Flux Limiter Methods in 3D Numerical Relativity

Abstract

New numerical methods have been applied in relativity to obtain a numerical evolution of Einstein equations much more robust and stable. Starting from 3+1 formalism and with the evolution equations written as a FOFCH (first-order flux conservative hyperbolic) system, advanced numerical methods from CFD (Computational Fluid Dynamics) have been successfully applied. A flux limiter mechanism has been implemented in order to deal with steep gradients like the ones usually associated with black hole spacetimes. As a test bed, the method has been applied to 3D metrics describing propagation of nonlinear gauge waves. Results are compared with the ones obtained with standard methods, showing a great increase in both robustness and stability of the numerical algorithm.

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arXiv:gr-qc/0202101v1 28 Feb 2002
FLUX LIMITER METHODS IN 3D
NUMERICAL RELATIVITY
C. Bona, C. Palenzuela
February 4, 2008
Abstract
New numerical methods have been applied in relativity to obtain a
numerical evolution of Einstein equations much more robust and stable.
Starting from 3+1 formalism and with the evolution equations written
as a FOFCH (first-order flux conservative hyperbolic) system, advanced
numerical methods from CFD (computational fluid dynamics) have been
successfully applied. A flux limiter mechanism has been implemented in
order to deal with steep gradients like the ones usually associated with
black hole spacetimes. As a test bed, the method has been applied to
3D metrics describing propagation of nonlinear gauge waves. Results are
compared with the ones obtained with standard methods, showing a great
increase in both robustness and stability of the numerical algorithm.
1Introduction
From the very beginning, 3D numerical relativity has not been an easy domain.
Difficulties arise either from the computational side (the large amount of vari-
ables to evolve, the large number of operations to perform, the stability of the
evolution code) or from the physical side, like the complexity of the Einstein
equations themselves, boundary conditions, singularity avoidant gauge choices,
and so on. Sometimes there is a connection between both sides. For instance,
the use of singularity avoidant slicings generates large gradients in the vicinity
of black holes. Numerical instabilities can be produced by these steep gradients.
The reason for this is that the standard evolution algorithms are unable to deal
with sharp profiles. The instability shows up in the form of spurious oscillations
which usually grow and break down the code.
Numerical advanced methods from CFD (Computational Fluid Dynamics)
can be used to avoid this. Stable codes are obtained which evolve in a more
robust way, without too much dissipation, so that the shape of the profiles of the
evolved quantities is not lost. These advanced methods are then specially suited
for the problem of shock propagation, but they apply only to strongly hyperbolic
systems, where one is able get a full set of eigenfields which generates all the
physical quantities to be evolved. In the 1D case, these methods usually fulfill
the TVD (Total Variation Diminishing) condition when applied to transport
equations. This ensures that no new local extreme appear in the profiles of the
eigenfields, so that spurious oscillations are ruled out ab initio (monotonicity
preserving condition). Unluckily, there is no general method with this property
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in the 3D case, mainly because the eigenfield basis depends on the direction of
propagation. We will show how this can be achieved at least in some cases.
The specific methods we will use are known as flux limiter algorithms. We
will consider plane waves in 3D as a first generalization of the 1D case, be-
cause the propagation direction is constant. This specific direction leads then
to an specific eigenfield basis, so that the 1D numerical method can be easily
generalized to the 3D case.
The algorithm will be checked with a ”Minkowski waves” metric. It can be
obtained by a coordinate transformation from Minkowski space-time. All the
metric components are transported while preserving their initial profiles. The
line element has the following form:
ds2= −H(x − t) dt2+ H(x − t) dx2+ dy2+ dz2
where H(x − t) is any positive function. We can choose a periodic profile with
sharp peaks so both the space and the time derivatives of H(x − t) will have
discontinuous step-like profiles. If we can solve well this case (the most extreme),
we can hope that the algorithm will work as well in more realistic cases where
discontinuities do not appear.
Minkowski waves are a nice test bed because the instabilities can arise only
from the gauge (there are pure gauge after all!). This is a first step to deal
with evolution instabilities in the Einstein equations by the use of flux limiter
methods. This will allow us to keep all our gauge freedom available to deal
with more physical problems, like going to a co-rotating frame or adapting to
some special geometry. Advanced numerical methods take care of numerical
problems so that ’physical’ gauge choices can be used to take care of physics
requirements.
(1)
2The system of equations
We will use the well known 3+1 description of spacetime [1, 2, 3] which starts
by decomposing the line element as follows:
ds2= −α2dt2+ γij(dxi+ βidt) (dxj+ βjdt)
where γij is the metric induced on the three-dimensional slices and βiis the
shift. For simplicity the case βi= 0 (normal coordinates) will be considered.
