Let's start from the concrete 2×2 system. In the Chapter 5 of the "Front tracking for hyperbolic conservation laws" – Holden, Risebro, 2011, we could find a shallow water system:
$$h_t+q_x=0,$$
$$q_t+(\frac{q^2}{h}+\frac{h^2}{2})_x=0,$$
with Riemann initial data:
$$
(h,q)(x,0)= \begin{cases}
(h_l,q_l), x<0 \\[2ex]
(h_r,q_r), x> 0,
\end{cases}
$$
Here $h_l , h_r, u_l, u_r$ are constants in $\mathbb{R}$ and $h \in \mathbb{R}$, $q \in \mathbb{R}$, $x \in \mathbb{R}$, $t \in [0,T]$.
One of the solutions of this problem is given with a shock wave:
$$(1) \hspace{1cm}
(h,q)(x,t)= \begin{cases}
(h_l,q_l), x< s t \\[2ex]
(h_r,q_r), x> s t,
\end{cases}
$$
where $s$ is the speed of the discontinuity.
My question is: how do we imagine $(1)$ in the 3D plain? Let me explain.
A conservation law in the area of PDEs is given with:
$$(2) \hspace{1cm} u_t+f(u)_x=0,$$
where $x \in \mathbb{R}$, $t \in [0,T]$, $u=u(x,t)=(u_1,u_2,…,u_n)$, $f=f(u)=(f_1,f_2,…,f_n)$. If $n=1$, we talk about scalar conservation law. If $n>1$, we talk about system of nxn conservation laws. A shallow water system represents 2×2 system of conservation laws.
In $(2)$ when $n=1$, shock wave is given with:
$$(3) \hspace{1cm}
u(x,t)= \begin{cases}
u_l, x<st \\[2ex]
u_r, x>st.
\end{cases}
$$
On the picture in "$x-t-u$" plane and for the positive speed $s$, shock wave given with $(3)$ would look like this:
Here by "variable" we mean "u".
But what happens in the case $n>1$? How do we imagine shock waves then? Of course in the case of $n>1$, $(3)$ would look the same, it would just have $n$ components of $u_l$ and $u_r$ (for example in $(1)$ $u_l=(h_l,q_l)$, $u_r=(h_r,q_r)$ – so they have two components each).
My thoughts:
-
In order to picture that we would need some "(n+2)D plane $x-t-u_1-…-u_n$". So that is not possible to imagine that way. A professor of mine have told me that in 3D we imagine all of the $n$ components on the same picture i.e. all in the "$x-t-u$" plane but that looks confusing even for the $n=2$.
-
OR maybe we could imagine it in $n$ planes "$x-t-u_1,…., x-t-u_n$" (but the problem with this is that $(3)$ as whole is a shock wave solution, and we do not think of every component as a shock waves).
-
OR maybe we could use some other type of a 3D plane instead of "$x-t-u$"?
At first I was thinking that answer to this questions was obvious but now I am not so sure. Help with this would be great (and help with pictures would be amazing).
Best Answer
The wave speeds will be the same for all the conserved variables. The only thing that will change from one variable to another are the left and right constant states (two of which are known from the initial condition). Indeed, it does not seem reasonable to combine several conserved variables on a single $x$-$t$-variable 3D plot, but one can still represent at most one variable on one 3D plot. Alternatively, some 2D plots could be produced:
One may plot the conserved variables or normalized conserved variables at fixed time $t$ with respect to the space variable $x$: (from P.L. Roe, J. Comput. Phys. 43, 1981)
Of course one could do the opposite by representing the conserved variables w.r.t. $t$ for a fixed position $x$, but it is less common in fluid dynamics.
Another classical representation corresponds to your 3D representation viewed from above. It is a plot of the wave location in the $x$-$t$ plane, such as (from K.V. Karelsky, A.S. Petrosyan, Fluid Dyn. Res. 38, 2006)
This graph displays the wave trajectories in $x$-$t$ coordinates. For shock waves, this may be interpreted as a map of the gradient's norm $\| h_x + q_x \|$, which can be evaluated numerically (see e.g. this post where the scalar case is presented).