ADVANCED METHODS IN PARTICLE HYDRODYNAMICS
SHU-ICHIRO INUTSUKA
National Astronomical Observatory
Mitaka, Tokyo 181-8588, Japan
We have developed numerical methods that use particles to solve
hydrodynamics and radiation hydrodynamics and reformulated Smoothed
Particle Hydrodynamics using a new action principle. The force acting on
each particle was determined by solving the Riemann problem. Use of the
Riemann Solver strengthened the method, making it accurate for the study
of strong shock phenomena. These techniques were implemented in conservation form. The method was also extended to multi-dimensional radiation hydrodynamics without invoking any diusion approximation, being
equally applicable to optically thin and thick regimes. The results of test
problems in 1D, 2D, and 3D are shown.
Abstract.
1.
Introduction
Since its invention by Lucy (1977) and Gingold & Monaghan (1977),
smoothed particle hydrodynamics (SPH) has been used to study a variety of astrophysical problems. A broad discussion of the method can be
found in a review by Monaghan (1992). However, some of its disadvantages
remain to be overcome. SPH is less accurate with strong shocks than stateof-the-art nite dierence schemes such as the Piecewise Parabolic Method
(PPM, Colella & Woodward, 1984). Especially in the three-dimensional calculation of colliding gases, particles often penetrate into the opposite side,
and large post-shock oscillations often appear, contaminating the numerical
results.
In this paper, a new method to handle shocks in particle hydrodynamics
is constructed (cf., Inutsuka, 1994, 1995). The force acting on each particle
is determined by solving the Riemann problem (RP). Use of the Riemann
Solver, the main ingredient of Godunov's method (1959), strengthens the
accuracy of the method for the study of strong shock phenomena. The
368
SHU-ICHIRO INUTSUKA
method may thus be regarded as a multi-dimensional Lagrangian sequel to
Godunov's method.
Section 2 provides a description of the method. A few numerical examples involving a strong shock are presented in Section 3. In Section 4, we
extend the application of the method to radiation hydrodynamics.
2.
The method
In particle methods, each particle has its own mass, and its density is
estimated by
X
i =
mj W (xi 0 xj ; h1 );
(1)
j
where subscripts denote particle labels, mj is the mass of the j -th particle,
and W (x; h1) is a spherically symmetric kernel function. Although there
are many possible forms for W (x; h1), a typical form is the following:
1 d 2 2
p e0x =h1 :
W (x; h1 ) =
h1 (2)
In the above equation, d is the number of dimensions, and h1 is a smoothing
length which can vary. In this brief paper, we will concentrate on a case
where the smoothing length is constant.
The standard procedure to derive sets of evolution equations for SPH
has been previously described (e.g., Monaghan, 1992). In this paper, we
will show how evolution equations are derived from action principles.
2.1.
NEW ACTION PRINCIPLE
Barotropic uid has a Lagrangian density L expressed in Lagrangian coordinates:
Z
Z 1
v 2 0 u dx:
(3)
L = Ldx =
2
The equation of motion for the uid can be derived by minimizing the
action. By analogy, a new Lagrangian for the system of particles can be
dened by the following equation:
X 1 2 Z
Lnew =
mk vk 0 u(x)W (x 0 xk ; h2 )dx :
(4)
2
k
Using the denition of density,
(x) =
X
j
mj W (x 0 xj ; h1 );
(5)
ADVANCED METHODS IN PARTICLE HYDRODYNAMICS
369
the Euler-Lagrange equation,
d @Lnew
dt @ x_ i
gives
x i = 0
+
X
j
X
j
mj
@
@ xi
@
mj
@ xj
Z
Z
0 @L@ x = 0;
new
i
(6)
P
W (x 0 xi ; h1 )W (x 0 xj ; h2 )dx
2
P
W (x 0 xi ; h2 )W (x 0 xj ; h1 )dx:
2
(7)
The space-symmetry of Lnew guarantees the conservation of linear momentum and angular momentum. In eq.(7), h1 and h2 are not necessarily the
same. The choice of h1 or h2 gives dierent equations of motion.
If we take h2 = 0, then W (x; h2 ) = (x), and the above equation
reduces to the standard SPH equation:
Case 1
x i = 0
X
j
mj
If we take h2 =
following equation:
Case 2
x i = 0
X
j
mj
@
@ xi
Pi
2i
h1
@
0 @x
+
Pj
2j
!
@
W (xi 0 xj ; h1 ):
@ xi
(8)
= h, then the above equation provides the
!Z
j
P (x)
W (x 0 xi ; h)W (x 0 xj ; h)dx
2 (x)
(9)
This equation has several good properties compared with equation (8). For
example, the right-hand side vanishes identically when P (x) is constant in
space, while in equation (8) it does not.
