8 Schedule

Class times: TR 10:15-11:30 in A226 (A building).

Tentative outline and homeworks (keep checking for changes):

• 08/30/05 T
• 09/01/05 R

• 09/06/05 T Due: 2.4
• 09/08/05 R Due: 2.2

• 09/13/05 T Due: 2.8a, 2.9a plus properties
• 09/15/05 R Due: verify the solution to the Laplace I.V. problem

• 09/20/05 T Due: Determine properly posedness of the I.V. problems for the given three P.D.E.s
• 09/22/05 R Due: 2.10 (swap u and v in the second equation), 2.12

• 09/27/05 T Due: 3.2
• 09/29/05 R Due: 3.12, 3.6

• 10/04/05 T
• 10/06/05 R Due: write the odd entries (3.46), (3.48), ... in table 3.2 in terms of operators and thus verify that the stated derivative is indeed approached. Project 1.

• 10/11/05 T Due: 3.1 (write finite difference formula in terms of mesh point values, find actual truncation error, not just its order.)
• 10/13/05 R Midterm

• 10/18/05 T Due: Examine consistency and accuracy (using the class' definitions, not the book's ones) for the so-called DuFort-Frankel scheme
  
  $$\frac{\bar\delta_t u_j^n}{\Delta t} = \alpha \frac{(\delta_x^2 - \delta^2_t) u_j^n}{(\Delta x)^2}$$
  
  
  for the heat equation $u_t=\alpha u_{xx}$.
• 10/20/05 R Due:

• 10/25/05 T Due: Project 3. Also, discuss the truncation error (and also everything you need before you can talk about truncation error) for the forward time, forward space scheme
  
  $$\frac{\Delta_t u_j^n}{\Delta t} + a \frac{\Delta_x u_j^n}{\Delta x} = 0$$
  
  
  for the one way wave equation
  
  $$u_t + a u_x = 0$$
  
  

• 10/27/05 R Due: Show that the solution $\vec u$ of
  
  $$A \vec u = \vec f \qquad \mbox{ with } A = \left( \begin... ...ft(\begin{array}{c} 2\Delta x 3\Delta x \end{array}\right)$$
  
  
  does not become zero for $\Delta x\to 0$, even though $\vec f$ does. However, show that the solution of the following system does:
  
  $$\strut A \vec u = \vec f \qquad \mbox{ with } A = \left( ... ...ft(\begin{array}{c} 2\Delta x 3\Delta x \end{array}\right)$$
  
  
  In what sense is this a simplistic model for finite difference schemes? In particular, what corresponds here to the scheme being ``consistent''? What is the ``order of accuracy''? What corresponds to ``stability''?

• 11/01/05 T Due: 3.27 for the one way wave equation $u_t + c u_x=0$, where $c$ is some constant. Some formulae for complex numbers and their magnitudes that may come in handy follow. In these, $a, b, c, \ldots$ are general complex numbers, while $a_r, b_r, c_r, \ldots$ and $a_i, b_i, c_i, \ldots$ are ordinary real numbers. Also, if complex number $a$ equals $a=a_r + {\rm i}a_i$, where ${\rm i}= \sqrt{-1}$ and $a_r$ and $a_i$ are ordinary real numbers, then $a_r$ is called the real part of $a$, and $a_i$ the imaginary part of $a$. For example, since
  
  $$e^{{\rm i}\xi} = \cos(\xi)+{\rm i}\sin(\xi)$$
  
  
  the real part of $e^{{\rm i}\xi}$ is $\cos(\xi)$ and the imaginary part is $\sin(\xi)$. Formulas for magnitudes:
  
  $$\vert a\vert=\vert a_r+{\rm i}a_i\vert = \sqrt{a_r^2+a_i^2}... ...\qquad \left\vert\frac1a\right\vert = \frac{1}{\vert a\vert}$$
  
  
  (Note that finding the square magnitude of $g$ is often more convenient than the magnitude of $g$ itself.) Euler-type formulae:
  
  $$\cos(\xi) = \frac{e^{{\rm i}\xi} + e^{-{\rm i}\xi}}{2} \qqu... ...i) = \frac{e^{{\rm i}\xi} - e^{-{\rm i}\xi}}{2{\rm i}} \qquad$$
  
  
  Use both direct substitution and operators to solve 3.27, and see whether you get the same result.
• 11/03/05 R Due: 3.25 for the one way wave equation $u_t + c u_x=0$, where $c$ is some constant. Note that the book is wrong in what it says about stability.

• 11/08/05 T Due: 3.30. Make sure to handle the cases of positive and negative discriminant separately. Project 4.
• 11/10/05 R Due: 3.31, but in both one and two spatial dimensions.

• 11/15/05 T Due: What does the CFL condition require for the forward time, center space scheme for the heat equation? Is it the same as can be concluded from Von Neumann stability analysis? If not, which gives the more accurate restriction on the scheme, CFL or Van Neumann? (Remember that the scheme is always consistent.) CFL for the FTFS scheme for the one-way wave equation (compare with the Von Neumann result.)
• 11/17/05 R Due: What does the CFL condition require for the backward time, center space scheme for the heat equation? Is it the same as Von Neumann analysis? How about the backward time, center space scheme for the one-way wave equation?

• 11/22/05 T Due: Project 5.
• 11/24/05 R THANKSGIVING

• 11/29/05 T Due: Reread notes on solution of multidimensional implicit problems. Work on theory part of the final project.
• 12/01/05 R Due:

• 12/06/05 T Due:
• 12/08/05 R Due:

• 12/12/03: Final Monday 7:30-9:30 am (ignore FAMU schedule).