[Paper Review] Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates

Ed Bueler|arXiv (Cornell University)|Sep 23, 2004

Numerical methods for differential equations28 references18 citations

TL;DR

This paper presents a Chebyshev collocation method for solving linear, periodic ordinary and delay differential equations (DDEs) with rigorous a posteriori error estimates. It introduces a novel a posteriori eigenvalue perturbation theorem for operators on Hilbert spaces, enabling verified bounds on the spectral radius of the monodromy operator, thus providing mathematically rigorous stability certification for DDEs with integer delays.

ABSTRACT

We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered.

Motivation & Objective

• To develop a reliable, verified numerical method for computing solutions and stability properties of linear, periodic DDEs with integer delays.
• To provide a posteriori error estimates for the solution of initial value problems in DDEs using spectral collocation.
• To derive a posteriori bounds on the distance between eigenvalues of the monodromy operator and those of its matrix approximation.
• To extend the method to systems of DDEs and demonstrate its utility on a delayed, damped Mathieu equation.
• To address technical challenges in polynomial evaluation, $L^∞$ norm estimation, and condition number control in spectral methods.

Proposed method

• Uses Chebyshev collocation on Gauss-Lobatto points to discretize DDEs and approximate fundamental solutions with high-order accuracy.
• Applies a posteriori error estimation techniques to bound the $L^\infty$-norm of the residual in initial value problems.
• Constructs a monodromy matrix approximation by solving sequential ODEs on subintervals using spectral collocation and interpolating solutions at collocation points.
• Generalizes the Bauer-Fike theorem to operators on Hilbert spaces (Theorem 11) to bound eigenvalue perturbations of the monodromy operator.
• Employs a posteriori bounds on the condition number of the eigenvector matrix to control eigenvalue accuracy in the discrete approximation.
• Validates the method on a delayed Mathieu equation, showing exponential convergence of error bounds with increasing $N$.

Experimental results

Research questions

• RQ1 How can a posteriori error estimates be rigorously derived for Chebyshev collocation solutions of linear DDEs with integer delays?
• RQ2 What is the relationship between the eigenvalues of the monodromy operator and those of its matrix approximation, and how can this be bounded a posteriori?
• RQ3 Can the spectral properties of the monodromy operator be verified with mathematically rigorous error bounds using spectral collocation?
• RQ4 How do the accuracy and condition number of the monodromy matrix approximation scale with the number of collocation points $N$?
• RQ5 To what extent can the method be generalized to non-integer delays, neutral DDEs, or functional differential equations?

Key findings

• The a posteriori error bound for the DDE initial value problem decays exponentially with the number of collocation points $N$, as demonstrated in numerical experiments.
• For the parameter set $(a,b)=(-1.1,1)$, the largest eigenvalue of the monodromy matrix is $0.9369$, and the a posteriori bound on the distance to the true eigenvalue is $r=0.0434$, proving stability since $|\mu_1| + r < 1$.
• The radius of the error discs around computed eigenvalues decays roughly exponentially with $N$, with floating-point accuracy limited around $10^{-5}$ for $N \gtrapprox 220$.
• The method successfully produces a stability chart for equation (1) by computing monodromy matrix eigenvalues and their verified error bounds pixel by pixel.
• The eigenvalue perturbation theorem (Theorem 14) provides a rigorous, computable bound on the distance between a true eigenvalue of the monodromy operator and the nearest computed eigenvalue of the matrix approximation.
• The approach is generalizable to systems of DDEs, as demonstrated on a delayed, damped Mathieu equation with similar convergence and verification properties.