Consider two problems:

$$(1) \hspace{1cm} u_t+f(u)_x = 0, $$

$$(2) \hspace{1cm} u_t+f(u)_x = g(u). $$

Problem (1) represents system of conservation laws, and problem (2) represents system of balance laws (or conservation laws with source term). If we have discontinuous initial data such as

$$(3)\hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases}$$

than we talk about Riemann problems. One type of solutions of (1)-(3) and (2)-(3) are shock waves given by

$$(4)\hspace{1cm} u(x,t)= \begin{cases} u_l, x<s \cdot t \\[2ex] u_r, x>s \cdot t \end{cases}$$

where $s$ is speed of the shock. We calculate it from the Rankine-Hugoniot conditions (in order not to complicate things, here $u_l$ and $u_r$ are constants, so shock speed $s$ is a constants; if we had $u_l(x,t)$ and $u_r(x,t)$, than we would have $s(t)$).

For problem (1)-(3) Rankine-Hugoniot conditions are given by

$$s\cdot (u_r - u_l) = f(u_r)- f(u_l)$$

I am interested in problem (2)-(3). By "C. Dafermos, Hyperbolic conservation laws in continuum physics, 2016", the fact that we have a source doesn't change nothing, i.e. Rankine-Hugoniot conditions are again given by

$$s\cdot (u_r - u_l) = f(u_r)- f(u_l)$$

and speed is the same than. (See start of a Chapter 3)

How is that possible? In my head it sounds reasonable that source affects the speed of the shock. At least I would expeced that speed changes with t, i.e. now we would have $s(t)$.

Also, in this paper the authors included source in the Rankine-Hugoniot conditions (see (12),(13),(14) in this paper). Similar things could be found in various papers that do numerical solving of pdes.

So my questions are:

How do Rankine-Hugoniot conditions look in the case of balance laws and how do we calculate the speed of the shock wave in that case?

I would really appreciate a help with this.

EDIT: I would like to know also what happens in the problems where source in $(2)$ isn't given with $g(u)$. Instead of $g(u)$ it could be written anything alse (e.g. $g(u)$=constant or $g(u)=\partial_{xt}^2 u$ or whatever). Of course we talk about Rankine-Hugoniot conditions only in the case of discontinous solutions. If we take $g(u)=\triangle u$, as I recall, we have smooth solutions and we do can't talk about Rankine-Hugoniot conditions.

  • $\begingroup$ The right-hand term doesn't change anything (see related post) $\endgroup$ – Harry49 Jun 23 '18 at 23:02
  • 1
    $\begingroup$ For @Harry49: Thank you. That post you point out plus two books (Garabedian - Partial Differential Equations, 1964 - Section 14.1 and Mishra - Numerical methods for conservation laws and related equations - Section 3.2) have helped me gain understanding why the right hand term doesn't change anything. Everything is easier to see when you put the equations in the integral form. Only I am not sure if that works for every type of deterministic or stochastic source (such as white noise). And in some cases speed changes with t. But I have far better understanding now. $\endgroup$ – Mark Jul 5 '18 at 11:38
  • $\begingroup$ @Harry49: I was discussing this question today with a friend, so I have an additional follow up question. You wrote above that right-hand term doesn't change anything. So if I have right-hand side that is given by h(x,t) or $\triangle u$ or $u^6$ for example, the same conclusion is valid i.e. nothing changes? What about if I have some stochastic white noise on the right-hand side? $\endgroup$ – Mark Dec 24 '18 at 15:32

I found the answer to the question: "How do Rankine-Hugoniot conditions look in the case of balance laws?"

It is in the book: Rational extended thermodynamics - Muller, Ruggeri, 1998

In Chapter 8 - Subsection 5.1 (Weak solutions) and Subsection 5.2 (Rankine-Hugoniot Equations). It is given in general terms but it is the proof I was looking for.

So the Rankine-Hugoniot condition stays the same in the case (2),(3) as in the case (1),(3).

But what happens to the speed?

In the conservation law case (1),(3) shock speed is constant. For the balance law case (2),(3), I think that we calculate speed in the same way but it isn't constant anymore. I came to this conclusion by solving Burgers equation with and without source with some concrete functions $f,g$ and concrete constants $u_l,u_r$. But I am not sure that could be generalized. If I am wrong about this, please correct me.

If I get any new info I'll post it here.


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