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\noindent{\bf Math 269B.} {\bf Homework \#3.} Due date: Friday Feb. 3 or Monday Feb. 6. 

\noindent(there are 4 exercises)

\noindent{\bf [1]} ({\it computational assignment}) 
Solve the one-way wave equation with variable coefficients 
$$u_t+(1+\alpha x)u_x=0$$ 
on the interval $[-3,3]$ and 
$0\leq t \leq 2$ with the Lax-Friedrichs scheme 
$$v_m^{n+1}=\frac{1}{2}(v_{m+1}^n+v_{m-1}^n)-\frac{1}{2}a(t_n,x_m)\lambda
(v_{m+1}^n-v_{m-1}^n),$$
where $a(t,x)=(1+\alpha x)$. Consider $\alpha=-0.5$, $\lambda=1$. Demonstrate
that the instability phenomena occur where $|(1+\alpha x_m)|>1$. Use the 
initial data 
$$u_0(x)=\left\{
\begin{array}{ll}
1-|x| & \mbox{ if } |x|\leq 1,\\
0 & \mbox{ otherwise}. 
\end{array}
\right.
$$
Specify the solution to be 0 at both boundaries. You can use a grid spacing 
of 0.1 ($h=0.1$). 

\noindent{\bf [2]} Show that the scheme 
$$\frac{v^{n+1}_{l,m}-\frac{1}{4}(v^n_{l+1,m+1}+v^n_{l-1,m+1}+v^n_{l+1,m-1}+v^n_{l-1,m-1})}{k}+a\triangle^x_0 v^n_{l,m}+b\triangle^y_0 v^n_{l,m}=0$$
for the equation $u_t+au_x+bu_y=0$, with $\triangle x=\triangle y=h$, is stable
if and only if $(|a|+|b|)\lambda\leq 1$. 

\noindent{\bf [3]} Consider the system 
$$\left\{ 
\begin{array}{l}
u_t=av_x\\
v_t=au_x. 
\end{array}
\right.
$$

(a) Give a single PDE equivalent with this system. 

(b) Apply the Lax-Friedrichs scheme to the system. Is the obtained system of
finite differences consistent with the system of PDE's ? 

(c) Apply von Neumann stability analysis to the vector scheme, in the case 
when $\lambda=\frac{k}{h}=constant$, substituting 
$$
\left(
\begin{array}{c}
u^n_m\\
v^n_m
\end{array}
\right)
=g^n e^{im\theta}
\left(
\begin{array}{c}
u^0\\
v^0
\end{array}
\right)
$$ 
in the difference vector equation, where $\left(
\begin{array}{c}
u^0\\
v^0
\end{array}
\right)$
is a constant vector both in space and in time. 

\noindent{\bf [4]} 
Consider the system of equations 
$$\left(
\begin{array}{c}
u\\
v
\end{array}
\right)_t
=
A
\left(
\begin{array}{c}
u\\
v
\end{array}
\right)_x;
\ \ \ \ 
A=\left(
\begin{array}{cc}
1 & \alpha \\
0 & 1
\end{array}
\right)
,
$$
which is approximated by the Lax-Friedrichs scheme, 
$$V^{n+1}_j=\frac{1}{2}(V^n_{j+1}+V^n_{j-1})
+\frac{\triangle t}{2\triangle x}A(V^n_{j+1}-V^n_{j-1}),$$
where $V^n_j\approx (u(x_j,t_n),v(x_j,t_n))$. 

Determine the stability properties
depending on whether $\alpha\neq0$ or $\alpha=0$. 

Hint: If $\alpha=0$ this 
becomes an uncoupled system of two scalar one-way wave equations, and the 
stability condition 
of the scalar Lax-Friedrichs scheme applies for each equation. If 
$\alpha\neq0$, show that the scheme is unstable (you can compute the 
amplification matrix $G(\theta)$, for constant $\lambda$, and show that 
the matrices $G^n$ are unbounded in some matrix norm, for at least one 
value of $\theta$). Compute an explicit 
form of $G^n$. 


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