Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 80 / 3

For a Runge-Kutta method with real stage matrix $A=(a_{ij})$ and weights $b_i$, defineThe method has algebraic stability of a Runge-Kutta method ifThe matrix condition means $z^{T}Mz\ge 0$ for every real vector $z$. Its significance is nonlinear contractivity: an algebraically stable method, on well-defined stage solutions, is B-stable for a dissipative vector field satisfying $⟨f(t,y)-f(t,z),y-z⟩\le 0$. This is distinct from scalar A-stability, which concerns the linear test equation. The contractivity proof in part (b) explains the matrix criterion directly.

The stated hypotheses are precisely the algebraic criterion in part (a). To prove its stability content, compare two Runge-Kutta method steps for the same dissipative vector field. Let $d=y_n-z_n$ be the input difference, $D_i=Y_i-Z_i$ the stage differences, and $F_i=f(t_n+c_{i}h,Y_i)-f(t_n+c_{i}h,Z_i)$. ThenExpanding the squared norm of $d_{+}$ givesIn each term of the linear sum, replace $d$ by $D_i-h{\sum }_j{a}_{ij}{F}_j$. Symmetrizing the resulting double sum proves the Runge-Kutta contractivity identity:Dissipativity makes $⟨D_i,F_i⟩\le 0$, and the weights are nonnegative. For the second term, positive semidefiniteness permits a factorization $M=C^{T}C$, givingFor $h\ge 0$ both contributions in the distance-change identity are nonpositive. Therefore $\Vert y_{n+1}-z_{n+1}\Vert \le \Vert y_n-z_n\Vert$. This proves the contractivity assertion behind algebraic stability, usually called the Butcher contractivity theorem. It assumes the compared implicit stage equations have solutions; the matrix condition by itself should not be substituted for a separate stage-solvability argument.

For the fourth-order two-stage Gauss collocation method, $b_1=b_2=1/2$, $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt{3}/6$ and $a_{21}=1/4+\sqrt{3}/6$. The weights are positive. For the matrix defining algebraic stability,andHence $M=0$ is positive semidefinite, so the method is algebraically stable. Its two-stage contractivity estimate follows from part (b). The PDF has $a_{22}=1/4$; the extra $\sqrt{3}/6$ attached to that entry in the converted TeX is a transcription error.