Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 69 / 3

Let $h=\Delta x$ and sample a sufficiently smooth periodic exact solution. The central finite differences have expansionsandThe semidiscrete right side therefore equals $u_{xx}+\alpha u_x+h^2(u_{xxxx}/12+\alpha u_{xxx}/6)+O(h^4)$. The spatial consistency of a numerical method is second order, for each fixed $\alpha$. There is no time discretization in the stated ordinary differential system, so no separate temporal approximation order is being assigned. The Taylor statement concerns smooth solutions; L2-compatible initialization of grid data handles nonsmooth initial data without undefined point sampling.

Use the discrete norm $\Vert U{\Vert }_h^2=h{\sum }_{m=1}^N|U_m{|}^2$, with periodic indices. Two independent arguments prove uniform stability of a numerical method.

For Fourier stability analysis, a discrete Fourier mode $U_m(t)={\hat{U}}_k(t)e^{im{\theta }_k}$ with ${\theta }_k=2\pi k/N$ has semidiscrete eigenvalueIts real part is nonpositive, so $|{\hat{U}}_k(t)|\le |{\hat{U}}_k(0)|$. The unitary discrete Fourier transform and discrete Parseval identity then give $\Vert U(t){\Vert }_h\le \Vert U(0){\Vert }_h$ for every $N$, $t\ge 0$ and fixed real $\alpha$.

For the energy method, denote the centered first difference by $D_0$ and the forward difference by $D_{+}$. Periodic discrete summation by parts makes $D_0$ skew-adjoint and gives $⟨U,D_{xx}U{⟩}_h=-\Vert D_{+}U{\Vert }_h^2$. ThereforeThe constant mode is preserved rather than strictly damped. Both proofs apply equally to differences of two solutions. The semidiscretization is contractive in the discrete L2 norm. A later choice of a time integrator would require its own stability analysis; no time-step condition is needed for this continuous-time result.