; Spijker, M.N., The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. numer. anal., 10, 31-48, (1990) · Zbl 0693.65056 |

[16] | Tavernini, L., One step methods for the numerical solution of Volterra functional differential equations, SIAM J. numer. anal., 8, 786-795, (1971) · Zbl 0231.65070 |

[17] | Tavernini, L., Linear multistep methods for the numerical solution of Volterra functional differential equations, Appl. anal., 1, 169-185, (1973) · Zbl 0291.65020 |

[18] | Torelli, L., Stability of numerical methods for delay differential equations, J. comput. appl. math., 25, 15-26, (1989) · Zbl 0664.65073 |

[19] | Torelli, L., A sufficient condition for GPN-stability for delay differential equations, Numer. math., 59, 311-320, (1991) · Zbl 0712.65079 |

[20] | Watanabe, D.S.; Roth, M.G., The stability of difference formulas for delay differential equations, SIAM J. numer. anal., 22, 132-145, (1985) · Zbl 0571.65075 |

[21] | Zennaro, M., P-stability properties of Runge-Kutta methods for delay differential equations with nonsmooth solutions, Numer. math., 49, 305-318, (1986) · Zbl 0598.65056 |

[22] | Zverkina, T.S., A modification of finite difference methods for integrating ordinary differential equations with nonsmooth solutions, Zh. vycisl. mat. i mat. fiz., 4, 149-160, (1964), (in Russian) |

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