The purpose of the following test is to map out the students' background in mathematics, numerical methods, and programming when they come to INF5620. Solve the exercises on a piece of paper and give it to Joakim Bø or Hans Petter Langtangen by Aug 26, 2013. Don't write your name on the solution -- the purpose is not to uncover the knowledge of the individual student but the group as a whole.
Mark the topics that have been covered in previous courses and indicate which course (if you remember).
1. Differentiate the function $$ \begin{equation*} u(x,t) = A\exp{(ik(x-ct))}\end{equation*} $$ with respect to \( t \). (\( i \) is the imaginary unit: \( i=\sqrt{-1} \)).
2. You know the three values \( y(0) \), \( y(0.25) \), and \( y(0.5) \) of some function \( y(t) \). How can you compute \( dy/dt \) at these three points?
3. Describe how you would compute the integral $$ \begin{equation*} \int_{0}^1 e^{-t^2}dt .\end{equation*} $$
4. Describe how you would solve the ordinary differential equation problem $$ \begin{equation*} u^{-1}u' + 1 = 0,\quad u(0)=1.\end{equation*} $$
5. Formulate Newton's method for solving $$ \begin{equation*} w + c\Delta t w^2 = u,\end{equation*} $$ with respect to \( w \).
6. Implement the algorithm in Exercise 5 in your favorite programming language (just sketch the program on paper).
7. Compute the divergence of the gradient of the scalar field \( u(x,y)=\sin\pi x\sin\pi y \), i.e., \( \nabla\cdot(\nabla u) \).
8. Formulate a numerical method for solving $$ \begin{equation*} \frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}, \quad x\in (0,1),\ t>0,\end{equation*} $$ where \( k>0 \) is a constant. The initial condition reads \( u(x,t)=1 \), while we set \( u(0)=u(1)=0 \) as boundary conditions.
9. What is the most complicated partial differential equation problem you have solved, or seen be solved by a lecturer?
10. Put a cross in front of the concepts that you are able to briefly define.
Mathematics:
11. Give a physical interpretation of \( u \) in the equations below. $$ \frac{du}{dt}=-au\] \[ \frac{d^2u}{dt^2} + \omega^2 u = 0\] \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\] \[ \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u\] \[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} =0\] \[ \frac{\partial^2 u}{\partial t^2} = (\lambda +\mu)\nabla(\nabla\cdot u) + \mu\nabla^2 u\] \[ \frac{\partial u}{\partial t} + u\nabla u = -\nabla p + \frac{1}{\hbox{Re}}\nabla^2 u,\quad \nabla\cdot u = 0$$
12. Given a vector field \( {\bf v}(x, t) \) describing the velocity of some flowing material, what is the physical interpretation of \( \int_\Gamma {\bf v}\cdot{\bf n}ds \)? \( \Gamma \) is some closed 3D surface and \( \bf n \) is the unit outward normal vector to this surface.
13. In Exercise 11, transform the surface integral to a volume integral and use the result to give a physical interpretation of the divergence of \( \bf v \).