CODEWITHSUNDEEP
matlabnumerical-integration
anonymous
anonymous

Reputation: 115

Numerical integration of a double integral using Gauss-Legendre quadrature

I already have a code for generating the weights and abscissas for the Gaussian-Legendre quadrature rule for integration but I am having difficulty in using it for double integrals.

To generate the weights and abscissas I use the following notation [w,x]=leg(x1,x2,n) where w are the weights, x are the abscissas, x1 is the lower limit of the integral, x2 is the upper limit, and n is the number of quadrature points.

I'll just give a simple example so that it can help me understand the idea. Suppose I have the following integral $\int_0^1 \int_0^1 x^2 y^2 dx dy$ How do I implement this?

Thanks in advance.

Upvotes: 0

Views: 1561

Answers (1)

Nico Schlömer
Nico Schlömer

Reputation: 58721

Double integrals are integrals over a rectangular domain. They can either be treated with dedicated rectangular schemes or – what you attempt to do – with product schemes from one-dimensional quadrature. You could even mix and match two different schemes here.

The points of the product schemes are the 1D points in a Cartesian product, i.e., for every x_i from scheme 1 and y_j from scheme 2, (x_i, y_j) is a point in the product scheme. The weights are the product of the two corresponding weights.

enter image description here

If you want to make it easy on you, you could use quadpy (a project of mine):

import numpy
import quadpy

quadpy.quadrilateral.integrate(
    lambda x: numpy.exp(x[0]),
    [[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0]],
    quadpy.quadrilateral.Product(quadpy.line_segment.GaussLegendre(4))
    )

Upvotes: 0

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