WaveEquationswith Numeric method MovingBoundaries Code tests - - PowerPoint PPT Presentation
Statement of the problem WaveEquationswith Numeric method MovingBoundaries Code tests Closing remarks NumericalSolutionand ApplicationtoCosmology SanjeevS.Seahra Departmentof Mathematics&Statistics
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 1/29
SanjeevS.Seahra
Departmentof Mathematics&Statistics Universityof NewBrunswick,Canada incollaborationwith:Antonio Cardoso,TakashiHiramatsu,Kazuya KoyamaandFabioPSilva
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 2/29
■ generalized Stefan problems
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 2/29
■ generalized Stefan problems ■ application to braneworld cosmology
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 2/29
■ generalized Stefan problems ■ application to braneworld cosmology ■ linearized Stefan problem
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 2/29
■ generalized Stefan problems ■ application to braneworld cosmology ■ linearized Stefan problem ■ characteristic numerical integration scheme
Statement of the problem Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 2/29
■ generalized Stefan problems ■ application to braneworld cosmology ■ linearized Stefan problem ■ characteristic numerical integration scheme ■ code tests
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 3/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 4/29
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem:
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
■ free boundary problems arise in many other situations:
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
■ free boundary problems arise in many other situations: ◆ biology (biofilm growth)
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
■ free boundary problems arise in many other situations: ◆ biology (biofilm growth) ◆ manufacturing (behaviour of steel during welding)
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
■ free boundary problems arise in many other situations: ◆ biology (biofilm growth) ◆ manufacturing (behaviour of steel during welding) ◆ finance (American put option pricing)
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29
■ classic Stefan problem: ◆ two phase thermal system where interface between the
media evolves in time (i.e., melting)
■ free boundary problems arise in many other situations: ◆ biology (biofilm growth) ◆ manufacturing (behaviour of steel during welding) ◆ finance (American put option pricing) ◆ braneworld cosmology
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate bacteria
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate bacteria anti-bacterialagent
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate anti-bacterialagent anti-bacterialdiffuses throughbiofilmand triestokillbacteria
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate anti-bacterialdiffuses throughbiofilmand triestokillbacteria bacteriaadaptsinto formthatconsumes anti-bacterial
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate anti-bacterialdiffuses throughbiofilmand triestokillbacteria bacteriaadaptsinto formthatconsumes anti-bacterial biocideactioncauses filmtoshrink,cell adaptionslowsrate
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29
substrate
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
evolutiongoverned byIsrealjunction conditions evolutiongoverned byEinsteinfield equations
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk
evolutiongoverned byIsrealjunction conditions evolutiongoverned byEinsteinfield equations
nastynonlinearPDEs nastynonlinearPDEs
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29
braneworldmodelssay
boundaryof a5Dbulk braneshapeevolvesin reponsetobulkgravity andbranematter
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
principalgoalin braneworldmodelsisto solvetheinitialvalue problem
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
initialtimeslice principalgoalin braneworldmodelsisto solvetheinitialvalue problem
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
initialtimeslice
selectinitial bulkgeometry
principalgoalin braneworldmodelsisto solvetheinitialvalue problem
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
initialtimeslice
selectinitial bulkgeometry selectinitial braneshape
principalgoalin braneworldmodelsisto solvetheinitialvalue problem
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
initialtimeslice
selectinitial bulkgeometry selectinitial braneshape
finaltimeslice evolutionof bulk geometrygivenby hyperbolicPDEs subjecttoBCson brane principalgoalin braneworldmodelsisto solvetheinitialvalue problem
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29
initialtimeslice
selectinitial bulkgeometry selectinitial braneshape
finaltimeslice evolutionof brane shapedetermined bybulkgeometry principalgoalin braneworldmodelsisto solvetheinitialvalue problem evolutionof bulk geometrygivenby hyperbolicPDEs subjecttoBCson brane
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
■ observationally interesting to study linear fluctuations about
cosmological solutions
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
■ observationally interesting to study linear fluctuations about
cosmological solutions
■ dynamical degrees of freedom in this case:
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
■ observationally interesting to study linear fluctuations about
cosmological solutions
■ dynamical degrees of freedom in this case: ◆ bulk field ψ ⇒ gravitational potential perturbations
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
■ observationally interesting to study linear fluctuations about
cosmological solutions
■ dynamical degrees of freedom in this case: ◆ bulk field ψ ⇒ gravitational potential perturbations ◆ brane field ∆ ⇒ matter density perturbations
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29
■ in general, equations of motion (EOMs) for braneworlds are
extremely difficult to deal with
■ can derive analytic solutions with high symmetry ◆ e.g. cosmology: three of the four spatial dimensions are
isotropic and homogeneous
■ observationally interesting to study linear fluctuations about
cosmological solutions
■ dynamical degrees of freedom in this case: ◆ bulk field ψ ⇒ gravitational potential perturbations ◆ brane field ∆ ⇒ matter density perturbations ■ Fourier decompose ψ and ∆ to reduce dimensionality
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
inlineartheory,braneworld cosmologicalperturbation problemreducestothefollowing:
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
b r a n e
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
b r a n e
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
b r a n e
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
b r a n e
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
whataboutthenotionof a “freeboundary”?