2.2.
THE INTRODUCTION OF THE RIEMANN SOLVER
At this point, the Riemann Solver is introduced into the particle method.
The (nite dierence) Godunov scheme uses the result of RP at each cell
interface (van Leer, 1979). Here, we also use the result of RP at the middle
point of the i-th particle and the j -th particle in the following:
x i = 0
X
j
mj P 3
@
@ xi
0
@
@ xj
!Z
1 W (x 0 x ; h)W (x 0 x ; h)dx (10)
i
j
(x)
2
370
SHU-ICHIRO INUTSUKA
where P 3 is the result of RP between the i-th particle and the j -th particle.
Similarly, the equation of energy becomes the following:
!Z
X
@
@
W (x 0 xi ; h)W (x 0 xj ; h)
3
u
i = 0 mj P (v i 0 vj ) @ x 0 @ x
dx:
2 (x)
i
j
j
(11)
3.
Numerical examples
The present method was tested on a variety of 1D, 2D, and 3D problems, a
few of which are described below. In all calculations, a variable smoothing
length was used. The computation time needed was quite similar to that
needed by standard SPH.
3.1.
SHOCK TUBES
3.1.1. Sod's problem
In Fig. 1, the results of 1D Sod's shock tube (Sod, 1978) are plotted. Solid
curves correspond to the analytic solution. The number of equal-mass particles in the x-direction is only 40 on the right-hand side of the initial
discontinuity. Shock surfaces are resolved by small number of particles.
Sod’s Shock Tube
Blast Wave
25
20
Velocity
Velocity
1
.5
15
10
5
0
0
8
1.2
1
Density
Density
6
.8
.6
.4
4
2
.2
0
0
-.4
-.2
0
Fig. 1
.2
.4
-.4
-.2
0
Fig. 2
.2
.4
3.1.2. Blast wave
Fig. 2 plots the result of an extremely strong shock-tube problem (Woodward & Colella, 1984) in which the Mach number was 200. Only 100 particles were used in the x-direction on each side of the initial discontinuity.
Initial pressure of the uid on the left-hand side was 105 times that of righthand side. Even in this kind of severe problem, the present method gave
stable and accurate results and was free from penetration problems.
371
ADVANCED METHODS IN PARTICLE HYDRODYNAMICS
3.2.
FIFTEEN-DEGREE WEDGE CHANNEL FLOW
This problem, evaluated by Levy, Powell, & van Leer (1993), was a useful
test to calibrate the method. The geometry was a two-dimensional channel
with a 15-degree wedge on the lower wall. A 15-degree expansion corner was
also included. The inow Mach number was two. It was noticed, however,
that this kind of problem is poorly suited for the particle method because
(1) a rigid-wall boundary condition must be set up at the wall of the tunnel;
and (2) in-owing particles have to be prepared at all times at the left-side
boundary.
15 Degree Wedge Channel Flow
1.0
0.5
0.0
-1.0
-0.5
0.0
0.5
Fig.3
Fig.4
1.0
1.5
2.0
Figure 3 shows our method's result. Mach-number contours from 1.0 to
2.0 are plotted. Figure 4 shows particle positions. In this calculation, we realized the rigid-wall boundary condition by placing \ghost particles" in the
wall. Initially, 32296 particles were owing inside the channel, which must
be compared to the 32296 grids nite dierence calculation by Levy et al.
Our results were satisfactory. The main dierence between the results obtained by Levy et al. and ours was the thickness of the numerical boundary
layer just above the lower wall. This was due to the fact that ghost particles
372
SHU-ICHIRO INUTSUKA
inside the expansion corner do not realize reecting boundary conditions
completely, which causes a relatively thick numerical boundary layer. It is
clear that a nite dierence method must be used in the actual study of
this kind of rigid-wall boundary problem. This result is presented only to
demonstrate the capability of this particle scheme.
4.
Extension to radiation hydrodynamics
Multi-dimensional calculations of radiation hydrodynamics (RHD) for modeling astrophysical phenomena constitute a frontier in theoretical astrophysics. Most RHD calculations presented to date invoke diusion approximation at some level to simplify calculations. In such calculations, the
direction of radiative energy ow is restricted to the direction of the temperature gradient of the media. However, that kind of treatment is inadequate
in optically thin media. In this paper, a self-consistent description of radiation and uid is presented by a Lagrangian particle scheme. This method
is equally applicable to optically thin and thick media in three-dimensional
calculations.
4.1.
