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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 123, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/123\hfil The Riemann problem in gasdynamics]
{The Riemann problem in gasdynamics}

\author[E. D. Conway, S. I. Rosencrans\hfil EJDE-2009/123\hfilneg]
{Edward Daire Conway, Steven I. Rosencrans}  % in alphabetical order

\address{Steven I. Rosencrans \newline
Mathematics Department\\
Tulane University\\
New Orleans, LA 70118, USA}
\email{srosenc@tulane.edu}

\thanks{Submitted August 2, 2009. Published September 29, 2009.}
\thanks{Supported in part by grant GP9019 from the National
Science Foundation}
\subjclass[2000]{35L03, 35L65, 35L67, 76L05}
\keywords{Riemann problem; shock wave; rarefaction wave}

\begin{abstract}
 In this note we give a proof of the existence of a solution to the
 Riemann problem in one-dimensional gasdynamics. Lax's 1957 paper on
 conservation laws leaves no
 doubt that such a solution exists, but it seems to us that there
 may be interest in a brief and explicit proof favorable to
 numerical computations. Our procedure also allows us to give a
 simple characterization of those problems in which a given wave is
 a shock or a rarefaction wave. In the final section we prove
 a result of Von Neumann's concerning
 the overtaking of two shocks.
 This paper was written in 1969 and is being published now at the 
 suggestion of Jerry Goldstein, whose editorial note is included.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

Editorial note by Jerry Goldstein:  This work was done in 1969 and
presented in the PDE Seminar at Tulane, where the two authors and
I were colleagues.  I was struck by the beauty and simplicity of
the result, although Ed and Steve felt that the experts understood
that the result was implicitly contained in Peter Lax's earlier
paper \cite{lax}.  Later these results were included in Joel
Smoller's book \cite{smoller1983}, but I still felt that the
Conway-Rosencrans global in time analysis of the Riemann problem
in gasdynamics deserved a wider audience.  Ed Conway died in 1985.
In March 2009, I met Steve Rosencrans in New Orleans and urged him
again to publish this lovely short note.  I am very pleased that
he finally agreed to do so.

\section{Introduction}

In this note we give a proof of the existence of a solution to the
Riemann problem in gasdynamics. Lax's paper \cite{lax} leaves no
doubt that such a solution exists, but it seems to us that there
may be interest in a brief and explicit proof favorable to
numerical computations. Our procedure also allows us to give a
simple characterization of those problems in which a given wave is
a shock or a rarefaction wave. In the final section we prove
 a result of Von Neumann's concerning
the overtaking of two shocks.

The proof we give follows the general method given by Lax although
a familiarity with that work will not be needed. In the general
case treated by Lax existence is proved only for data close to a
constant state (see below) but in the special case of gasdynamics,
it is possible to prove a global result.

The Riemann problem is of current interest because of several
recent papers (e.g., \cite{glimm}, \cite{smoller}) considering
weak solutions of conservation laws, in which existence is proved
by approximating the given Cauchy problem by a sequence of Riemann
problems. The Riemann problem is the Cauchy problem for
one-dimensional gasdynamics with the initial data
\[
\mathbf{v}(0,x)=\begin{cases}
\mathbf{v}_r&x\ge0,\\
\mathbf{v}_l&x<0,
\end{cases}
\]
where $\mathbf{v}_r$ and $\mathbf{v}_l$ are any 3D vectors and
generally
\[
\mathbf{v}(t,x) = \begin{pmatrix}
\rho(t,x)\\ p(t,x)\\ u(t,x) \end{pmatrix}
\]
($t\ge0$ and $-\infty<x<\infty$) and $\rho$, $p$, $u$ are the
density, pressure, and velocity.  The function $v$ must satisfy
the equations of gasdynamics  and the Rankine-Hugoniot conditions
\cite[eqns. 34.06 and 54]{cf1948}. We assume that the gas is
polytropic, i.e.,
\[
p=\text{Const.}\times e^{S/c_v}\rho^\gamma,
\]
where $c_v$ is the specific heat at constant volume, $S$ is the
entropy per unit mass, and $1 < \gamma < 2$. The sound speed is
then $c=\sqrt{\gamma p/\rho}$.

