Editing Runge–Kutta methods (section)

==B-stability==
The ''A-stability'' concept for the solution of differential equations is related to the linear autonomous equation <math>y'=\lambda y</math>. {{harvtxt|Dahlquist|1963}} proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition. The corresponding concepts were defined as ''G-stability'' for multistep methods (and the related one-leg methods) and ''B-stability'' (Butcher, 1975) for Runge–Kutta methods. A Runge–Kutta method applied to the non-linear system <math>y'=f(y)</math>, which verifies <math>\langle f(y)-f(z),\ y-z \rangle\leq 0</math>, is called ''B-stable'', if this condition implies <math>\|y_{n+1}-z_{n+1}\|\leq\|y_{n}-z_{n}\|</math> for two numerical solutions.

Let <math>B</math>, <math>M</math> and <math>Q</math> be three <math>s\times s</math> matrices defined by
<math display=block>
\begin{align}
B & =\operatorname{diag}(b_1,b_2,\ldots,b_s), \\[4pt]
M & =BA+A^TB-bb^T, \\[4pt]
Q & =BA^{-1}+A^{-T}B-A^{-T}bb^TA^{-1}.
\end{align}
</math>
A Runge–Kutta method is said to be ''algebraically stable''<ref>{{harvnb|Lambert|1991|p=275}}</ref> if the matrices <math>B</math> and <math>M</math> are both non-negative definite. A sufficient condition for ''B-stability''<ref>{{harvnb|Lambert|1991|p=274}}</ref> is: <math>B</math> and <math>Q</math> are non-negative definite.