Godunov's method and several other methods for computing solutions to the equations of gas dynamics use Riemann solvers to resolve discontinuities at the interface between cells. A new method is proposed here for solving the Riemann problem based on a global existence proof for the solution to the Riemann problem. The method is found to be very reliable and computationally efficient

Dutt, P.

English

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NASA Contnetor Report 178042
lCASE REPORT NO. 86-3 NASA-CR-I7804219860011681
leASE
A RIEMANN SOLVER BASED ON A GLOBAL EXISTENCE PROOF
FOR THE RIEMANN PROBLEM
Pravir Dutt .-- -.-
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I, • ,
. - ,,,'" .'
Contract Nos. NASl-17070 and NASl-18107
January 1986
'. -\ • _ _.- •• 1.
INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665
Operated by the Universities Space Research Association
NI\SI\
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23665
https://ntrs.nasa.gov/search.jsp?R=19860011681 2020-03-20T15:15:58+00:00Z

A RIEHANN SOLVER BASED ON A GLOBAL EXISTENCE PROOF
FOR THE RIEMANN PROBLEM
Pravir Dutt
Institute for Computer Applications in Science and Engineering
ABSTRACT
Godunov's method and several other methods for computing solutions to the
equations of gas dynamics use Riemann solvers to resolve discontinuities at
the interface between cells. A new method is proposed here for solving the
Riemann problem based on a global existence proof for the solution to the
Riemann problem. The method is found to be very reliable and computationally
efficient.
Research was supported by the National Aeronautics and Space Administration
under NASA Contract Nos. NASI-17070 and NAS1-18107 while the author was in
residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225.
i

