A spectral collocation method for functional and delay differential equations

Abstract

A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by non-smooth initial data. Geometric convergence is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous, and state-dependent delay. The framework is a natural extension of standard spectral collocation methods based on polynomial interpolants and can be readily incorporated into existing spectral discretisations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear functional and delay differential equations.

Figures (17)

• Figure 1: Left: MATLAB code for solving the ODE \ref{['eqn:example1']} using Chebyshev spectral collocation. The line [t,,w] = chebpts(n, [0,1]) (from Chebfun) returns the n-point Chebyshev grid on the interval $[0,1]$ and the associated barycentric weights (\ref{['eqn:chebwts']}). Centre: Exact solution (solid) and approximate solution (dots) using an $n =$ 12-point discretisation. Right: The solid line shows geometric convergence of the solution down to around machine precision, $M_\varepsilon = 2^{-52}$, as $n$ is increased. Here, and throughout, the reported error is the $\ell^\infty$ error at the interpolation points. The dashed line shows (again, here and throughout) $M_\varepsilon\times{\rm cond}(A)$ as a function of $n$.
• Figure 2: Left: MATLAB code for solving the DDE \ref{['eqn:example2']} using Chebyshev spectral collocation. The code is a simple modification of that from Figure \ref{['fig:example1']} for solving the ODE \ref{['eqn:example1']}. Centre: Solution using a 12-point Chebyshev grid. Right: Convergence of the solution (solid) and $M_\varepsilon\times{\rm cond}(A)$ (dashed) as $n$ is increased. As in the ODE case, geometric convergence to around the level of machine precision is observed. The condition number of the discretised DDE behaves similarly to the ODE case.
• Figure 3: Left: Block linear system resulting from the spectral collocation discretisation of \ref{['eqn:example3']}. The overline notation is used to indicate that the first row of the corresponding matrix is removed in order to implement boundary/continuity constraints via boundary bordering. The system is block lower triangular, and one could first solve for $\bm{y}_L$ and then $\bm{y}_R$, akin to the method of steps. However, this will not be the case in context of FDEs as we discuss shortly. Centre: Solution using 10- and 11-point Chebyshev grids on $[0,1/2]$ and $[1/2,1]$, respectively. Right: Convergence as $n$ is increased, where $n$ and $n+1$ are the discretisation sizes of the left and right subintervals, respectively. Using a multidomain discretisation on $[0,1/2]\cup[1/2,1]$ so that the solution is analytic on each subdomain results in geometric convergence (solid line). If only a single Chebyshev grid on $[0, 1]$ is used, then algebraic convergence is observed (dash-dotted line) due to the discontinuity in the derivative of the solution at $t = 1/2$.
• Figure 4: Left: Block discretisation of the DDE (\ref{['eqn:example3']}) on the interval $[0, 2]$ with 'breakpoints' at $0.5, 1, 1.5$. Here $P_{j,k}$ is given by $P(\bm{t}_j-\tfrac{1}{2};\bm{t}_k)$ and ${\bm{e}_n}_k$ denotes the $j$th column of the $n\times n$ identity matrix. Each nonzero entry will result in an essentially dense $n\times n$ block. Centre: Solution using 10-, 11-, 12-, 13-point grids on the respective subintervals (see Remark \ref{['remark1']}). Right: Geometric convergence as discretisation size is increased ($n$, $n+1$, $n+2$, $n+3$-point discretisations on each subinterval, respectively.)
• Figure 5: Left: Sparsity plot of the 12-point Chebyshev spectral discretisation. Here, even though the same value of $n$ is used on each subinterval, the interpolation operator is non-trivial (because the intervals are of differing widths), resulting in dense off-diagonal blocks of the discretisation. Centre: Analytic and computed solution to (\ref{['eqn:historyexample']}) on a 12-point grid. Right: Geometric convergence as the discretisation size is increased.

Theorems & Definitions (17)

• Example 1
• Example 2
• Example 3
• Remark 1
• Example 4
• Remark 2
• Remark 3
• Example 5
• Example 6
• Remark 4