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Butcher Tableau

A Butcher tableau is a compact display of the coefficients of a Runge-Kutta method:

cA
b^T

For an s-stage method applied to y^'=f(t,y), the stages k_i for 1<=i<=s and the update are

k_i=f(t_n+c_ih,y_n+hsum_(j=1)^(s)a_(ij)k_j)
(1)
y_(n+1)=y_n+hsum_(i=1)^(s)b_ik_i.
(2)

The method is explicit when a_(ij)=0 for j>=i, in which case A is strictly lower triangular.

See also

Numerical Integration, Ordinary Differential Equation, Runge-Kutta Method

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References

Butcher, J. C. Numerical Methods for Ordinary Differential Equations, 3rd ed. Chichester, England: Wiley, 2016.Hairer, E.; Nørsett, S. P.; and Wanner, G. Solving Ordinary Differential Equations I. New York: Springer-Verlag, 1987.

Cite this as:

Weisstein, Eric W. "Butcher Tableau." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ButcherTableau.html

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