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**An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays.**
*(English)*
Zbl 1064.65078

For the numerical solution of (ordinary or partial) differential equations with fixed or distributed delay, it is important to have detailed knowledge of the stability properties of the discretization methods. In particular, the delay-dependent stability regions need to be investigated. The authors begin by discussing the ordinary differential equation setting and first analyze the analytical stability region. Then they prove that the trapezoidal rule preserves many properties of this analytical stability region, including the particularly important feature of asymptotic stability.

In the second part of the paper, the case of parabolic partial differential equations (PDEs) is investigated. Here the analytical, semi-discrete and fully discrete stability regions are discussed. The derivative with respect to the space variable is discretized by a standard central difference and the time derivative is approximated by the trapezoidal rule. It turns out that, in contrast to the case of PDEs without delay, the spatial discretization leads to a reduction of the size of the stability region. As a consequence, the fully discrete scheme cannot preserve the asymptotic stability completely.

In the second part of the paper, the case of parabolic partial differential equations (PDEs) is investigated. Here the analytical, semi-discrete and fully discrete stability regions are discussed. The derivative with respect to the space variable is discretized by a standard central difference and the time derivative is approximated by the trapezoidal rule. It turns out that, in contrast to the case of PDEs without delay, the spatial discretization leads to a reduction of the size of the stability region. As a consequence, the fully discrete scheme cannot preserve the asymptotic stability completely.

Reviewer: Kai Diethelm (Braunschweig)

### MSC:

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

35R10 | Partial functional-differential equations |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

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