= Two-node collocation A-stability criterion
{title2=$c_1+c_2\geq1$}
For distinct nodes $c_1,c_2\in[0,1]$, the two-stage <collocation Runge-Kutta method> has <stability function>
$$
R(z)=\frac{1+(1-a)z+rz^2}{1-az+dz^2},\qquad a=(c_1+c_2)/2,\quad d=c_1c_2/2,\quad r=(1-c_1)(1-c_2)/2.
$$
The denominator has no zeros in the closed left half-plane. On the imaginary axis the denominator's squared modulus minus the numerator's is $(d^2-r^2)y^4$. Thus $d\geq r$, equivalently $c_1+c_2\geq1$, is necessary and sufficient for <A-stability>, using the <maximum modulus principle> and the bound at infinity.
Back to article page
