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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 
$$u_t=u_{xx}+bu_{xy}+u_{yy}\ \mbox{ for } t>0, \ 0< x  < 1, 
 \ 0<y < 1 ,$$
with $u=0$ on the boundary, and $u(0,x,y)=\phi(x,y)$, a smooth function. 

{\bf (a)} For what values of $b$ can you obtain a convergent, unconditionally 
stable finite difference scheme ? 

{\bf (b)} Construct such a scheme. Explain your answers. 

Note that this is a particular case of the more general problem, given next. 

\bigbreak 

{\bf [5]} Consider the PDE 
$$u_t=Au_{xx}+2Bu_{xy}+Cu_{yy},$$ 
with associated smooth initial and boundary conditions. 

(a) For what constants $A,B,C$ is this a well-posed parabolic equation ? 
(answer: if $A>0,\ C>0,\ B^2<AC$; explain). 

(b) Apply the $\Theta$-method to this problem (as outlined below). 

Let $\Phi^n_{j,l}=\frac{A}{(\triangle x)^2}(u^n_{j+1,l}-2u^n_{j,l}+u^n_{j-1,l})
+\frac{B}{2\triangle x \triangle y}(u^n_{j+1,l+1}-u^n_{j-1,l+1}-u^n_{j+1,l-1}+u^n_{j-1,l-1})+\frac{C}{(\triangle y)^2}(u^n_{j,l+1}-2u^n_{j,l}+u^n_{j,l-1})$.

The general $\Theta$-method, for $0\leq \Theta\leq 1$, is 
$$\frac{u^{n+1}_{j,l}-u^n_{j,l}}{\triangle t}=\Theta \Phi^{n+1}_{j,l}+(1-\Theta)\Phi^n_{j,l}.$$ 

(c) Obtain the stability conditions of the scheme, function of $\Theta$. 

Hint: 

- Use $u^n_{j,l}=g^ne^{ij\theta_1}e^{il\theta_2}$ as usual. 

- Compute the amplification factor $g$ of the scheme, and write it in the form $g=\frac{1+(1-\Theta)\Psi}{1-\Theta\Psi}$, for $\Psi=\Psi(\theta_1,\theta_2)$ to be determined. 

- Notice that $\Psi$ is a real analytic periodic function of $(\theta_1,\theta_2)$, therefore $g(\theta_1,\theta_2)$ is also real.

- To see under what conditions $|g(\theta_1,\theta_2)|\leq 1$, find first the 
minimum and maximum values of $\Psi$ (to find the extema of $\Psi$, write
a linear system of two equations obtained from $\frac{\partial\Psi}{\partial\theta_1}=\frac{\partial\Psi}{\partial\theta_2}=0$; consider the unknowns $\sin\theta_1$ and $\sin\theta_2$). 

- Using the lower and upper bounds of $\Psi$, you will obtain the stability 
conditions of the scheme (function of $\Theta$). 

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