I. INTRODUCTION
The introduction of a finite-difference scheme based on Riemann solvers by
Godunov permitted the development of robust numerical methods for problems
involving strong shocks. He proposed to consider all quantities in a
computational cell as given by cell averages and to resolve the resulting
discontinuities at the cell edges by the solution to a Riemann problem. The
method has the important advantage that it automatically handles strong shocks
and interactions and is able to predict cavitation should it occur.
The use of Godunov's method has attracted particular interest in the field
of aerodynamics. Here one expects to encounter only ideal gas (polytropic)
equations of state. A Riemann solver for the equations of gas dynamics was
first proposed by Godunov. It was subsequently modified and improved by
Chorin [2].
In this paper we take advantage of a _lobal existence proof for the
Riemann problem to propose an altogether different method for solving the
Riemann problem. This method has the advantage that it is computationally
very efficient and is guaranteed to converge to the solution of the Riemann
problem.
2. THE RIEMANN PROBLEM IN GAS DYNAMICS
The gas dynamics equations in Eulerian coordinates are
Pt + (PU)x= 0
(PU)t + (pu2 + P)x = 0
-2-
et + ((e + p)U)x = 0
where p is the density of the gas, u is the velocity, pu is the momentum,
e is the energy per unit volume, and p is the pressure.
We assume that the gas is ideal and polytropic so that the equation of
state is given by
p = pRT
where p is the pressure, p the density, T the temperature, and R the
universal gas constant. Further, we assume that y > i, where y is the
ratio of specific heats.
We seek a solution to the gas dynamics equations with initial data
H[ I= for x < 0LPLJ
and
= for x > O.
Lp j
The solution to the Riemann problem is illustrated in Figure 1 on a
x - t graph.
-3-
2-contactdiscontinuity/
# u /
/
p_ / ./ PR'u,
/
/ P _ckor
/ rarefaction/
PL'UL' //
PL PR'UR'
PR
:_:='--X
Figure |: Solution to the Riemann problem.
The initial discontinuity is resolved by a system of waves consisting of a
l-shock or rarefaction, a 2-contact discontinuity, and a 3-shock or rare-
faction. If cavitation is excluded the pressure and velocity on either side
of the contact discontinuity are equal and are given by p and u .
3. THE GLOBAL EXISTENCETHEOREM
A parametric representation of the solution to the Riemann problem and
global existence proof are discussed in Smoller [3]. We outline the
discussion to relate it to our numerical scheme. Let
B =Y + 1
y - 1
and
"f- 1
T - 27 •
-4-
We denote the sound speed by c, where c = (YP)I/2 . Thus cL is the
sound speed for the state (PL,UL,PL). The solution to the Riemann problem is
obtained by the following three one-parameter family of curves.
1 - family, for xI E ]P_
* -x1
P---= e
PL
-Xl/Y
, e , Xl>_0
PL
PL fl (Xl) - x1
B+e
xI ' x1<0
1+Be
I _zx 1
u* - y 2 1 (i - e ), Xl_> 0uL
CL = hi (Xl) - /o -Xl
2(_) i0= 1 - e < 0
x 1--X
_ _ [ (i + Be 1) I/2
2 - family, for x2 €
OR x2
-- = e
PL
-5-
3 - family, for x3 E
PR x3
p, - e
OR 1
, - f3(x3) : f1(x3)
PR
2 Tx3
* y - 1 (e - I), x3 _>0
uR - u
, = h3(x3) -
cR
x3
2(T) I/2 (e - I)
x ' x3 <0.
y - 1 I1 + 8e 3)1/2
To obtain a solution of the Riemann problem, we need to solve the above
system of equations for the parameters Xl, x2, and x3. It should be noted
that x I < 0 corresponds to a 1-shock and x I > 0 to a 1-rarefaction wave.
A similar statement holds for the parameter x3. Using the above representa-
tion the following theorem can be proved.
Theorem: Let (OL,UL,PL) an___d (PR,UR,PR) be any two states (not
necessarily close). Then there is a unique solution to the Riemann problem
with these initial states if and only if
2
uR - uL <- (cL + cR).y- 1
If this condition is violated, then a vacuum is present in the solution.
-6-
4. THE NUMERICAL METHOD
We first check whether
2
uR - uL < _ (cL +y - I CR)"
If the condition is violated, then cavitation occurs.
To obtain the values Xl, x2, and x3 explicitly, we define
0 RA-
PL
PR
B =
PL
uR - uL
C -
cL
We first solve for xl, where xI satisfies the equation:
g(x1) m hl(Xl) + (_)1_hl(Xl + log B) - C = 0.
It should be noted that the function g is a monotone function. Hence, the
equation can be solved efficiently by a numerical routine. We first locate an
interval [a,b] such that g(a) < 0 and g(b) > 0. With a and b as our
initial guesses, a few iterations of the regula-falsi method are needed to
obtain an accurate value for xI.
We then have
x3 = xI + log B.
-7-
Finally, we obtain x2 from the equation:
x2
fl(xl)e f3(x3) = A.
Having obtained xI, x2, and x3 we substitute them into the parametric
representaion for the solution to the Riemann problem and obtain the values
u ' P ' PL' and PR"
5. NUMERICAL RESULTS
We found that over a range of y and on a variety of shock tube problems
our Riemann solver converged to the exact solution within about five
iterations of the regula-falsi method.
6. CONCLUSIONS
A numerical method for solving the Riemann problem in gas dynamics has
been proposed. The method is based on an existence proof for the solution to
the Riemann problem. Its advantages are that it can predict the occurrence of
cavitation and it is computationally very efficient.
-8-
APPENDIX
Computer Program for the Riemann Solver
subroutine riemann
common/input/ul,ur,rhol,rhor,pl,pr
common/output/pstar,ustar
common/parm/gamma,beta,tau,a,b,c
c input:ul,ur--velocltles to the left and right of interface
c rhol,rhor--densities
c pl,pr--pressures
c output:pstar--interface pressure
c ustar--interface velocity
beta=(gamma+l.)/(gamma-l.)
tau=(gamma-l.)/(2.*gamma)
cl=sqrt(gamma*pl/rhol)
cr=sqrt(gamma*pr/rhor)
c check for cavltalton
if((ur-ul).gt.(2.*(cl+cr)/(gamma-1.)))go to 3
a=rhor/rhol
b=pr/pl
c=(ur-ul)/cl
x=0.
c locate interval on which g(x) has a zero
9 if((g(x).gt.0.).or.(g(x+l.).it.0.))then
if(g(x).gt.0.)then
x_x-l.
else
x=x+l.
end if
end if
if((g(x).gt.0.).or.(g(x+l.).it.0.))go to 9
c do regula-falsi iterations
xmax=x+l.
xmln=x
n=l
8 x=xmax-g(xmax)*(xmax-xmin)/(g(xmax)-g(xmln))
if(g(x).it.0.) then
xmin=x
else
xmax=x
end if
n=n+]
if((abs(xmax-xmin).gt.0.0001).and.(n.le.10)) go to 8
xl=xmax
c calculate ustar,pstar
pstar=exp(-l.*xl)*pl
ustar=ul+cl*h1(xl)
return
-9-
3 print *,'cavitation occurs"
return
end
function hl(x)
common/parm/gamma,beta,tau,a,b,c
if(x.gt.O.)then
hl=(2./(gamma-1.))*(1.-exp(-1.*tau*x))
else
h1=2.*sqrt(tau)*(l.-exp(-1.*x))/(gamma-l.)
h1=hl/sqrt(1.+beta*exp(-1.*x))
end if
return
end
function g(x)
external hl
common/parm/gamma,beta,tau,a,b,c
y=x+alog(b)
g=h1(x)+sqrt(b/a)*hl(y)-c
return
end
-10-
_FE_N_S
1. J. K. DUKOWICZ, J. Comput. Phys. 61 (1985), 119.
2. A. J. CHORIN, J. Comput. Phys. 22 (1976), 517.
3. J. SMOLLER, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag,
New York, 1983.



