Nonlinear stability of general linear methods for neutral delay differential equations.

*(English)*Zbl 1167.65046The authors consider the numerical solutions of neutral delay differential equations in the general nonlinear form,

\[ \begin{cases} y'(t) = f(t,y(t), y(t-\tau), y'(t-\tau)), & t\geq 0,\\ y(t) = \phi(t), & t\leq 0. \end{cases} \] The main purpose is to study the stability of general linear methods, such as backward differentiation formulae (BDF’s), extended and modified extended BDF’s, parallel multi-value hybrid methods, Runge-Kutta, multi-step Runge-Kutta. A general class of methods to which the results can be applied is described in section 3 of the paper, while in section 2 the main theoretical results attending to some required properties of the components of the differential equation are given.

Section 3 also contains many definitions and comments related to stability that are enlarged with the introduction of the new concepts of GS(p), GAS(p), and weak GAS(p) stability of general linear methods with linear interpolation. Many implications between these and other concepts of stability, as well as the relation with \((k,p,0)\)-algebraic stability are investigated, specially in section 4, where the main results and proofs of the paper are given.

The paper ends with some examples of implementation of multi-step Runge-Kutta methods and some numerical experiments to support the results of the work.

\[ \begin{cases} y'(t) = f(t,y(t), y(t-\tau), y'(t-\tau)), & t\geq 0,\\ y(t) = \phi(t), & t\leq 0. \end{cases} \] The main purpose is to study the stability of general linear methods, such as backward differentiation formulae (BDF’s), extended and modified extended BDF’s, parallel multi-value hybrid methods, Runge-Kutta, multi-step Runge-Kutta. A general class of methods to which the results can be applied is described in section 3 of the paper, while in section 2 the main theoretical results attending to some required properties of the components of the differential equation are given.

Section 3 also contains many definitions and comments related to stability that are enlarged with the introduction of the new concepts of GS(p), GAS(p), and weak GAS(p) stability of general linear methods with linear interpolation. Many implications between these and other concepts of stability, as well as the relation with \((k,p,0)\)-algebraic stability are investigated, specially in section 4, where the main results and proofs of the paper are given.

The paper ends with some examples of implementation of multi-step Runge-Kutta methods and some numerical experiments to support the results of the work.

Reviewer: Josep J. Masdemont (Barcelona)

##### MSC:

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

65L05 | Numerical methods for initial value problems |

34K40 | Neutral functional-differential equations |

34K50 | Stochastic functional-differential equations |

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

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

##### Keywords:

nonlinear neutral delay differential equations; general linear methods; numerical stability; algebraic stability; backward differentiation formulae; multi-step Runge-Kutta methods; numerical experiments
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\textit{W.-S. Wang} et al., J. Comput. Appl. Math. 224, No. 2, 592--601 (2009; Zbl 1167.65046)

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