\section{Method of solution of the Riemann problem}

Following Lax we restate the results of \cite{cf1948} as follows:
given any state $\mathbf{v}_l$ there exist three one-parameter
families of states
\[
\{B(y)\mathbf{v}_l\},\quad \{C(y)\mathbf{v}_l\},\quad
\{F(y)\mathbf{v}_l\}
\]
$-\infty<y<\infty$  such that
 \begin{itemize}
\item $\mathbf{v}_l$ can be joined to $B(y)\mathbf{v}_l$ on
  the right by a backward-facing shock (if $y < 0$)
  or a centered backward-facing rarefaction wave (if $y > 0$)

\item $\mathbf{v}_l$ can be joined to $C(y)\mathbf{v}_l$ on the
 right by a contact discontinuity.

\item $\mathbf{v}_l$ can be joined to $F(y)\mathbf{v}_l$ on the
 right by a forward-facing shock (if $y < 0$) or a centered
 forward-facing rarefaction wave (if $y > 0$).
\end{itemize}

We solve the Riemann problem by finding $y_1,y_2,y_3$ such that
\[
F(y_3)C(y_2)B(y_1)\mathbf{v}_l= \mathbf{v}_r.
\]
The solution then consists of constant states separated by
rarefaction waves, shocks, or contact discontinuities
(see \cite{cf1948}).

In the following sections we determine the maps $B,C,F$. Actually,
they are given almost explicitly in \cite{cf1948},
and we use these results. The choice of the logarithm of the
pressure ratio as a parameter makes the formulas explicit except for
the inversion of one scalar function.

\section{One-parameter family of forward-facing waves}

First we shall derive the set of all states reached from
$\mathbf{v}_l$ by a centered forward-facing rarefaction wave. In
this case  $p_r>p_l$, see \cite[Section 81]{cf1948} so we can
write
\[
\frac{p_r}{p_l}=e^y, \quad y\ge0.
\]
Then
\[
\frac{p_r}{p_l}=\Big(\frac{\rho_r}{\rho_l}\Big)^{1/\gamma}
=e^{y/\gamma}
\]
(see \cite[Eqn. 40.10]{cf1948}). Finally \cite[Eqn. 40.09]{cf1948}
implies
\[
\frac{u_r-u_l}{c_l}=\frac{2}{\gamma-1}(e^{\tau y}-1),
\]
where
\[
\tau=\frac{\gamma-1}{2\gamma}
\]
and $c_l$ is the left-hand value of the sound speed.
Now we compute the set of states reached from $\mathbf{v}_l$
by a forward-facing shock. We need the following two equations,
\cite[Eqn. 67.02]{cf1948}
\begin{equation} \label{67.02}
\frac{\rho_1}{\rho_0}=\frac{p_1+\mu^2p_0}{p_0+\mu^2p_1},\quad
\mu^2=\frac{\gamma-1}{\gamma+1}
\end{equation}
and \cite[Eqn. 71.05]{cf1948}
\begin{equation} \label{71.05}
\frac{|u_1-u_0|}{c_0}=\frac{1-\mu^2}{\sqrt{1+\mu^2}}
\frac{p_1-p_0}{p_0}\sqrt{\frac{p_0}{p_1+\mu^2p_0}}
\end{equation}
where the subscript $0$ stands for the state in front of
(i.e., ``before") the shock, and $1$ stands for the state in
back of (i.e., ``after") the shock. In a forward-facing wave,
particles cross the shock from right to left, so $1 = l$ and $0 = r$.
Furthermore in a forward-facing shock
$p_r\le p_l$, so we can write
\[
\frac{p_r}{p_l}=e^x\quad (x\le0).
\]
Then \eqref{67.02} implies
\[
\frac{\rho_r}{\rho_l}=\frac{e^x+\mu^2}{1+\mu^2e^x}\,.
\]
In a forward-facing shock $u_1 \ge u_0$  so \eqref{71.05} implies
\[
\frac{u_r-u_l}{c_l}=-\frac{1-\mu^2}{\sqrt{1+\mu^2}}
\frac{1-e^x}{\sqrt{\mu^2+e^x}}.
\]
The above calculations show that
\[
F(y)\mathbf{v}=\begin{pmatrix}
f_3(y)v_1\\e^yv_2\\v_3+\sqrt{\frac{\gamma
v_2}{v_1}}h_3(y)\end{pmatrix},
\]
where
\[
h_3(y)=
\begin{cases}
\frac{2}{\gamma-1}(e^{\tau y}-1)&y\ge0\\[4pt]
\sqrt{\frac{2}{\gamma(\gamma+1)}}\frac{e^y-1}{\sqrt{\mu^2+e^y}}&y<0
\end{cases}
\]
and
\[
    f_3(y)=\begin{cases}
    e^{y/\gamma}& y\ge0\\
    \frac{\mu^2+e^y}{1+\mu^2e^y}& y<0.
    \end{cases}
\]

