Butcher Table

Butcher tableau is essentially a compact recipe card that defines a specific Runge-Kutta method for solving ordinary differential equations. It neatly organizes all the coefficients you need for the calculation into a single table.

Definition

A Butcher Table represents a Runge-Kutta method for solving ordinary differential equations $y^{\prime }=f(t,y)$. It encodes how to compute intermediate stages and combine them to advance the solution.

General Form

$$\begin{array}{cc}\mathbf{c}& A\\ & {\mathbf{b}}^T\end{array}=\begin{array}{ccccc}{c}_1& a_{11}& a_{12}& \cdots & a_{1s}\\ c_2& a_{21}& a_{22}& \cdots & a_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ c_s& a_{s1}& a_{s2}& \cdots & a_{ss}\\ & b_1& b_2& \cdots & b_s\end{array}$$

• $C$→ time increments for intermediate stages
• $a$ → coefficients for combining previous $k$ values in each stage
• $b$ → final weights for combining all $k$ values to get $y_{i+1}$
• $k$ → intermediate slope evaluations