Klecka's tau

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.

All Gauss–Legendre methods are A-stable.

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:

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

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

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

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

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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