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{\bf MATH 269B: Final assignment (due at the end of the week of finals)}
You can leave it with Jacquie Bowens in MS 7619, or slide it under the door of my office, or in my mailbox.
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{\bf [1]} Consider the heat equation
$$u_t=bu_{xx},$$
to be solved for $t>0$, with smooth initial data $u(x,0)=u_0(x)$ and real $x$.
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{\bf (a)} Under what condition on the parameter $b$ is this a well-posed
problem ? Explain.
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{\bf (b)} For the same problem, consider the scheme
$$\frac{u^{n+1}_j-u^n_j}{\triangle t}=b\Big[\theta \triangle^2_x u^{n+1}_j+(1-\theta)\triangle^2_x u^{n}_j\Big],$$
where
$$\triangle ^2_x u^{n}_j=\frac{u^{n}_{j+1}-2u^n_j+u^n_{j-1}}{h^2}.$$
(i) What well known schemes do we obtain for $\theta=0$, $\theta=1$ and $\theta=\frac{1}{2}$ ?
(ii) Show that for $\frac{1}{2}\leq \theta \leq1$, this is an unconditionally
stable scheme.
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{\bf [2]} For the equation
$$u_{tt}+u_t=u_{xx}+u_x$$
to be solved for $t>0$, with smooth initial data
$$u(x,0)=u_0(x),\ \ u_t(x,0)=u_1(x)$$
{\bf (a)} Restate this problem as an equivalent system of two first order
equations
(first order in both time and space), using a ``factored'' form of the initial
second order equation. Thus obtain an equivalent system of the form
$$\vec{U}_t=A\vec{U}_x+B\vec{U},$$
with $A$ and $B$ two 2x2 matrices (note that the term $B\vec{U}$ will be a lower order term).
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{\bf (b)} Give a convergent finite difference approximation of your choice to
this first order system. Justify your answers.
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{\bf [3]} To solve
$$u_t+au_x=0 \ \mbox{ for } t>0, \ 0\leq x \leq 1,$$
$u(x,0)=\phi(x)$ smooth, $u$ periodic in $x$, $u(x+1,t)=u(x,t)$, we use:
$$\frac{1}{2\triangle t}\Big[(v_{j}^{n+1}+v_{j+1}^{n+1})
-(v_{j}^{n}+v_{j+1}^{n})\Big]+\frac{a}{2\triangle x}\Big[v_{j+1}^{n+1}-v_{j}^{n+1}+v_{j+1}^{n}-v_{j}^{n}\Big]=0.$$
For what values of $\frac{\triangle t}{\triangle x}$, if any, does this
converge ? At what rate ? Explain your answers.
(Recall that for smooth initial data, the order of accuracy of the scheme
gives the order of accuracy of the solution.)
(Note that sometimes the computation of $|g(\theta)|^2$ is simplified if you
use $e^{i\theta}=(e^{i\theta/2})^2$, $e^{-i\theta}=(e^{-i\theta/2})^2$, and
$1=e^{i\theta/2}e^{-i\theta/2}$. These could have been used for the angled derivative method.)
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\underline{Optional problems}
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{\bf [4]} Consider the differential equation
$$u_t=u_{xx}+bu_{xy}+u_{yy}\ \mbox{ for } t>0, \ 0< x < 1,
\ 00,\ C>0,\ B^2