For the Cauchy problem, insert the <Fourier mode> $u_m=e^{im\vartheta}$. The spatial symbol is
$$
\lambda_h(\vartheta)
=-\frac4{\Delta x^2}\sin^2\frac\vartheta2
+i\frac{\alpha}{\Delta x}\sin\vartheta.
$$
Its real part is nonpositive for every $\vartheta$, so each mode has modulus $e^{t\operatorname{Re}\lambda_h}\leq1$. By the discrete <Fourier transform> and <Parseval identity>, the scheme is stable in the discrete $L^2$ norm.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-341/4/b/solution