---
title: Gauss–Legendre Collocation
url: https://www.emergentmind.com/topics/gauss-legendre-collocation
type: topic
---

# Gauss–Legendre Collocation

Gauss–Legendre Collocation is a high-order spectral discretization strategy that exploits the properties of Legendre polynomial roots (the Gauss–Legendre nodes) for accurate, stable approximation of solutions to ODEs, PDEs, optimal control problems, and integral equations. The method is underpinned by exact quadrature properties, diagonal mass matrices, generalized summation-by-parts (SBP) operators, and strong connections to symplectic time integrators and entropy stability. Gauss–Legendre collocation forms the foundation of efficient and accurate schemes for nonlinear conservation laws, high-order discontinuous Galerkin (DG) methods, and direct collocation in trajectory optimization, and has well-quantified error and convergence properties.

## 1. Definition and Key Algebraic Structures

Gauss–Legendre collocation relies on collocation at the roots of the Legendre polynomial $P_N(x)$ in $(-1,1)$, denoted $\{x_i\}_{i=1}^N$. These nodes, together with their positive quadrature weights $\{w_i\}_{i=1}^N$,
satisfy
\[
\int_{-1}^{1} f(x)\,dx = \sum_{i=1}^N w_i\,f(x_i)
\]
for all polynomials $f$ of degree $\le 2N-1$ [2203.06062, 1809.01178, 1802.00915].

**Lagrange interpolation** uses basis polynomials $\ell_j(x)$:
\[
\ell_j(x) = \prod_{\substack{1\le i\le N \\ i\neq j}} \frac{x-x_i}{x_j-x_i}
\]
which interpolate data at the nodes. This generates an interpolation operator and a **discrete derivative (differentiation) matrix**,
\[
D_{ij} = \ell_j'(x_i)
\]
so that, for nodal values $u_j = u(x_j)$,
\[
(\partial_x u_N)(x_i) = \sum_{j=1}^N D_{ij} u_j \approx u'(x_i)
\]
[2203.06062, 1809.01178, 1304.0974].

The mass matrix $W=\operatorname{diag}(w_1,\ldots,w_N)$, and the differentiation matrix $D$, together satisfy a **generalized summation-by-parts (SBP) property**,
\[
W D + (W D)^T = B
\]
with $B_{11}=-1$, $B_{NN}=+1$, $B_{ij}=0$ otherwise; equivalently, $D+D^T = W^{-1}B$ [2203.06062, 1809.01178].

## 2. Applications in Discontinuous Galerkin and Spectral Element Methods

Gauss–Legendre collocation underpins discontinuous Galerkin spectral element methods (DGSEM) with volume nodes at Gauss points, exploiting the diagonal mass matrix and high quadrature accuracy.

The elementwise semidiscrete DGSEM collocation for a conservative law
\[
\partial_t\mathbf{u} + \partial_x \mathbf{f}(\mathbf{u}) = 0
\]
is, at node $i$,
\[
J\,w_i\,\dot{\mathbf{u}}_i + \sum_{j=1}^N \hat{S}_{ij}\,\mathbf{f}^*_{(i,j)}
- \ell_i(-1)\,\mathbf{f}^*_{i,L} + \ell_i(+1)\,\mathbf{f}^*_{i,R} = 0
\]
where $\hat{S}=2Q-\hat{B}$, $Q=WD$, the $\mathbf{f}^*_{(i,j)}$ are symmetric, consistent two-point volume fluxes, and face fluxes use entropy projections [2203.06062, 1809.01178, 2504.00173].

**Entropy-stable extensions** are constructed via entropy-conservative volume and surface fluxes satisfying the Tadmor condition, with dissipation at interfaces to enforce entropy inequalities. The SBP property is essential for discrete entropy conservation [2203.06062, 1809.01178, 2504.00173].

On curvilinear meshes, Jacobian and metric terms are embedded via isoparametric mappings, and **discrete metric identities** must be enforced for free-stream preservation and entropy stability, as in hexahedral elements [2203.06062].

## 3. Comparison with Legendre–Gauss–Lobatto Collocation and Numerical Performance

Gauss–Legendre collocation (Gauss) contrasts with Legendre–Gauss–Lobatto (LGL) collocation in several aspects:
- **Degree of precision**: Gauss quadrature integrates to degree $2N-1$ with $N$ points; LGL integrates to the same degree with $N+1$ points (endpoints included).
- **Mass matrix**: Both have diagonal mass matrices under polynomial collocation.
- **SBP enforcement**: LGL is “classical” since endpoints coincide with boundaries; Gauss SBP is “generalized” and requires face interpolation [2203.06062, 1809.01178, 2504.00173].
- **Accuracy and efficiency**: For a fixed number of degrees of freedom, Gauss-based schemes achieve lower error constants than LGL, especially on curved or non-affine meshes. Gauss schemes have a more severe timestep restriction but outperform LGL in terms of error per computational cost for prescribed accuracy [2203.06062, 2504.00173, 1809.01178].

