EGN 5456 Computational Mechanics 10/30/98
Closed book Van Dommelen 8:35-9:25 am

Show all reasoning and intermediate results leading to your answer.

The following PDE describes convection of entropy in a pipe:

$${\partial s\over\partial t} + u {\partial s\over\partial x} = 0$$

where s=s(x,t) is entropy, t time, u>0 a given constant flow velocity in the pipe, and x the distance along the pipe. The boundary and initial conditions are:

$$s(0,t) = 1 \qquad s(x,0)=0$$

The length of the pipe is 2.

We want to solve this problem numerically using the following scheme:

$${s^{n+1}_j - s^n_j\over \Delta t} + {u\over 2} {s^{n+1}_j -... ...ver\Delta x} + {u\over 2} {s^n_{j+1} - s^n_j\over\Delta x} = 0$$

where s_jⁿ is the value of the entropy at mesh point number j and time step number n, $\Delta t$ the time increment, and $\Delta x$the spacing between mesh points.

Discuss this scheme in as much detail as possible, discussing the following questions fully (in any order):

1.

Does the scheme make sense? (25%)

2.

Will the scheme work? (25%)

3.

How well will the scheme work? (25%)

4.

Exactly how would you need to perform the computation? (25%)

To save time, in case you feel the need to do any Taylor series expansions, you may represent terms involving third order derivatives and higher by order symbols.

Solution page 1 Solution page 2 Solution page 3