Insert the Fourier mode $u_m^n=G^ne^{im\theta}$ into the scheme. <Von Neumann stability analysis> gives
$$
G(\theta)=\frac1{1+2\mu-2\mu\cos\theta}
=\frac1{1+4\mu\sin^2(\theta/2)}.
$$
For every physical <Courant number> $\mu\geq0$, one has $0<G(\theta)\leq1$. Thus the method is unconditionally stable: the range is $\boxed{\mu\geq0}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/3/b/solution