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
whataboutthenotionof a “freeboundary”? inlineartheory, fluctuationsinbrane positionaresmall
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
whataboutthenotionof a “freeboundary”? inlineartheory, fluctuationsinbrane positionaresmall
branebendingmode
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
whataboutthenotionof a “freeboundary”? inlineartheory, fluctuationsinbrane positionaresmall viaacoordinatechange,onecan treatthebranebendingmodeas ascalarfunctiononthebrane
branebendingmode
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29
whataboutthenotionof a “freeboundary”? inlineartheory, fluctuationsinbrane positionaresmall inthispicture,thebraneboundary isfixedtoitsbackgroundpos’n
branebendingmode
viaacoordinatechange,onecan treatthebranebendingmodeas ascalarfunctiononthebrane
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem
Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29
b r a n e
eventhoughbraneshapeisfixed, problemisstillcomplicated...
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 12/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems:
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
◆ mapping the brane to a stationary position and using
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
◆ mapping the brane to a stationary position and using
■ works, but slow
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
◆ mapping the brane to a stationary position and using
■ works, but slow ■ doesn’t handle brane fields
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
◆ mapping the brane to a stationary position and using
■ works, but slow ■ doesn’t handle brane fields
◆ others . . .
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29
■ other people have attacked similar problems: ◆ Fourier spectral decomposition with time dependent
coefficients (Koyama 02)
■ leads to integral equations that have to be solved
numerically (poor convergence)
◆ decomposition of bulk field in terms of Tchebychev
polynomials with time dependent coefficients (Hiramatsu et al 03)
■ leads to (many) ODEs to solve (works, but slow)
◆ mapping the brane to a stationary position and using
■ works, but slow ■ doesn’t handle brane fields
◆ others . . . ■ need a fast and accurate algorithm to facilitate comparison
to observations
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e tosolveproblemnumerically,weneed tospecifycomputationaldomain
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e tosolveproblemnumerically,weneed tospecifycomputationaldomain
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e tosolveproblemnumerically,weneed tospecifycomputationaldomain goodidea:designdomain basedonthecausalproperties
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e mostapplicationsonlycare aboutvalueof fieldsonbrane betweenanintialandfinaltime
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e mostapplicationsonlycare aboutvalueof fieldsonbrane betweenanintialandfinaltime
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e mostapplicationsonlycare aboutvalueof fieldsonbrane betweenanintialandfinaltime
e v
u t i
f i e l d s
r a n e
p l e t e l y s p e c i f i e d
y
n i t i a l
a t a
e
t h e s e
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e mostapplicationsonlycare aboutvalueof fieldsonbrane betweenanintialandfinaltime
initialdatahere
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e mostapplicationsonlycare aboutvalueof fieldsonbrane betweenanintialandfinaltime
initialdatahere
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
note:amore traditionalchoiceof domainmayhave lookedlikethis
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29
b r a n e
note:amore traditionalchoiceof domainmayhave lookedlikethis
weonlycalculate thebulkfieldwhere weneedit
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
nowneedtopartition domainintofinitesegments
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
nowneedtopartition domainintofinitesegments breakbrane upintosmall segments
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
nowneedtopartition domainintofinitesegments breakbrane upintosmall segments constructgridby drawingfutureandpast nullraysfrombrane
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
nowneedtopartition domainintofinitesegments breakbrane upintosmall segments constructgridby drawingfutureandpast nullraysfrombrane bulk“diamond”cell
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
nowneedtopartition domainintofinitesegments breakbrane upintosmall segments constructgridby drawingfutureandpast nullraysfrombrane bulk“diamond”cell brane “triangle” cell
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
some“real”computationalgridsusedinapplications:
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29
some“real”computationalgridsusedinapplications: sinceourgridsarebasedonthecharacteristics
“characteristicintegrationscheme”
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 16/29
asinallnumeric methods,bothevolution lawswillbeapproximate rulesderivedfromEOMs
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 17/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 17/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 17/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 17/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 17/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
whathappensif we integratewaveequation
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
whathappensif we integratewaveequation
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
whathappensif we integratewaveequation
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
linearapproximation
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
divergence theorem
linearapproximation
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 18/29
“traditional cell” “nullcell”
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
linearapprox
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
linearapprox
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
linearapprox
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
linearapprox
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 19/29
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 20/29
someboundaryterms havebeencalled “nonlocal”inliterature
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 20/29
someboundaryterms havebeencalled “nonlocal”inliterature
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 20/29
someboundaryterms havebeencalled “nonlocal”inliterature keeptrackof branefieldsat“half- nodes”andusehigher-order stencilsinapproximations
