Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 38A (Numerical Analysis)

Write $h=\Delta x$, $k=\Delta t=\mu h$, and $D_{0}u=[u(x+h)-u(x-h)]/(2h)$. The scheme is the trapezoidal time rule with a centered spatial derivative:Insert a smooth exact solution of $u_t=u_x$. Taylor expansion of the unnormalized one-step residual givesAll repeated time derivatives equal the corresponding spatial derivatives. The first two orders cancel, leavingThus the one-step local error is $O(h^3)$ at fixed Courant number. If “local truncation error” is instead defined after division of the residual by $\Delta t$, its order is $O(h^2)$; these are two different normalizations of the same second-order method.

For the Fourier mode $u_m^n=g^n{e}^{im\xi }$, substitution givesThe denominator is never zero for real $\mu ,\xi$, and numerator and denominator are complex conjugates, so $|g|=1$. Parseval's identity therefore makes each step norm-preserving on square-summable grid data. The method is von Neumann stable for every $\mu >0$, with no upper Courant restriction. It is an implicit nondissipative scheme; unconditional stability does not remove phase error or guarantee accuracy at a large time step.