\section{One-parameter family of backward-facing waves}

The procedure is as in the previous section. The results are
\[
B(y)\mathbf{v}=\begin{pmatrix}f_1(y)v_1\\e^{-y}v_2\\
v_3+\sqrt{\frac{\gamma v_2}{v_1}}h_1(y)\end{pmatrix},
\]
where
\[
f_1(y)=1/f_3(y)\quad -\infty<y<\infty
\]
and
\[
h_1(y)= \begin{cases}
\frac{2}{\gamma-1}(1-e^{-\tau
y})&y\ge0\\[4pt]
\sqrt{\frac{2}{\gamma(\gamma+1)}}
\frac{1-e^{-y}}{\sqrt{\mu^2+e^{-y}}}&y<0.
\end{cases}
\]

\section{One-parameter family of contact discontinuities}

In this case the density suffers an arbitrary change while $p$ and
$u$ are constant. Thus we may take
\[
C(y)\mathbf{v}=\begin{pmatrix}
e^yv_1\\v_2\\v_3
\end{pmatrix}.
\]

\section{Solution of the Riemann problem}

We need the following observation:
\begin{equation} \label{e1}
\frac{h_3(x)}{\sqrt{f_1(x)}}\frac{e^{-x/2}}{h_1(x)}=1\quad
-\infty<x<\infty.
\end{equation}
We define
\begin{gather*}
A_\rho=\frac{\rho_r}{\rho_l},\\
A_p=\frac{p_r}{p_l},\\
A_u=\frac{u_r-u_l}{c_l}.
\end{gather*}
It is easy to see that $h_1$ is monotonic increasing and that
\[
h_1([-\infty,\infty])=[-\infty,\frac{2}{\gamma-1}].
\]
Therefore, the equation
\begin{equation} \label{e2}
h_1(y_1)+\sqrt{\frac{A_p}{A_\rho}}h_1\left(y_1+\log A_p\right)=A_u
\end{equation}
has a unique solution $y_1$ if and only if
\begin{equation} \label{e3}
A_u < \frac{2}{\gamma-1}\Big( 1+\sqrt{A_p/A_\rho}\Big).
\end{equation}
Let
\begin{gather*}
y_3=y_1+\log A_p \\
y_2=\log\Big( \frac{A_\rho f_1(y_3)}{f_1(y_1)}  \Big).
\end{gather*}
It is then a straightforward calculation to show that
\[
F(y_3)C(y_2)B(y_1)\mathbf{v}_l=\mathbf{v}_r
\]
so that we have a solution to the Riemann problem for arbitrary
 $\mathbf{v}_r$ and $\mathbf{v}_l$ satisfying \eqref{e2}.
\footnote{Physical interpretation: if \eqref{e3} is violated, the two
sections of the gas are moving away from one another so fast
that a vacuum is formed}

Note that this solution is explicit except for the inversion of
only one function, i.e. \eqref{e2}. Thus the method is well-suited
to computations.

We also have some information about the solution.
The backward wave is a rarefaction wave if and only if $y_1>0$,
which by \eqref{e2} is equivalent to
\[
\sqrt{\frac{A_p}{A_\rho}}h_1(\log A_p)< A_u<\frac{2}{\gamma-1}
\Big(1+\sqrt{A_p/A_\rho}\Big)
\]
 and is a shock otherwise. Similarly, we find that the forward-facing
wave is a rarefaction wave if and only if
$y_3>0$, which by \eqref{e2} is equivalent to
\[
h_1(-\log A_p) < A_u <\frac{2}{\gamma-1}
\Big(1+\sqrt{A_p/A_\rho}\Big)
\]
and is a shock otherwise. These explicit criteria seem to be new.

\section{Overtaking of two weak forward-facing shock waves}

We now apply the above results to the interaction of two forward-facing
shock waves. This problem has been treated by von Neumann (unpublished).
(However, see related material in \cite{vn}.)  He showed that
if $\gamma \le 5/3$, then the resulting configuration after the
waves meet consists of a backward-facing rarefaction wave,
a contact discontinuity, and a forward-facing transmitted shock.
The proof is included in the report \cite{cf1943}.

