I am interested in solving for the Inviscid Burgers Equation on a ring.
$$ u_t + uu_x=0;\quad(x,t) \in [0,L)\times \mathbb{R}_+,\\ u(x,0)=\phi(x),\\ u(0,t)=u(L,t). $$
This problem I believe is well posed as the single x-derivative yields an integration constant that can be resolved by imposing periodic boundary conditions, provided that $\phi(x)$ is also periodic:
$$ \phi(x) = \phi(x+L). $$
In other words:
$$ \phi(x) = a_0+\sum_{n=1}^{\infty}\Bigg(a_n\cos\Big(\frac{2\pi n}{L}x\Big)+b_n\sin\Big(\frac{2\pi n}{L}x\Big)\Bigg) $$
Choosing $\phi(x)=\cos(2\pi x)$ (setting $L=1$), I can attempt to solve this equation by method of Characteristics
$$ (1)\;\frac{dt}{d\tau} = 1;\;t(s,0)=0 \\ (2)\;\frac{dx}{d\tau} = u;\;x(s,0)=s \\ (3)\;\frac{du}{d\tau} = 0;\; u(s,0)=\cos(2\pi s) $$
Solving $(3)$ and $(1)$ are rather simple: $t=\tau$ and $u(s,\tau) = \cos(2\pi s)$. Solving for $(2)$ leads to a nasty implicit equation
$$ (4)\; x(s,t) = \cos(2\pi s)t + s $$
I don't believe solving for $s(x,t)$ from $(4)$ has a closed form solution and it would need to be determined numerically. My concern has to do with:
- Where characteristic lines intersect on a ring
- How to account for intersecting characteristic lines on a ring
- How do I generate a table of $(x,t,u)$ from this and plot the solution
Any ideas on how to move forward? Have I made any incorrect assumption?