1

I am trying to solve the characteristic equations of $u_t + u^2u_x = 0$ without initial conditions in order to show graphically how profiles get smoothed or develop a shock depending on initial data.

I have that the characteristic equations are $\frac{dt}{ds}=1$, $\frac{dx}{ds} = u^2$ and $\frac{dz}{ds} = 0$.

Then that $t=s+c_1$, where $c_1$ is a constant, $u =\frac{-1}{s+c_2}$

Is this right and what do I need to do next in order to get a general solution?

Any help much appreciated, thanks.

user149891
  • 13

1 Answers1

1

Your first solution is correct: $t(s) = s + t_0$, but you must also solve the $x$ equation: $x(s)=u_0^2s + x_0$ (where $u_0 = u(x(s),t(s)) = u(x_0,t_0)$). You now know that the fluid flow is constant on the line $$(x(s),t(s)) = (u_0^2 s + x_0,s+t_0)$$

Now you just plug in initial data and follow the flow.

Winther
  • 24,478