AVOIDING GAUGE AND CONSTRAINT SHOCKS IN NUMERICAL RELATIVITY

1
AVOIDING GAUGE AND CONSTRAINTSHOCKSIN
NUMERICAL RELATIVITY
Bernd Reimann
ICN, Mexico City AEI, Golm
AEI, 30/09/2004
2
Aim and Topics

• The aim of this talk is to give a brief summary
  of joint work done together with M. Alcubierre,
  D. Núñez and J.A. González.
• It is work in progress, and a paper is in
  preparation. The topics are
• FORMATION and AVOIDANCE of SHOCKS
• What kind of shocks arise by what mechanisms for
  systems of PDE's?
• gradient catastrophe gt (indirect)
  linear degeneracy
• blow-up in finite time gt source
  criteria
• Numerical examples (equations of wave-type)
• EINSTEINS EQUATIONS in SPHERICAL SYMMETRY
• What shocks can form in the particular case of
  Einsteins equations?
• gauge shocks gt
  shock-avoiding lapse
• constraint shocks gt adjustments
  using constraints
• Numerical examples (robust stability test,
  evolutions of Gaussian perturbations on top of
  Minkowski)

3
Formation of Shocks for Wavelike Equations

• We use mathematical theory concerning
  shock-/singularity-formation for hyperbolic
  systems of PDEs with wavelike character
• Shock formation mechanisms
  
  S. Alinhac, Blowup
  for Nonlinear Hyperbolic Equations, 1995
• Gradient catastrophe, linear degeneracy
  
  P. Lax, Hyperbolic
  Systems of Conservation Laws and the Mathematical
  Theory of Shock Waves, 1973
• Blowup in finite time, analysis of source terms
  
  F. John, Nonlinear Wave
  Equations, Formation of Singularities, 1979 F.
  John, Formation of Singularities in
  One-Dimensional Nonlinear Wave Propagation, Comm.
  Pure Appl. Math., vol. 27, p. 377-405, 1974
• Gauge shocks in Einsteins equations
  
  M. Alcubierre,
  The Appearance of Coordinate Shocks in Hyperbolic
  Formulations of General Relativity, Phys. Rev. D
  55, p. 5981-5991, 1997
• This theory typically states that for
  perturbations of order
  shocks form
  on a timescale of order , whereas
  in shock- avoiding cases one can obtain
  timescales

4
Evolution Equations

• We are interested in scalar PDEs like
• and their generalization being u-v-evolution
  systems with PDEs of the
• form together with
• Here and are n- and m-dimensional
  vectors, and A is a strongly hyperbolic mxm
  matrix depending on the us only. Its real
  eigenvalues and its linearly independent
  eigenvectors are
• and
• Using matrices R and R-1 consisting of the
  eigenvectors, the matrix A can be diagonalized by
  introducing the eigenfields ,

us lapse, metric components vs partial
derivatives of the us w.r.t. x and t (denoted by
Ds and Ks)
5
Geometric Blow-Up and Linear Degeneracy

• The blow-up mechanism is due to focusing of
  characteristics. Here derivatives of u become
  infinite whereas u itself remains finite.
• Example Burgers equation
  yields
• Here particles move with zero acceleration.
  Shocks form as particles with high velocity
  collide with others ahead having lower velocity.
• This gradient catastrophe occurs at time
• The corresponding criteria for shock-avoidance
  is linear degeneracy, which demands
  and generalizes for systems of PDEs to
• By replacing , M. Alcubierre
  in the 90ies developed for the u-v-system of
  PDEs the condition of indirect linear degeneracy
  ,

6
ODE-Mechanism and the Source Criteria

• Due to self-increase in its influence domain u
  blows up in finite time.
• Example ODEs of the form
  have solutions
  
• which become infinite at the time
• For the u-v-system of PDEs when writing the
  evolution of the main system in terms of
  eigenfields, one finds
• In particular, for the time derivative along a
  characteristic follows
• Hence, for the source criteria we demand in
  general
• or, if we identify in a particular case more
  dangerous terms ,

7
Equations of Wave-Type

• As a first example we study equations of
  wave-type having the form
• which by introducing and
  we write as first order system
• Eigenvalues
  Eigenfields
• For indirect linear degeneracy from
• follows that c is a constant (which we set to 1).
• For the source criteria from
• follows that q has to be free of terms
  (i.e. no term, and only quadratic
  terms
  are allowed).

8
Numerical Examples (1)
Convergence factors
INDIRECT LINEAR DEGENERACY

NO
YES
vs
Gradient catastrophe (1) c u, q 0
Standard wave equation c 1, q 0
Y E S
S O U R C E
?x u
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
?t u
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Initial perturbation
9
Numerical Examples (2)

INDIRECT LINEAR DEGENERACY

NO
YES
vs
Standard wave equation c 1, q 0
Gradient catastrophe (1) c u, q 0
Y E S
S O U R C E
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Wavelike, shock-free behavior!
10
Numerical Examples (3)

INDIRECT LINEAR DEGENERACY

NO
YES
vs
Standard wave equation c 1, q 0
Gradient catastrophe (1) c u, q 0
Y E S
S O U R C E
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Blow-up in u, code crashes at t 7!
11
Numerical Examples (4)

INDIRECT LINEAR DEGENERACY

NO
YES
vs
Standard wave equation c 1, q 0
Gradient catastrophe (1) c u, q 0
Y E S
S O U R C E
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Spikes form in D and K, run with highest
resolution crashes at t 12!
12
Numerical Examples (5)