The intrinsic curvature of the slices is then given by the three-dimensional Ricci
tensor(3)Rij, whereas their extrinsic curvature Kijis given by:
i,j = 1,2,3(2)
∂tγij= −2 α Kij
(3)
In what follows, all the geometrical operations (index raising, covariant deriva-
tions, etc) will be performed in the framework of the intrinsic three-dimensional
geometry of every constant time slice. With the help of the quantities defined
in (2,3), the ten fourdimensional field equations can be expressed as a set of six
evolution equations:
∂tKij= −∇iαj+ α [(3)Rij− 2K2
ij+ tr K Kij− 8π (Tij−T
2γij)] (4)
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plus four constraint equations
(3)R − tr(K2) + (tr K)2= 16πα2T00
∇kKki− ∂i(tr K) = 8πα T0
(5)
i
(6)
The evolution system (4) has been used by numerical relativists since the very
beginning of the field (see for instance the seminal work of Eardley and Smarr
[4]), both in spherically symmetric (1D) and axially symmetric (2D) spacetimes.
By the turn of the century, the second order system (4) has been rewritten as a
first-order flux conservative hyperbolic (FOFCH) system [5, 6, 7] in order to deal
with the generic (3D) case, where no symmetries are present. But the second
order system (4) is still being used in 3D numerical calculations [8], mainly when
combined with the conformal decomposition of Kij as introduced by Shibata
and Nakamura [9, 10]. In the system(3,4) there is a degree of freedom to be
fixed because the evolution equation for the lapse function α is not given. In
the study of Black Holes, the slicing is usually chosen in order to avoid the
singularity [11]:
∂tlnα = −α Q
where:
Q = f(α)trK
(7)
(8)
Three basic steps are needed to obtain a FOFCH system from the ADM system.
First, one must introduce some new auxiliary variables to express the second
order derivatives in space as first order. These new quantities correspond to the
space derivatives:
Ak= ∂klnα ,Dkij= 1/2 ∂kγij
(9)
The evolution equations for these variables are:
∂tAk+ ∂k(α f trK)
∂tDkij+ ∂k(α Kij)
=
=
0
0
(10)
(11)
At the second step the system is expressed in a first order balance law form
∂t? u + ∂kFk(? u) = S(? u) ,(12)
where the array ? u displays the set of independent variables to evolve and both
”fluxes” Fkand ”sources” S are vector valued functions. At the third step
another additional independent variable is introduced to obtain a strongly hy-
perbolic system [11]:
Vi= Dirr− Drri
and its evolution equation is obtained using the definition of Kij from (3) and
switching space and time derivatives in the momentum constraint (6). The
result is an independent evolution equation for Viwhile the previous definition
(13) in terms of space derivatives can be instead be considered as a first integral
of the extended system. The extended array ? u will then contain the following
37 functions ? u = (α, γij, Kij, Ai, Dkij, Vi).
(13)
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3The numerical algorithm
Due to the structure of the equations, the evolution(represented by the operator
E(∆t)) described by (12) can be decomposed into two separate processes; the
first one is a transport process and the second one is the contribution of the
sources.
The sources step (represented by the operator S(∆t)) does not involve space
derivatives of the fields, so that it consists in a system of coupled non-linear
ODE (Ordinary Differential Equations):
∂t? u = S(? u) (14)
The transport step (represented by the operator T(∆t)) contains the principal
part and it is given by a set of quasi-linear transport equations:
∂t? u + ∂kFk(? u) = 0(15)
The numerical implementation of these separated processes is quite easy. Second
order accuracy in ∆t can be obtained by using the well known Strang splitting.
E(∆t) = S(∆t/2)T(∆t)S(∆t/2)(16)
According to (3,7) the lapse and the metric have no flux terms. It means that
a reduced set of 30 quantities ? u = (Kij,Ai,Dkij,Vi) are transported in the
second step over an inhomogeneous static background composed by (α,γij).
The equations for the transport step (15) are given by:
∂tKij+ ∂k(α λkij)
∂tAk+ ∂k(α f(α) trK)
∂tDkij+ ∂k(α Kij)
=0 (17)
=
=
0
0
(18)
(19)
∂tVk
=0 (20)
where:
λkij= Dk
ij−m
2Vkγij+1/2δk
i(Aj+2Vj−Djrr)+1/2δk
j(Ai+2Vi−Dirr) (21)
and m is an arbitrary parameter.
To evolve the transport step, we will consider flux-conservative numeric al-
gorithms [12], obtained by applying the balance to a single computational cell.
In the 1D case the cell goes from n to n+1 in time (t = n · ∆t) and from j-1/2
to j+1/2 in space (xj= j · ∆x), so that we have:
j−∆t
Un+1
j
= Un
∆x[Fn+1/2
j+1/2− Fn+1/2
j−1/2](22)
Interface fluxes can be calculated in many different ways, leading to different
numerical methods. We will use here the well known MacCormack method.
This flux-conservative standard algorithm works well for smooth profiles, as it
can be appreciated in Figure 1.
But this standard algorithm is not appropriate for step-like profiles because
it produces spurious oscillations near the steep regions, as it can be appreciated
in Figure 2.
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Figure 1: Plot of Kxx for the initial metric given by (1) with H(x − t) =
1 + A cos[ω (x − t))] with periodic boundaries.Continuous line is the initial
condition.Dashed line is after 40 iterations
Figure 2: Same as in Fig. 1 with the step-like initial data for Kxx. Continuous
line is the initial condition. Dashed line is after 10 iterations. Note the spurious
oscillations around the corners
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More advanced numerical methods must be used to eliminate (or at least
to reduce) these oscillations. These advanced methods use information about
the eigenfields and the propagation direction, so the flux characteristic matrix
along the propagation direction must be diagonalized.