The criteria derived at the end of Section 6 immediately enable
us to formulate the problem as an inequality. As a simple
application, we prove von Neumann's result for sufficiently
\textbf{weak} interacting shocks: the resulting backward wave
\textbf{is} a rarefaction wave if $\gamma<5/3$ and \textbf{is not}
a rarefaction wave if $\gamma >5/3$.

The problem is formulated as follows. We are given three states,
$\mathbf{v}_l$, $\mathbf{v}_m$, $\mathbf{v}_r$ such that
\begin{gather*}
\mathbf{v}_m=F_s(\mathbf{v}_l)\quad s<0\\
\mathbf{v}_r=F_t(\mathbf{v}_m)\quad t<0
\end{gather*}
and we wish to solve the Riemann problem with initial states
$\mathbf{v}_l$, $\mathbf{v}_r$. From these equations we see
\[
\mathbf{v}_r=F_t(F_s(\mathbf{v}_l))
\]
which implies
\begin{gather*}
A_\rho =f_3(t)f_3(s)\\
A_p =e^{t+s}\\
A_u =h_3(s)+h_3(t)\sqrt{e^s/f_3(s)}.
\end{gather*}
 From Section 6 we know that (assuming \eqref{e3}) the backward-facing
 wave is a rarefaction wave if and only if
\[
A_u>\sqrt{\frac{A_p}{A_\rho}}h_1(\log A_p).
\]
Using \eqref{e1} we see that this is equivalent to
\begin{equation} \label{e4}
h_1(t+s)<h_1(t)+\frac{h_1(s)h_1(t)}{h_3(t)}.\end{equation}
Let
\begin{gather*}
\beta:=\frac{\gamma+1}{\gamma-1}\\
\psi(x):=\frac{1-e^{-x}}{\sqrt{1+\beta e^{-x}}}\\
G(t,s):=\psi(t+s)\psi(-t)-\psi(t)\psi(-t)+\psi(t)\psi(s)\\
\phi(x):=\psi(x)/x
\end{gather*}
Then \eqref{e4} is equivalent to
\[
G(t,s)<0 \quad \text{for } t<0, \; s<0.
\]
The following rearrangement:
\begin{align*}
\frac{G(t,s)}{t^2(s+t)}&=2\big\{\frac{\phi(s)-\phi(-t)}{s+t}\big\}
\big\{\frac{\phi(t)-\phi(-t)}{s+t}\big\}\\
&\quad -\phi(-t)\Big\{\frac{\frac{\phi(s+t)-\phi(t)}{s}
-\frac{\phi(s)-\phi(-t)}{s+t}}{t}\Big\}
\end{align*}
allows us to calculate the limit
\begin{align*}
\lim_{s,t\uparrow0}\frac{G(t,s)}{t^2s(s+t)}
&=2\phi'(0)^2-\frac{3}{2}\phi(0)\phi''(0)\\
&=\frac{3}{64}\frac{\gamma^2-1}{\gamma^3}
\big(\gamma-\frac{5}{3}\big),
\end{align*}
which proves the stated result.


\begin{thebibliography}{0}

\bibitem{cf1943}
R. Courant and K. O. Friedrichs;
\emph{Interaction of shock and rarefaction waves in one-dimensional media},
National Defense Research Committee,
Applied Mathematics Panel Report 38.1R, PB32196, AMG-1, 1943.

\bibitem{cf1948} R. Courant and K. O. Friedrichs;
\emph{Supersonic Flow and Shock Waves}, Interscience, New York, 1948.

\bibitem{glimm} J. Glimm;
\emph{Solutions in the large for nonlinear hyperbolic systems
of equations}, Comm. Pure Appl. Math. \textbf{18}, 697-715, 1965.

\bibitem{lax} P. D. Lax;
\emph{Hyperbolic systems of conservations laws II},
Comm. Pure Appl. Math. \textbf{10}, 537-566, 1957.

\bibitem{vn} J. von Neumann;
\emph{The theory of shock waves}, in Collected Works, Pergamon Press,
Oxford, 1963.

\bibitem{smoller} J. A. Smoller and J. L. Johnson;
\emph{Global solutions of hyperbolic systems of conservation laws},
Arch. Rational Mech. Anal. \textbf{32}, 169-189, 1969.

\bibitem{smoller1983} J. A. Smoller;
\emph{Shock Waves and Reaction-Diffusion Equations},
Springer, New York, 1983.

\end{thebibliography}

\end{document}