INDIRECT LINEAR DEGENERACY

NO
YES
vs
Standard wave equation c 1, q 0
Gradient catastrophe (1) c u, q 0
Y E S
S O U R C E
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Spikes form in D and K, code crashes at t 13!
13
Numerical Examples (6)

INDIRECT LINEAR DEGENERACY

NO
YES
vs
Standard wave equation c 1, q 0
Gradient catastrophe (1) c u, q 0
Y E S
S O U R C E
Gradient catastrophe (2) c u, q 2uD2-K2/u
Wavelike equation c 1, q 2D2-K2
Time derivative of Burgers equation c u, q
2uD2
Blow-up in finite time c 1, q 2D2
N O
Gradient in u and hence D and K become infinite,
crash at t 7!
14
Einsteins Equations us and vs

• For Einsteins equations in spherical symmetry,
  we adopt the Bona-Maso evolution equation for the
  lapse,
• where f(?) is an arbitrary function. We consider
  no shift, and work with the metric components
• These variables are combined in the vector
• As vs we introduce spatial and time derivatives
  of these quantities
• However, rather than working with DA and KA, we
  introduce for convenience D DA - 2DB and K KA
  2KB
• These derivatives build the vector

15
Constraint and Evolution Equations

• In terms of these variables we obtain the
  Hamiltonian constraint
• and the momentum constraint
• The PDEs for the evolution of the us and vs
  are given by
• and, when adding constraints to the
  ADM-equations,

hD? ?/A1/2 mD? ?
hD ?/A1/2 (-8mD) ? hDB ?/A1/2
(2mDB) ? (2hK) ?/A
mK ?/A1/2 (1/2hKB) ?/A mKB ?/A1/2
sources quadratic in D and K
16
Eigenvalues and Eigenfields

• Demanding integrability for the us, , the
  6 adjustments in the evolution equations of the
  Ds have to vanish hDs mDs 0
• In addition we set the adjustments in the
  evolution equation for KB to
  and
• in order to obtain the following eigenvalues and
  eigenfields
• At this point, only the adjustments hK and mK
  and the free parameter ? (scaling the eigenspeed
  of ) remain undetermined.
• In the following we fix these 3 free parameters
  demanding a strongly hyperbolic principal part
  (restricting ) and indirect
  linear degeneracy and/or the source criteria.

This also follows from the source criteria!
17
Gauge Shocks

• For the eigenpair traveling along the time
  lines, no shocks are found as
• For the gauge eigenpair indirect
  linear degeneracy yields
• and the source criteria gives
• Hence, both conditions agree in pointing out the
  formation of gauge shocks unless f(?) satisfies
• Harmonic slicing, , is a member of
  this shock-avoiding family. Furthermore, M.
  Alcubierre pointed out that 1log-slicing,
  is an approximate solution often
  more useful in NR.

Identical gauge shocks occur in toy 11
relativity!
18
Constraint Shocks

• For the constraint eigenpair,
  indirect linear degeneracy gives
• This result hence depends on the gauge choice,
  but the adjustments .
  together with hK -2 and mK 0 seem to be
  preferred.
• The source criteria yields
• Here we find
  and arbitrary ?.
• Alternatively, demanding for particular cases
  the coefficient of the mixed term to
  vanish, the adjustments found for indirect linear
  degeneracy, hK -2 and mK 0, are favored.
• Hence, both conditions agree in detecting gauge
  shocks unless f(?) satisfies

Constraint shocks do not exist in 11!
19
Robust Stability Test ?-family (1)



exponential growth for ADM and for ? ? 0 or
? ? 1
essentially no error growth for 0 ltlt ? ltlt
1 and ? gtgt 1
crossing times
20
Robust Stability Test ?-family (2)


eigenspeeds for wf? and w0 become similar
as ?f? ? 0
ADM (no adjustments, not a member of the
?-family)
eigenspeeds for wf? and wc? become similar
as ?f? ? 1
initial average error
?
21
Robust Stability Test hK (1)



exponential growth for hK ? -2 (e.g. for ADM)
essentially no error growth for hK ? -2
crossing times
22
Robust Stability Test hK (2)

ADM (no adjustments, i.e. hK 0)

adjustment suggested by indirect linear
degeneracy or by eliminating mixed terms in the
sources
initial average error
hK
23
hK-hKB Parameter Search

Initial data Minkowski Gaussian perturbation
in KB Evolution Harmonic slicing (f?1) 2D
parameter search Output for the 2-norm of the
Hamiltonian and momentum constraints is shown at
times t 10, 20 and 30 as a
function of the adjustments hK and hKB (black
crash
dark-gray exp. growth.. light-gray linear
growth white no growth)

exponentially growing error for ADM
shock-avoiding valley for the ?-family
(source criteria)
shock-avoiding point at hK -2 (indirect
linear degeneracy)
24
Conclusions and Outlook

• Mechanisms for shock formation have been
  identified (gradient catastrophe blow-up in
  finite time), and shock-avoiding conditions
  (indirect linear degeneracy source criteria)
  have been derived and tested numerically for
  equations of wave-type.
• For Einsteins equations in 11 dimensions gauge
  shocks, and in spherical symmetry in addition
  constraint shocks have been found. Here gauge
  shocks can be avoided by choosing for the
  Bona-Maso lapse the free function as f 1
  const/alpha2, and the constraint shocks by adding
  the constraints in a suitable way to the
  evolution equations. By doing so,
  shock-avoiding evolution systems with only linear
  growth in the constraints are obtained.
• Our recommendation for evolution systems in NR
  hence is
• strong hyperbolic principal part
  shock-avoiding source terms
• In future work we want to analyze in spherical
  symmetry different systems (ADM, BSSN, KST)
  and/or generalize this analysis to 31.