The Rankine-Hugoniot jump conditions for conservation and balance laws

Question

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 the shock speed $s$ is a constant.

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 given above and how do we calculate the speed of the shock wave in that case?

To be honest I maybe have misunderstood something in the C. Dafermos book. Maybe the author just wanted to say that the form of the Rankine-Hugoniout conditions stays the same. And the speed is changing because the source affects left and right states. (That sounds reasonable to me)

Additionaly, 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.

I would really appreciate a help with this.

Answer 1

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.

Answer 2

As specified in this post the Rankine-Hugoniot condition which locally links the jumps to the shock speed is unchanged when an arbitrary r.h.s. $g(u)$ is included in the balance law. Now, let us consider the Riemann problem -- i.e. the initial value problem (3) -- and assume that the admissible solution is a shock wave. The method of characteristics makes it possible to identify the values of $u$ on both sides of the shock path $x = s(t)$. To illustrate this feature analytically, consider $g(u) = a u$. The method of characteristics gives $$ u(x,t) = \left\lbrace \begin{aligned} & u_l e^{at}, & & x< s(t) ,\\ & u_r e^{at}, & & x> s(t) . \end{aligned}\right. $$ The first line is the left state in the Rankine-Hugoniot (RH) condition, and the second line is the right state in the RH condition. Thus, the RH condition yields $$ s'(t) = \frac{f(u_r e^{at}) - f(u_l e^{at})}{u_r - u_l} $$ with the initial condition $s(0) = 0$. Note that the speed $s'$ is constant if $a=0$.

If the r.h.s. consists of a differential operator, then it may be possible that the Riemann solution can't be a shock wave. The viscous Burgers equation $u_t + u u_x = \epsilon u_{xx}$ is such an example.