METHOD
In this section, we consider the following equations of RHD in Lagrangian
coordinates:
d
+ r 1 v = 0;
(12)
dt
dv
dt
du
dt
Z
1
1
= 0 rP 0 r8 +
= 0 P r 1 v 0 4
Z
c
F d
S d + c
Z
E d
(13)
(14)
= ( 0 1)u;
(15)
where c, u, , S , E , and F are the speed of light, specic energy, mass
opacity, source function, radiation energy density, and radiative ux, respectively. Other symbols have their usual meanings. The transfer equation
is
P
dI
ds
= (S 0 I ):
(16)
Hereafter, we suppress explicit mention of -dependence for the sake of
brevity. Note that the present scheme is not restricted to gray problems.
ADVANCED METHODS IN PARTICLE HYDRODYNAMICS
373
4.1.1. Algorithm
From the formal solution of eq.(16), we can construct the radiation energy
density at the position of the i-th particle as
1 X m S Z e0 (x;x ) W (x 0 x )dx;
Ei =
(17)
j j j
j
c j
R(x; xi )2
where (x; xi ) and R(x; xi ) are the optical depth and geometrical distance
from the position x to the j -th particle. Similarly, the following is found
for radiative ux:
i
Fi =
X
j
j mj S j
Z e0 (x;x )
i
R(x; xi )3
(xi 0 x)W (x 0 xj )dx:
(18)
4.1.2. 1D Analogue
Considering a medium which is uniform in the y - and z -directions, onedimensional analogues of eqs.(17) and (18) reduce to
"Z 1
#
Z1
2
X
e0
Ei =
j mj Sj
W (x 0 xj )
d dx;
(19)
c
j
Fix
= 2
X
j
j mj Sj
Z1
01
01
(x;xi )
sign(xi 0 x) W (x 0 xj )
"Z 1
(x;xi )
#
e0
d dx:
2
(20)
In uniform optically thick medium, where S (= B : black body radiation), , and are constant in space, Ei reduces to the energy density of
black body radiation:
Z 1 "Z 1 e0 #
4
4 B:
E =
B
d d =
(21)
i
4.2.
c
0
c
TEST ON THERMAL RELAXATION MODES
To test the ability of the present scheme, we considered temperature perturbations imposed on a static, gray, LTE ambient medium, initially in
radiative equilibrium. The governing equation for gas energy reduced to
CV
dT
dt
= 04S + cE:
(22)
The standard normal mode analysis (T = T0 + T1 = T0 + eikx+nt ) provided
the dispersion relation for the linear uctuation (Spiegel, 1957):
16SB T03 1 0 cot01 ;
n(k ) = 0
(23)
CV
k
k
374
SHU-ICHIRO INUTSUKA
which was the exact solution of the problem. Likewise, the Eddington approximation provided the following results (Unno & Spiegel, 1966):
"
2 #01
16
SB T03
n(k ) = 0
1+3 k
:
C
(24)
V
Figures 5 and 6 plot the result obtained in our scheme. The left panel
shows a 1D case in which Np = = 100, and the right panel shows a 3D case
in which Np= = 3:25. The solid curve denotes the exact solution (eq.[23]),
while the dashed curve denotes the Eddington approximation (eq.[24]). Circles correspond to the numerical result of our scheme, which shows excellent
agreement with the exact solution. Note that the Eddington approximation
produces errors at k= > 1 .
Fig.6 Thermal Relaxation Modes (3D)
1.2
1
1
Damping Rate × tR
Damping Rate × tR
Fig.5 Thermal Relaxation Modes (1D)
1.2
.8
.6
.4
.2
.8
.6
.4
.2
0
0
-4
-2
0
log10 [ k / (κρ) ]
2
4
-2
-1
0
1
2
log10 [ k / (κρ) ]
References
Colella, P., & Woodward, P. (1984) JCP, 54, 174.
Godunov, S. K. (1959) MSB, 47, 271.
Gingold, R. A., & Monaghan, J. J. (1977) MNRAS, 181, 375.
Inutsuka, S. (1994) Mem. S. A. It., 65, 1027.
Inutsuka, S. (1995) in
Makino & P. Hut, (Tokyo: Kluwer)
Levy, D. W., Powell, K. G., & van Leer, B. (1993) JCP, 106, 201.
Lucy, L. (1977) AJ, 82,1013.
Monaghan, J. J. (1992) ARAAP, 30, 543.
Sod, G. A. (1978) JCP, 27, 1.
Spiegel, E. A. (1957) ApJ, 126, 202.
Unno, W., & Spiegel, E. A. (1966) PASJ, 18, 85.
van Leer, B. (1979) JCP, 32, 101.
Woodward, P., & Colella, P. (1984) JCP, 54, 115.
. ed. J.
Dynamical Evolution of Star Clusters, IAU Symp. 174