Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations
Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations
复制标题
计算时滞微分方程周期解的分段多项式配置的稳定性
DOI：
10.1007/s002110100313
复制
发表时间：
2002
影响因子：
2.1
通讯作者：
E. Doedel
中科院分区：
文献类型：
--
作者：
K. Engelborghs;E. Doedel
总结。证明了计算线性延迟微分方程渐近稳定和不稳定周期解的一类分段多项式配置方法在非均匀网格上的数值稳定性
点y(T)=a(T)y(T)+b(T)y(t-\tau)+f(T)$(周期)边值方法。例如，在研究计算非线性延迟方程周期解的配置法的数值稳定性时，就出现了这个方程。我们得到了标准配置算法和两个变种的收敛结果。特别地，给出了配置解与真解之差的估计式。对于标准配置格式，收敛结果是“无条件的”，也就是说，它们不需要网格比限制。文中还给出了支持理论结果的数值结果。
Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation
$\dot y(t) = a(t)y(t)+b(t)y(t-\tau) + f(t)$ by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given.