High-accuracy stable difference schemes for well-posed NBVP.

*(English)*Zbl 1179.65056Adamyan, Vadim (ed.) et al., Modern analysis and applications. The Mark Krein centenary conference. Volume 2: Differential operators and mechanics. Papers based on invited talks at the international conference on modern analysis and applications, Odessa, Ukraine, April 9–14, 2007. Basel: Birkhäuser (ISBN 978-3-7643-9920-7/v. 2; 978-3-7643-9921-4/ebook; 978-3-7643-9924-5/set). Operator Theory: Advances and Applications 191, 229-252 (2009).

Summary: The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP)

$${v}^{\text{'}}\left(t\right)+Av\left(t\right)=f\left(t\right)(0\le t\le 1),v\left(0\right)=v\left(\lambda \right)+\mu ,\phantom{\rule{1.em}{0ex}}0<\lambda \le 1$$

or the differential equation in an arbitrary Banach space $E$ with the strongly positive operator $A$ are presented. The construction of these difference schemes is based on the Padé difference schemes for the solutions of the initial-value problem for the abstract parabolic equation and the high order approximation formula for

$$v\left(0\right)=v\left(\lambda \right)+\mu \xb7$$

The stability, the almost coercive stability and coercive stability of these difference schemes are established.

##### MSC:

65J08 | Abstract evolution equations (numerical methods) |

34G10 | Linear ODE in abstract spaces |

35K90 | Abstract parabolic equations |

65M06 | Finite difference methods (IVP of PDE) |

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

65L12 | Finite difference methods for ODE (numerical methods) |

47D06 | One-parameter semigroups and linear evolution equations |

65L20 | Stability and convergence of numerical methods for ODE |