Gauss–Legendre method

In numerical analysis and scientific computing, the Gauss–Legendre methods are a family of numerical methods for ordinary differential equations. Gauss–Legendre methods are implicit Runge–Kutta methods. More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s.[1]

All Gauss–Legendre methods are A-stable.[2]

The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is:

1/2	1/2
	1

The Gauss–Legendre method of order four has Butcher tableau:

${\tfrac {1}{2}}-{\tfrac {1}{6}}{\sqrt {3}}$	${\tfrac {1}{4}}$	${\tfrac {1}{4}}-{\tfrac {1}{6}}{\sqrt {3}}$
${\tfrac {1}{2}}+{\tfrac {1}{6}}{\sqrt {3}}$	${\tfrac {1}{4}}+{\tfrac {1}{6}}{\sqrt {3}}$	${\tfrac {1}{4}}$
	${\tfrac {1}{2}}$	${\tfrac {1}{2}}$

The Gauss–Legendre method of order six has Butcher tableau:

${\tfrac {1}{2}}-{\tfrac {1}{10}}{\sqrt {15}}$	${\tfrac {5}{36}}$	${\tfrac {2}{9}}-{\tfrac {1}{15}}{\sqrt {15}}$	${\tfrac {5}{36}}-{\tfrac {1}{30}}{\sqrt {15}}$
${\tfrac {1}{2}}$	${\tfrac {5}{36}}+{\tfrac {1}{24}}{\sqrt {15}}$	${\tfrac {2}{9}}$	${\tfrac {5}{36}}-{\tfrac {1}{24}}{\sqrt {15}}$
${\tfrac {1}{2}}+{\tfrac {1}{10}}{\sqrt {15}}$	${\tfrac {5}{36}}+{\tfrac {1}{30}}{\sqrt {15}}$	${\tfrac {2}{9}}+{\tfrac {1}{15}}{\sqrt {15}}$	${\tfrac {5}{36}}$
	${\tfrac {5}{18}}$	${\tfrac {4}{9}}$	${\tfrac {5}{18}}$

The computational cost of higher-order Gauss–Legendre methods is usually too high, and thus, they are rarely used.[3]

Notes

Iserles 1996, p. 47
Iserles 1996, p. 63
Iserles 1996, p. 47

gollark: Because modern silicon processes are *really* energy-intensive.

gollark: Do the energy cost estimates for cryptocurrency factor in the cost of manufacturing GPUs (and ASICs)?

gollark: https://www.iea.org/commentaries/the-carbon-footprint-of-streaming-video-fact-checking-the-headlines

gollark: I looked this up quickly, and apparently the estimates use unrealistically high power consumption figures.

gollark: And they're actively working to decrease power consumption since it costs them things, while, again, mining basically just consumes as much as possible.

References

Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.

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[1] Iserles 1996, p. 47

[2] Iserles 1996, p. 63

[3] Iserles 1996, p. 47