On highly distorted or curved meshes, Gauss–Legendre collocation avoids loss of convergence order that may occur with LGL points due to under-integration of geometric terms [1809.01178].

## 4. Extensions: Optimal Control, Hamiltonian Integration, and Fractional Problems

Gauss–Legendre collocation is a cornerstone of pseudospectral and direct collocation methods in optimal control. The approach transcribes the continuous OCP into a finite-dimensional NLP using the differentiation matrix, quadrature, and boundary constraints. Key features include:
- **Exponential convergence** for analytic solutions in the sup-norm, as established in optimal control theory [1507.08263, 2406.06824].
- **Modified Gauss–Legendre collocation** for nonsmooth/bang–bang optimal control introduces switch-time variables, extra control variables at endpoints, and additional collocation equations to accurately locate discontinuities [2406.06824].
- **Second-order trajectory optimization** uses a Gauss–Legendre pseudospectral discretization that maintains consistency between states and their derivatives at collocated points, crucial in robotic and mechanical systems [2302.09036].

**Hamiltonian problems:** The Gauss–Legendre collocation scheme corresponds to the classical Gauss Runge–Kutta methods, having order $2s$ for $s$ nodes, perfect A-stability, and symplecticity. These conserve quadratic invariants exactly and preserve energy to high accuracy for low-degree polynomial Hamiltonians. Energy-preserving symplectic perturbations and extensions (HBVMs) further exploit the collocation structure [1002.2727, 1005.1930, 1304.0974].

**Fractional and B-spline collocation:** Gauss–Legendre collocation is also employed to solve fractional integral equations with exponential convergence in $L^2$, and for Schrödinger-type boundary value and eigenvalue problems, where it excels—provided the function can be well resolved by the chosen basis [1802.00915, 2007.11312].

## 5. Superconvergence, Error Analysis, and Implementation

Superconvergence phenomena are documented in various contexts:
- **1D elliptic problems:** The $C^1$ Gauss collocation method achieves solution and derivative errors of order $2k-2$ at all mesh nodes for degree $k$. At selected internal Jacobi or Lobatto nodes, higher pointwise superconvergence rates are proved [2002.02266].
- **DG and spectral element methods:** Both theoretical and numerical results show spectral (exponential) accuracy for smooth problems, and favorable convergence constants for Gauss–Legendre over equispaced or nonlinearly distributed points [1002.2727, 1809.01178, 2504.00173].

**Implementation** leverages the structure of the differentiation matrix, diagonal mass matrix, and sparsity. Efficient splitting strategies for Newton-type solvers in Runge–Kutta or collocation context exploit the block structure of the Butcher matrix in Gauss–Legendre schemes and HBVMs [1304.0974].

## 6. Influence on High-Order PDE Discretization and Modern Research

Gauss–Legendre collocation undergirds cutting-edge research in entropy-stable high-order DG methods for compressible turbulence and MHD:
- **Entropy-stable DGSEM** built on Gauss nodes yields higher accuracy per degree of freedom than LGL-based methods for compressible turbulence, with sharper shock-capturing and reduced dissipation [2504.00173, 2203.06062].
- **Stabilization techniques**: Convex FV/DG blending and entropy-conservative/splitting form discretizations harness the SBP properties of Gauss collocation for robust high-order simulation of wall-bounded and separated flows [2504.00173].
- **SBP operators on Gauss nodes** are portable to tensor-product elements in multiple dimensions, enabling robust, accurate methods for curved geometries with discrete geometric conservation [1809.01178, 2203.06062].

Gauss–Legendre collocation is pivotal for contemporary high-order numerical PDE and optimal control solvers, with ongoing development in generalized SBP operators and adaptation to broader problem classes.

---

**References:**  
- [2203.06062]  
- [1809.01178]  
- [2504.00173]  
- [1002.2727]  
- [1304.0974]  
- [1005.1930]  
- [1802.00915]  
- [1507.08263]  
- [2406.06824]  
- [2302.09036]  
- [2002.02266]  
- [2007.11312]  
- [2011.02850]

Source: https://www.emergentmind.com/topics/gauss-legendre-collocation