Chebyshev-Weighted Collocation Scheme

• Chebyshev-weighted collocation scheme is a spectral method that leverages Chebyshev polynomials' weighted orthogonality for exponential convergence in solving differential equations.
• The method uses collocation at Chebyshev-Lobatto points to achieve robust discretization with minimal interpolation error and flexible enforcement of boundary conditions.
• The framework extends to multivariate, spline-based, and least squares formulations, enabling efficient uncertainty quantification and handling of singular or unbounded domains.

The Chebyshev-weighted collocation scheme constitutes a central class of high-order numerical methods for ordinary and partial differential equations (ODEs and PDEs), exploiting the optimal approximation properties and weighted orthogonality of Chebyshev polynomials. Collocation at Chebyshev points induces spectral (exponentially fast) convergence for analytic functions, robust conditioning, and flexibility in handling singular coefficients, boundary constraints, or unbounded domains. Chebyshev-weighted schemes extend to multivariate, multidomain, spectral tensor, or spline-based frameworks, and serve as a paradigm for both direct solvers and weighted least squares projection in uncertainty quantification.

1. Chebyshev Polynomials, Orthogonality, and Weighted Quadrature

Chebyshev polynomials of the first kind, $T_n(x)$, are defined on $[-1,1]$ by

$T_n(x) = \cos(n \arccos x), \quad n=0,1,2,\ldots.$

They satisfy a three-term recurrence and an orthogonality relation with respect to the Chebyshev weight $w(x) = (1-x^2)^{-1/2}$: $$\int_{-1}^1 T_r(x) T_s(x) w(x) \, dx = \begin{cases} 0, & r \ne s, \ \pi, & r = s = 0, \ \pi/2, & r = s > 0. \end{cases}$$ This weighted orthogonality directly informs quadrature and interpolation accuracy. Chebyshev-Gauss and Gauss-Lobatto nodes and weights are rooted in this structure, yielding exact quadrature for polynomials up to degree $2N-1$ or $2N$ depending on the grid (Bhowmik, 2014, Gheorghiu, 2020, Amodio et al., 2022).

2. Chebyshev Collocation Grids and Differentiation Matrices

Collocation at Chebyshev-Lobatto points,

$$x_j = \cos\biggl(\frac{j\pi}{N}\biggr), \quad j=0,\ldots,N,$$

distributes nodes with exponential density near the endpoints, minimizing interpolation error (Runge phenomenon) and providing stability for high-degree expansions (Bhowmik, 2014, Gheorghiu, 2020). The spectral differentiation matrix $D$ at these points has explicit entries: $$D_{ij} = \begin{cases} \frac{c_i}{c_j} \frac{(-1)^{i+j}}{x_i - x_j}, & i \ne j, \ -\sum_{k \ne i} D_{ik}, & i = j, \end{cases}$$ with $c_0 = c_N = 2$, $c_j = 1$ otherwise, and generalizations for higher derivatives via $D^{(m)} = D^m$ (Gheorghiu, 2020, Adak et al., 17 Dec 2025). This structure facilitates efficient enforcement of boundary conditions and operator discretization.

3. Spectral Collocation Formulation for ODEs and PDEs

In the Chebyshev-weighted collocation framework, the solution is represented as a truncated Chebyshev series: $u_N(x) = \sum_{k=0}^N a_k T_k(x)$ or in a barycentric/Lagrange interpolant form. The differential operator or residual is enforced to vanish at the collocation nodes,

$$\sum_{j=0}^{N} D_{ij} u_j = f(x_i, u_i), \quad i = 0, \ldots, N,$$

with explicit row modification to impose boundary or initial conditions (Bhowmik, 2014, Gheorghiu, 2020, Amodio et al., 2022, Hale, 2024). For nonlinear problems, these algebraic equations are solved by Newton-type methods, exploiting spectral conditioning.

For multivariate or tensor grids, Kronecker products of 1D differentiation and mass matrices are employed, extending the approach to high-dimensional PDEs and enabling direct space-time discretization (Adak et al., 17 Dec 2025, Adak et al., 2024).

