Relativistic Flows Using Spatial And Temporal Adaptive Structured Mesh Refinement. I. Hydrodynamics Page: 2 of 13
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2
ing solved before we give details on the different solvers
we have implemented. Section 4 discusses the adap-
tive mesh refinement strategy and implementation. We
then move on to describe various test problems for rele-
vant combinations of solvers, reconstruction schemes in
one, two and three dimensions, with and without AMR.
Section 6 presents an application of our code to three-
dimensional relativistic and supersonic jet propagation
problem before we summarize in section 7.
2. EQUATIONS OF SPECIAL RELATIVISTIC
HYDRODYNAMICS
The basic equations of special relativistic hydrodynam-
ics (SRHD) are conservation of rest mass and energy-
momentum:(pv);p 0,
and
T - 0,
(1)
(2)where p is the rest mass density measured in the fluid
frame, u W(1, v2) is the fluid four-velocity (assum-
ing the speed of light c 1), W is the Lorentz factor,
v2 is the coordinate three-velocity, T " is the energy-
momentum tensor of the fluid and semicolon denotes co-
variant derivative.
For a perfect fluid the energy-momentum tensor is
T " =phu'u" + pg"", (3)
where h 1 + E + p/p is the relativistic specific enthalpy,
E is the specific internal energy, p is the pressure and g"">
is the spacetime metric.
SRHD equations can be written in the form of conser-
vation lawstU 3 &Fj
at + xj ',
j=1(4)
3. NUMERICAL SCHEMES FOR SRHD
3.1. Time Integration
We use method of lines to discretize the system (4)
spatially,dU2,j,k F+1/2,jk F-l/2,j,k
dt Ax
Fij+1/2,k -Fj-1/2,k
FIj,k+1/2 - j,k-1/2
Az(12)
L(U), (13)
where i, j, k refers to the discrete cell index in x, y, z di-
rections, respectively. Fp11/2,j,k, F 1/2,k and Fzjk 1/2
are the fluxes at the cell interface.
As discussed by Shu & Osher (1988), if using a high
order scheme to reconstruct flux spatially, one must also
use the appropriate multi-level total variation diminish-
ing (TVD) Runge-Kutta schemes to integrate the ODE
system (13). Thus we implemented the second and third
order TVD Runge-Kutta schemes coupled with AMR.
The second order TVD Runge-Kutta scheme reads,US)= Un + +tL(U!),U
Un+1 _ U2 + 2 U) + -AtL(US)),(14)
(15)and the third order TVD Runge-Kutta scheme reads,
U(= U + AtL(Un)
U = 4Un+ US) + !AtL(Uw)
4 4 4
1 2 2
Un+1 _ U"+ -U(2) +-A (U ),
3 3 3(16)
(17)
(18)where the conserved variable U is given by
U (D, S1 S2 g3 7)T, (5)
and the fluxes are given by
F - (Dvy, Siv +pio1, S2ri+pbi2, S3i2+p& 3, Si-Dv)T .
(6)
Anile (1989) has shown that system (4) is hyperbolic
for causal EOS, i.e., those satisfying cs < 1 where the
local sound speed c8 is defined as
c2 = + . (7)
h[&p Kp2)&cjThe eigenvalues and left and right eigenvectors of the
characteristic matrix &F/&U, which are used in some of
our numerical schemes, are given by Donat et al. (1998).
The conserved variables U are related to the primitive
variables by
D=pW, (8)
SJ=phW2vJ, (9)
w=phW2-p--pW, (10)
where j = 1, 2, 3. The system (4) are closed by an equa-
tion of state (EOS) given by p = p(p, e). For an ideal
gas, the EOS is,
p (F - 1)pe, (11)
where F is the adiabatic index.where Un+1 is the final value after advancing one time
step from U.
For an explicit time integration scheme, the time step is
constrained by the Courant-Friedrichs-Lewy (CFL) con-
dition. The time step is determined as(19)
dt = Cmin ( A
where C is a parameter called CFL number and a1 is
the local largest speed of propagation of characteristics
in the direction i whose explicit expression can be found
in Donat et al. (1998).3.2. Reconstruction Method
Generally speaking, there are two classes of spatially
reconstruction schemes (see e.g. LeVeque 2002). One
is reconstructing the unknown variables at the cell in-
terfaces and then use exact or approximate Riemann
solver to compute the fluxes. Another is direct flux re-
construction, in which we reconstruct the flux directly
using the fluxes at cell center. To explore the coupling of
different schemes with AMR as well as exploring which
method is most suitable for a specific astrophysical prob-
lem, we have implemented several different schemes in
both classes.
To reconstruct unknown variables, we have imple-
mented piecewise linear method (PLM, Van Leer 1979),
piecewise parabolic method (PPM, Colella & Woodward
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Wang, Peng; Abel, Tom; Zhang, Weiqun & /KIPAC, Menlo Park. Relativistic Flows Using Spatial And Temporal Adaptive Structured Mesh Refinement. I. Hydrodynamics, report, April 2, 2007; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc882066/m1/2/: accessed April 2, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.