Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 63 / 2

The complete Butcher tableau in the PDF suppliesThese are the two Gauss nodes and the corresponding Gauss--Legendre Runge-Kutta method. We can verify its order directly rather than infer it from its name. Write $e=(1,1{)}^T$, $C=\mathrm{diag}(c)$ and let powers of $c$ be componentwise. The Butcher order conditions through order four areFor these coefficients $Ae=c$, $Ac=c^2/2$ and $c^2=c-e/6$. Also $b^T{c}^j=1/(j+1)$ for $j=0,1,2,3$, as follows by averaging the two values $1/2\pm s$. These identities give all eight displayed Butcher order conditions; for example $b^{T}CAc=b^T{c}^3/2=1/8$, $b^{T}A{c}^2=b^T(c^2/2-c/6)=1/12$, and $b^T{A}^{2}c=b^{T}A{c}^2/2=1/24$. They establish order at least four for smooth nonlinear ordinary differential equations.

To exclude order five, apply the method to the Dahlquist test equation. Solving its stage equations gives the stability functionThe exact solution has coefficient $1/120$ at degree five. Hence the method has order exactly four; its one-step defect is $O(h^5)$.

Use the stability function just calculated. Its denominator has its two zeros at $3\pm i\sqrt{3}$, so there is no pole in the closed left half-plane. Put $z=x+iy$, $N=1+z/2+z^2/12$ and $D=1-z/2+z^2/12$. A direct calculation yieldsFor $x\le 0$ this is nonnegative, and therefore $|R(z)|\le 1$. The method is A-stable. Equality holds on the imaginary axis, consistently with the absence of numerical damping for those scalar oscillatory modes. Since $R(z)\to 1$ as $z\to \infty$, the A-stability established here does not imply strong damping of very stiff decaying modes.

For an implicit Runge-Kutta method, algebraic stability means that $b_i\ge 0$ and the symmetric matrixis positive semidefinite. Here $b_1=b_2=1/2$ and $A+A^T=e{e}^T/2$, soThe method is algebraically stable. This also explains why the property matters for nonlinear problems. If a vector field satisfies $⟨f(v)-f(w),v-w⟩\le 0$, the Runge-Kutta contractivity identity for two solutions, with stage differences $D_i$ and vector-field differences $F_i$, isThe first sum is nonpositive and the second vanishes. Thus, whenever the implicit stage equations have the relevant solutions, their updates are contractive in the Hilbert space norm.