Welcome to our community

Be a part of something great, join today!

[SOLVED] Shock wave

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,836
Hello!!! (Wave)

I want to solve the equation $u_t+uu_x=0$ with the initial condition $u(x,0)=1$ for $x \leq 0$, $1-x$ for $0 \leq x \leq 1$ and $0$ for $x \geq 1$. I want to solve it for all $t \geq 0$, allowing for a shock wave. I also want to find exactly where the shock is and show that it satisfies the entropy condition.

I have tried the following.

We get that $u(x(t),t)=c$.

The characteristic line that passes through the points $(x,t)$ and $(x_0,0)$ has slope

$\frac{x-x_0}{t-0}=\frac{dx}{dt}=u(x,t)=u(x_0,0)=\left\{\begin{matrix}
1, & \text{ if } x_0 \leq 0,\\
1-x_0, & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$


$\Rightarrow x-x_0=\left\{\begin{matrix}
t, & \text{ if } x_0 \leq 0,\\
t(1-x_0), & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$

$\Rightarrow u(x,t)=\left\{\begin{matrix}
1, & \text{ if } x \leq t,\\
\frac{1-x}{1-t}, & \text{ if } t \leq x \leq 1,\\
0, & \text{ if } x \geq 1.
\end{matrix}\right.$


Do we have a shock at the time when $u$ is not continuous? But how can we find such a time?
 
Last edited:

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
9,416
Hello!!! (Wave)

I want to solve the equation $u_t+uu_x=0$ with the initial condition $u(x,0)=1$ for $x \leq 0$, $1-x$ for $0 \leq x \leq 1$ and $0$ for $x \geq 1$. I want to solve it for all $t \geq 0$, allowing for a shock wave. I also want to find exactly where the shock is and show that it satisfies the entropy condition.

I have tried the following.

We get that $u(x(t),t)=c$.

The characteristic line that passes through the points $(x,t)$ and $(x_0,0)$ has slope

$\frac{x-x_0}{t-0}=\frac{dx}{dt}=u(x,t)=u(x_0,0)=\left\{\begin{matrix}
1, & \text{ if } x_0 \leq 0,\\
1-x_0, & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$


$\Rightarrow x-x_0=\left\{\begin{matrix}
t, & \text{ if } x_0 \leq 0,\\
t(1-x_0), & \text{ if } 0 \leq x_0 \leq 1,\\
0, & \text{ if } x_0 \geq 1.
\end{matrix}\right.$

$\Rightarrow u(x,t)=\left\{\begin{matrix}
1, & \text{ if } x \leq t,\\
\frac{1-x}{1-t}, & \text{ if } t \leq x \leq 1,\\
0, & \text{ if } x \geq 1.
\end{matrix}\right.$


Do we have a shock at the time when $u$ is not continuous? But how can we find such a time?
Hey evinda !!

What will happen to $u$ when $t$ reaches $1$? (Wondering)

Btw, your solution doesn't cover $t>1$ does it?
The second condition breaks down, and the first and third condition will overlap. (Worried)
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,836
The shock wave happens when $t=1$.

Which is the entropy condition? (Thinking)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
9,416
The shock wave happens when $t=1$.

Which is the entropy condition?
I don't know what an entropy condition is. (Crying)

Do you have a definition? (Wondering)
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,836

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
9,416

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,836