$1\mathrm{.}1$ Instructions

Model a heated bar using the $1$D transient heat equation.

$\backslash [$

$\backslash$frac$\{\backslash$partial T$\}\{\backslash$partial t$\}=$c$^\{2\}\backslash$frac$\{\backslash$partial$^\{2\}$ T$\}\{\backslash$partial x$^\{2\}\}$

$\backslash ]$

Write a code to complete a finite difference analysis of the heat equation using central differencing in space and backward differencing in time.

$(1)$ First solve the heat equation with the following parameters:

$-$ A bar of unit length and thermal diffusivity, $\backslash ($ L$=1\backslash$mathrm$\{$~m$\}\backslash )$ and $\backslash ($ c$^\{2\}=1\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\}\backslash ).$

$-$ Homogeneous temperature boundary conditions.

$-$ An initial temperature of $100$ K $.$

$-$ The discretization in space should be fine enough to allow smooth solution plots.

$-$ The discretization in time should yield stable results.

$(2)$ Adjust these parameters and observe the changes in your solution. Specifically investigate at lease one change each in boundary$/$initial conditions and physical properties. What happens to the time to reach steady state? Does the necessary time discretization change?

Some examples that could be studied include:

$-$ Determining how $\backslash ($ c$^\{2\}\backslash )$ affects time to reach steady state.

$-$ Investigating initial conditions such as linear or parabolic temperature distributions.

$************$CANNOT USE OR REFERENCE CFL NUMBERS OR UPWINDING OR CRANK$-$NICKELSON METHODS.$*******$