| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
81 |
II |
2003 |
Numerical Analysis |
|
A3.19 B3.20 |
2003 |
(i) The diffusion equation
$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x) \frac{\partial u}{\partial x}\right), \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$$
with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$ and zero boundary conditions at $x=0$ and $x=1$, is solved by the finite-difference method
$$\begin{gathered}
u_{m}^{n+1}=u_{m}^{n}+\mu\left[a_{m-\frac{1}{2}} u_{m-1}^{n}-\left(a_{m-\frac{1}{2}}+a_{m+\frac{1}{2}}\right) u_{m}^{n}+a_{m+\frac{1}{2}} u_{m+1}^{n}\right] \\
m=1,2, \ldots, N
\end{gathered}$$
where $\mu=\Delta t /(\Delta x)^{2}, \quad \Delta x=\frac{1}{N+1}$ and $u_{m}^{n} \approx u(m \Delta x, n \Delta t), a_{\alpha}=a(\alpha \Delta x)$.
Assuming sufficient smoothness of the function $a$, and that $\mu$ remains constant as $\Delta x>0$ and $\Delta t>0$ become small, prove that the exact solution satisfies the numerical scheme with error $O\left((\Delta x)^{3}\right)$.
(ii) For the problem defined in Part (i), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}, \quad 0 \leqslant x \leqslant 1$. Prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.
[Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix $A=\left(a_{k, l}\right){k, l=1, \ldots, M}$ are contained in $\bigcup{k=1}^{m} \mathbb{S}_{k}$, where
$$\left.\mathbb{S}{k}=\left{z \in \mathbb{C}:\left|z-a{k, k}\right| \leqslant \sum_{\substack{l=1 \ l \neq k}}^{m}\left|a_{k, l}\right|\right}, \quad k=1,2, \ldots, m . \quad\right]$$