4. Chebyshev-Weighted Collocation in Extended and Singular Settings

Several extensions of the core scheme adapt the Chebyshev-weighted approach to singularly perturbed, degenerate, or fractional-order operators:

• Rational Chebyshev and semi-infinite domains: Mapping $[0, \infty) \to [-1,1]$ and employing rational Chebyshev functions (e.g., $U_n^*(x)$ via $s=(x-1)/(x+1)$) and the corresponding induced weight $w^*(x)$ to maintain orthogonality and super-algebraic convergence (Parand et al., 2015).
• Canonical Complete Chebyshev (CCC) Spline Spaces: Construction of Chebyshev-weighted B-splines for collocation in singular or non-standard differential operators, incorporating underlying variable weights $u_1(x)$, and enabling banded-matrix solves with optimal conditioning (Bosner, 2021).
• Fractional and functional DDEs: Caputo derivatives of shifted Chebyshev polynomials and spectral collocation for delay/fractional equations, including weighted differentiation matrices and multidomain/stepwise procedures (Mousa-Abadian et al., 2019, Hale, 2024).

5. Stability, Conditioning, and Spectral Convergence

The Chebyshev-weighted collocation method is characterized by:

• Spectral convergence: For analytic solutions, Chebyshev coefficients $a_k$ obey $|a_k| = O(\rho^{-k})$ ($\rho>1$), and the maximum-norm error decays exponentially in $N$ (Bhowmik, 2014, Gheorghiu, 2020, Amodio et al., 2022, Adak et al., 17 Dec 2025).
• Conditioning: The collocation matrix's condition number grows algebraically ($O(N^2$) (Bhowmik, 2014)), far superior to equispaced grids. For multivariate least squares with Chebyshev points, quadratic scaling in the number of samples relative to polynomial space dimension ensures robust stability (Zhou et al., 2014).
• Arbitrary boundary and interface treatment: Collocation at Chebyshev points naturally accommodates both standard and non-standard boundary/initial conditions, including multidomain and piecewise-analytic scenarios (Hale, 2024).

6. Advanced Algorithms and Multivariate Weighted Least Squares

The Chebyshev-weighted framework extends to deterministic (Weil-type) multivariate collocation grids, enabling stable weighted least squares approximation in tensor or total-degree polynomial spaces. The construction uses deterministic phase-based grids asymptotically equidistributed with the Chebyshev measure, and explicit weights $w_k = \rho(\mathbf y_k)/\rho_c(\mathbf y_k)$ to target arbitrary measures or polynomial bases (Zhou et al., 2014).

Basis	Node set	Weight function
$T_n$ (univariate)	Lobatto, Gauss	$(1-x^2)^{-1/2}$
Tensor Chebyshev	Phase-deterministic	Product of $w(x)$
CCC-splines	Partition + weight	$u_1(x)$-induced

In least squares, $A=D^{T}WD$ is well-conditioned for $m \gtrsim N^2$ grid points ($N=$ dimension); convergence and stability are fully deterministic and extend to non-Chebyshev targets via preconditioning (Zhou et al., 2014).

7. Applications and Extensions

Chebyshev-weighted collocation has widespread impact:

• High-order BVPs and PDEs: Modeling fluid mechanics, wave propagation, and Maxwell's equations with space-time spectral accuracy (Bhowmik, 2014, Adak et al., 17 Dec 2025, Adak et al., 2024).
• Nonlinear and stiff problems: Fast Newton-Kantorovich solvers, multi-branch bifurcation computation, and robust handling of nonlinearity (Gheorghiu, 2020).
• Large-scale and high-dimensional systems: Use of tensor-train (TT) and Kronecker operators for exponential compression and scalable solvers in 4D space-time settings (Adak et al., 17 Dec 2025, Adak et al., 2024).
• Uncertainty quantification: Chebyshev-weighted multivariate polynomial least squares for deterministic uncertainty propagation and approximation with explicit convergence guarantees (Zhou et al., 2014).
• Singular problems and nonuniform weights: CCC-splines and product-form operators for singularly weighted or degenerate ODEs (Bosner, 2021, Parand et al., 2015).

In summary, the Chebyshev-weighted collocation scheme is foundational in contemporary spectral methods, combining analytically motivated node selection, operator discretization with weighted orthogonality, and extensions to multivariate, singular, and high-dimensional settings, all backed by explicit convergence theory and robust conditioning (Bhowmik, 2014, Parand et al., 2015, Gheorghiu, 2020, Amodio et al., 2022, Adak et al., 17 Dec 2025, Zhou et al., 2014, Bosner, 2021, Adak et al., 2024, Hale, 2024, Mousa-Abadian et al., 2019).