MATH76108/29/2018

Problem.

Consider the IBVP

${\displaystyle \{\begin{array}{ll}{q}_t=\alpha q_{xx}& t>t_0,x\in (a,b)\\ q(t=t_0,x)=f\left(x\right)& x\in [a,b]\\ q(t⩾t_0,x=a)=Q_0& q_x(t⩾t_0,x=b)=1-\cos t\end{array}..}$

1. Determine time, space scaling to state problem as $q_t=q_{xx},x\in (0,1)$ with initial conditions at $t=0$, i.e. redefine $x,t$. State the intrinsic diffusion scale of the problem. (1pt)

2. Solve the problem analytically by superposition of solutions obtained by separation of variables for each type of boundary condition. (2pts)

3. Solve the problem to $𝒪(k^r,h^p)$ by finite differences for $r=1,2,3,4$, $p=2,4$ (all eight combinations). For each $(r,p):$

   1. state the finite difference formulas (you could determine 1 or two formulas by hand computation and then use Mathematica scripts from lessons) (1pts)

   2. Plot the region of stability for each choice of $(r,p)$. (2pts)

   3. Generate the Fortran code implementation for each $(r,p)$ (2pts)

   4. Obtain numerical approximations by execution of the Fortran code with time steps $t=\tau$, $t=0.1\tau$ where $\tau$ is the maximal stability limit. Skip any $(r,p)$ combination you've found to be unstable. (2pts)

   5. Plot the convergence behavior for stable computations $k\in \{k_0,k_0/2,\dots ,k_0/16\}$, $h\in \{h_0,h_0/2,\dots ,h_0/16\}$ with $(k_0,h_0)$ some initial choice of time, space steps. (2pts)

Remark.

1. Feel free to communicate with other students on approach to solve above problem, but draft the final report independently.

2. Draft your report by completing this file

3. The above tasks are impossible to solve within the allotted time without use of automation in generation of formulas, code, plots through utilities such as Mathematica, Gnuplot, make, Python. The appropriate techniques are shown in class. The homework requires you use these techniques on a different, but similar problem.

4. Start the homework immediately after it being posted. The homework requires assimilation of course theoretical and practical concepts and about 12-15 hours of effort during the two-week alloted time.

5. All results should be analyzed and commented. In particular, comment on whether numerical experiments conform to theoretical predictions.