George Mason University

Conor Philip Nelson

(8.4.1) Solve Burgers’ equations (8.63) on [0,1] with initial condition\( f(x)=\sin(2x\pi)\) and boundary conditions \(l(t)=r(t)=0\), using step sizes:
(a) \(h=k=0.1\)
(b) \(h=k=0.02\)
Plot the approximate solutions for \(0 \leq t \leq 1\).
Which equilibrium solution does the solution approach as time increases?

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We will utilize burgers.8.4.1.m for this problem. Consider Burger's equation with Dirichlet boundary conditions: (8.63) \[f(x,t)= \begin{cases} u_t +u*u_x=Du_{xx} & \\ u(x,0)=f(x) & \text{for } x_l \leq x \leq x_r\\ u(x_l,t)=l(t) & \text{for all } t \geq 0\\ u(x_r,t)=r(t) & \text{for all } t \geq 0\ \end{cases} \] (a) When \(h=k=0.1\). We have:


(b) When \(h=k=0.02\). We have:

It is clear to note that with this \(k\) value, these mesh lines are so close together that it causes a skew of vision where there will be blackness overtaking the depictions. Hence as to why we used the \(shading\) interpretion.

Note that as time increases, \(x\) goes to 0:

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