论文标题
抛物线偏微分方程的稳定歧管的验证数值近似
Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations
论文作者
论文摘要
本文开发了经过验证的计算方法，用于研究抛物线PDE的平衡溶液的无限尺寸稳定歧管，从而综合了由于数值近似而导致的不同误差。为了构建我们的近似值，我们将稳定的歧管分解为三个组件：有限的尺寸慢组分，快速限制的尺寸组件和强大的无限无限尺寸“尾巴”。我们在有限的尺寸投影中采用参数化方法来近似慢稳定的歧管以及所附的有限尺寸不变矢量束。这种近似提供了坐标的变化，该坐标在很大程度上消除了慢速稳定方向的非线性项。在此改编的坐标系中，我们应用Lyapunov-Perron方法，导致近似误差的数学严格界限。结果，我们获得的明显更加明显的界限比仅使用特征方向给出的线性近似所获得的明显范围。作为一个具体的示例，我们说明了1D Swift-Hohenberg方程的技术。
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional "tail". We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.
