Gaussian Quadrature
Integration rules that achieve the highest possible accuracy for a given number of points by optimizing both nodes and weights.
Optimizing the sample points
A quadrature rule approximates an integral as a weighted sum of the integrand at chosen points. Rules like the trapezoidal or Simpson's rule fix the points at equal spacing and choose only the weights. Gaussian quadrature treats both the points (nodes) and the weights as free parameters, and chooses them to integrate polynomials of as high a degree as possible exactly. With n points it integrates all polynomials up to degree 2n minus 1 exactly, twice the degree a fixed-point rule of the same size achieves.
The connection to orthogonal polynomials
The optimal nodes turn out to be the roots of a family of orthogonal polynomials matched to the integration interval and weight function. For the standard interval with unit weight these are the Legendre polynomials, giving Gauss-Legendre quadrature. Other weight functions give other families: Gauss-Hermite for integrals over the whole line with a Gaussian weight, Gauss-Laguerre for the half-line with an exponential weight, and Gauss-Chebyshev for a specific singular weight.
- Gauss-Legendre: finite interval, unit weight, general smooth integrands
- Gauss-Hermite: whole line with Gaussian weight, natural for velocity-space moments
- Gauss-Laguerre: half-line with exponential weight
- Gauss-Lobatto: includes the endpoints, useful when boundary values are needed
Strengths and limits
Gaussian quadrature is extraordinarily accurate for smooth integrands, converging exponentially fast as points are added. Its weakness is that the nodes for n points are unrelated to those for a different n, so refining the rule discards all previous evaluations. Gauss-Kronrod rules fix this by extending a Gauss rule with points that reuse the originals, enabling the error estimation used in adaptive quadrature.
In fusion computation
Gaussian quadrature is the standard for the element integrals in the finite element method and for velocity-space integrals in kinetic plasma theory, where Gauss-Hermite and Gauss-Laguerre rules match the Maxwellian weight and compute moments such as density, flow, and pressure accurately with few points.