| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
80 |
II |
2002 |
Numerical Analysis |
|
A2.19 B2.19 |
2002 |
(i)
Given the finite-difference method
$$\sum_{k=-r}^{s} \alpha_{k} u_{m+k}^{n+1}=\sum_{k=-r}^{s} \beta_{k} u_{m+k}^{n}, \quad m, n \in \mathbb{Z}, n \geqslant 0$$
define
$$H(z)=\frac{\sum_{k=-r}^{s} \beta_{k} z^{k}}{\sum_{k=-r}^{s} \alpha_{k} z^{k}}$$
Prove that this method is stable if and only if
$$\left|H\left(e^{i \theta}\right)\right| \leqslant 1, \quad-\pi \leqslant \theta \leqslant \pi .$$
[You may quote without proof known properties of the Fourier transform.]
(ii) Find the range of the parameter $\mu$ such that the method
$$(1-2 \mu) u_{m-1}^{n+1}+4 \mu u_{m}^{n+1}+(1-2 \mu) u_{m+1}^{n+1}=u_{m-1}^{n}+u_{m+1}^{n}$$
is stable. Supposing that this method is used to solve the diffusion equation for $u(x, t)$, determine the order of magnitude of the local error as a power of $\Delta x$.