1. Report No. NASA CR-178042 12. Government Accession No. 3. Recipient's Catalog No.
ICASE Report No. 86-3
4. Title and Subtitle 5. Report Date
A RIEMANN SOLVER BASED ON A GLOBAL EXISTENCE January 1986
PROOF FOR THE RIEMANN PROBLEM 6. Performing Organization Code
7. Author(s)
8'ffff!.fjrming Organization Report No.Pravir Dutt
9. Performing Organization Name and Address 10. Work Unit No.
Institute for Computer Applications in Science
and Engineering 11. Contract or Grant No.
Mail Stop 132C, NASA Langley Research Center NASI-17070; NASI-18107
U",~~"nn \7" ,)'l("(.." ."')'),, 13. Type of Report and Period Covered12. SpOlisoring Agency Name and Address Contractor Report
National Aeronautics and Space Administration 14. Sponsoring Agency CodeWashington, D.C. 20546
505-31-83-01
15. Supplementary Notes
Langley Technical Monitor: Submitted to J. Comput. Phys.
J. C. South
Final Report
16. Abstract
Godunov's method and several other methods for computing solutions to the
equations of gas dynamics use Riemann solvers to resolve discontinuities at
the interface between cells. A new method is proposed here for solving the
Riemann problem based on a global existence proof for the solution to the
Riemann problem. The method is found to be very reliable and computationally
efficient.
17. !{ey Words (Suggested by Authors(s)) 18. Distribution Statement
Riemann solver 59 - Mathematics & Computer Science
(General)
6!. - Numerical Analysis
Unclassified - unlimited
10. Security CIa.9sif.(of this report) 120. Security CIa.9sif.(of this page) 21. No. ~f2pages122. P~i~;
Unclassified Unclassified
For sale by the National Technical Information Service, Springfield, Virginia 22161
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A Riemann solver based on a global existence proof for the Riemann problem

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