4Eigenfields
We will use a convenient method to compute the eigenfields. Let us study the
propagation of a step-like discontinuity in the transported variables ? u which will
move along a specific direction n with a given velocity v. Information about
the corresponding eigenfields can be extracted from the well known Rankine-
Hugoniot shock conditions :
v[u] = nk[Fk(u)] (23)
where [ ] represents the jump in the discontinuity. In our case
v[Kij]
v[Ak]
v[Dkij]
v[Vk]
=nr[α λrij]
nk[α f(α) trK]
nk[α Kij]
0
(24)
=
=
(25)
(26)
=(27)
where we must note that both the background metric coefficients and the prop-
agation direction are supposed to be continuous, so they are transparent to the
[ ] symbol.
If we develop this expressions we arrive at the following conclusions, where
Sn= nrSris the projection of the quantity S over n and S⊥= Sk− Snnkare
the transverse components:
1) [Vk],[A⊥],[D⊥ij] and [An− f trDn] propagate along n with speed v = 0.
There are 18 such eigenfields. For the line element given by (1) nkis along the
x axis and all these fields are actually zero.
2) [λnij−trλnninj] and [Kij−trKninj] do generate eigenfields propagating
along n with speed v = ±α (light cones). There are only 10 such eigenfields
because all of them are traceless. For Minkowski waves, where there is only
gauge, all these combinations are zero. This indicates that the correct way
to get the traceless part of a given tensor Sij in this context is just to take
Sij− trS ninj, so that the contribution of gauge modes will disappear.
3) [An] and [trK] do generate eigenfields propagating along n with speed
v = ±√fα (gauge cones). There are 2 such eigenfields corresponding to the
gauge sector. For Minkowski waves, there are the only non-zero components.
We are left with:
v[trK]=α[An]
α f(α)[trK] nk
(28)
v[Ak]=(29)
so that [Ak] is proportional to nk. Now we can get the gauge eigenfields:
?
f nkFk(trK) ± F(An) (30)
These eigenfields propagate along n according simple advection equations, a
familiar situation in the 1D case. Although this decomposition and diagonaliza-
tion is trivial in 1D, it is very useful in the multidimensional case for a generic
direction n.
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Figure 3: Same as Fig.2 where the methods presented in this paper are applied.
Continuous line is the initial condition. Dashed line is after 10 iterations
5Flux limiter methods
The flux limiter methods [12] we will use can be decomposed into some basic
steps. First of all the interface fluxes have to be calculated with any standard
second order accurate method (MacCormack in our case). Then, the propaga-
tion direction n and the corresponding eigenfields can be properly identified at
every cell interface. Two advection equations (one for every sense of propaga-
tion) are now available for the gauge eigenfluxes (30).
Let us choose for instance the eigenflux which propagates to the right (an
equivalent process will be valid for the other eigenflux propagating to the left).
This interface eigenflux Fn+1/2
some increment ∆j+1/2= Fn+1/2
to use of a mixture of the increments ∆j+1/2and ∆j−1/2to ensure monotonicity.
In our case we are using a robust mixture which goes by applying the well known
minmod rule to ∆j+1/2and 2∆j−1/2. In that way, the limiter acts only in steep
regions, where the proportion between neighbouring increments exceeds a factor
of two.
We can apply this method to the step-like initial data propagating along
the x axis. We can see in Figure 3 that the result is much better than before.
It can be (hardly) observed a small deviation from the TVD condition, which
is produced by the artificial separation produced by the Strang splitting into
transport and non-linear source steps.
This method can be applied, with the general decomposition described in
section 4, to discontinuities which propagate along any constant direction, and
not only to the trivial case, aligned with the x axis, that we have considered
until now. To prove it, we have rotated the metric of Minkowski waves in the
x-z plane to obtain a diagonal propagation of the profile. The line element in
this case has the following form:
j+1/2can be understood as the grid point flux Fn
j+1/2−Fn
jplus
j. In general, the purpose of the limiter is
ds2=
−
H(x + z
√2
− t) dt2+1
2[1 + H(x + z
√2
− t)] (dx2+ dz2) + dy2
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Figure 4: 3D plot of Kxx.
periodic boundary conditions until one full period (about 80 iterations in this
case) has elapsed and it has returned to the initial position
The step-like profile has been propagated with
+
1
2[−1 + H(x + z
√2
− t)] (dx dz + dz dx)(31)
We show the results in the Figure 4. We can also see in Figure 5 a z=constant
section of the same results to allow a more detailed comparison with the initial
data.
Acknowledgements: This work has been supported by the EU Programme
’Improving the Human Research Potential and the Socio-Economic Knowledge
Base’ (Research Training Network Contract (HPRN-CT-2000-00137) and by
a grant from the Conselleria d’Innovacio i Energia of the Govern de les Illes
Balears
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Figure 5: Section z=const. from Fig. 4. Continuous line is the initial condition.
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