| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
103 |
II |
2009 |
Numerical Analysis |
|
Paper 3, Section II, B |
2009 |
Prove that all Toeplitz tridiagonal $M \times M$ matrices $A$ of the form
$$A=\left[\begin{array}{rrrrr}
a & b & & & \\
-b & a & b & & \\
& \ddots & \ddots & \ddots & \\
& & -b & a & b \\
& & & -b & a
\end{array}\right]$$
share the same eigenvectors $\left(\boldsymbol{v}^{(k)}\right){k=1}^{M}$, with the components $\boldsymbol{v}{m}^{(k)}=i^{m} \sin \frac{k m \pi}{M+1}, m=$ $1, \ldots, M$, where $i=\sqrt{-1}$, and find their eigenvalues.
The advection equation
$$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t \leqslant T,$$
is approximated by the Crank-Nicolson scheme
$$u_{m}^{n+1}-u_{m}^{n}=\frac{1}{4} \mu\left(u_{m+1}^{n+1}-u_{m-1}^{n+1}\right)+\frac{1}{4} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right)$$
where $\mu=\frac{\Delta t}{(\Delta x)^{2}}, \Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m \Delta x, n \Delta t)$. Assuming that $u(0, t)=u(1, t)=0$, show that the above scheme can be written in the form
$$B \boldsymbol{u}^{n+1}=C \boldsymbol{u}^{n}, \quad 0 \leqslant n \leqslant T / \Delta t-1$$
where $\boldsymbol{u}^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{T}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu$ for which the scheme is stable. [Fourier analysis is not acceptable.]