Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 40D / a / Solution

Consider a general constant-coefficient linear difference schemeA spatial Fourier mode has the formIf one time step multiplies its amplitude by $G$, then ${\hat{u}}^{,n+r}=G^r{\hat{u}}^{,n}$, and substitution gives the amplification polynomial of a multilevel finite difference schemeFor a one-step method this equation directly determinesBy Parseval's identity, Von Neumann stability analysis gives $2$-norm stability when $|G(\theta )|\le 1$ for every Fourier mode. For a multilevel method, every root must obey the corresponding root condition.