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 21/29
■ with O(h4) diamond and O(h3) triangle evolution laws, we
should have a quadratically convergent algorithm
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 21/29
■ with O(h4) diamond and O(h3) triangle evolution laws, we
should have a quadratically convergent algorithm
■ since computational time ∝ number of cells ∝ h−2, the
cumulative error in the output will be inversely proportional to the time
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 21/29
■ with O(h4) diamond and O(h3) triangle evolution laws, we
should have a quadratically convergent algorithm
■ since computational time ∝ number of cells ∝ h−2, the
cumulative error in the output will be inversely proportional to the time
◆ this is better than most conventional finite differencing
schemes
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 21/29
■ with O(h4) diamond and O(h3) triangle evolution laws, we
should have a quadratically convergent algorithm
■ since computational time ∝ number of cells ∝ h−2, the
cumulative error in the output will be inversely proportional to the time
◆ this is better than most conventional finite differencing
schemes
■ evolution in the bulk can be accomplished with half as many
function calls as ordinary 2nd order PDE solvers
Statement of the problem Numeric method
Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 21/29
■ with O(h4) diamond and O(h3) triangle evolution laws, we
should have a quadratically convergent algorithm
■ since computational time ∝ number of cells ∝ h−2, the
cumulative error in the output will be inversely proportional to the time
◆ this is better than most conventional finite differencing
schemes
■ evolution in the bulk can be accomplished with half as many
function calls as ordinary 2nd order PDE solvers
■ the computational domain is the minimum size needed to get
answers for fields on the brane
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 22/29
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code:
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code: ◆ reproduce an exact solution
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code: ◆ reproduce an exact solution ◆ verify that output converges as h → 0
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code: ◆ reproduce an exact solution ◆ verify that output converges as h → 0 ■ to quantitatively address either issue, it is useful to define a
“distance” between functions f1 and f2 on the brane:
b =
ηf − ηi ηf
ηi
[f1(η) − f2(η)]2 1/2
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code: ◆ reproduce an exact solution ◆ verify that output converges as h → 0 ■ to quantitatively address either issue, it is useful to define a
“distance” between functions f1 and f2 on the brane:
b =
ηf − ηi ηf
ηi
[f1(η) − f2(η)]2 1/2
■ for any brane quantity, true quadratic convergence implies
b ∝ h2
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 23/29
■ two ways to test the code: ◆ reproduce an exact solution ◆ verify that output converges as h → 0 ■ to quantitatively address either issue, it is useful to define a
“distance” between functions f1 and f2 on the brane:
b =
ηf − ηi ηf
ηi
[f1(η) − f2(η)]2 1/2
■ for any brane quantity, true quadratic convergence implies
b ∝ h2
■ also, if f1 and f2 are numeric results with h =
√ 2h0 and h0 respectively: ζ(h0) ≡ f1 − f2 b ∝ h2
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
inRandall-Sundrumbraneworld models,itispossibletogetexact solutionswhenthebranehasa lineartrajectory
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
inRandall-Sundrumbraneworld models,itispossibletogetexact solutionswhenthebranehasa lineartrajectory
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
inRandall-Sundrumbraneworld models,itispossibletogetexact solutionswhenthebranehasa lineartrajectory
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
differentlinescorrespondto braneswithdifferentslopes
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
differentlinescorrespondto braneswithdifferentslopes
explicitquadratic convergencefor smallcellsize
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 24/29
differentlinescorrespondto braneswithdifferentslopes
explicitquadratic convergencefor smallcellsize
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
inRandall-Sundrummodel,wehavecalculatedthe behaviourof gravitationalwavesaboutbranes withnon-lineartrajectories
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
inRandall-Sundrummodel,wehavecalculatedthe behaviourof gravitationalwavesaboutbranes withnon-lineartrajectories
gravitationalwaveamplitude
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
inRandall-Sundrummodel,wehavecalculatedthe behaviourof gravitationalwavesaboutbranes withnon-lineartrajectories
gravitationalwaveamplitude
noanalyticsolution,but cantestconvergenceof GWamplitudeonbrane
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
different brane shapes
Statement of the problem Numeric method Code tests
Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 25/29
different brane shapes
quadraticconvergence
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 26/29
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically)
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom ◆ explicitly tested against analytic results
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom ◆ explicitly tested against analytic results ■ principal application is to braneworld cosmology
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom ◆ explicitly tested against analytic results ■ principal application is to braneworld cosmology ◆ can also be applied to any (1 + 1) hyperbolic system with
an irregular (timelike) boundary
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom ◆ explicitly tested against analytic results ■ principal application is to braneworld cosmology ◆ can also be applied to any (1 + 1) hyperbolic system with
an irregular (timelike) boundary
■ need to generalize the code to deal with a genuinely
dynamic boundary
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 27/29
■ we have developed a novel numerical algorithm to solve
wave equations in the presence of complicated boundaries
◆ based on characteristics ◆ quadradically convergent (theoretically and practically) ◆ capable of evolving boundary degrees of freedom ◆ explicitly tested against analytic results ■ principal application is to braneworld cosmology ◆ can also be applied to any (1 + 1) hyperbolic system with
an irregular (timelike) boundary
■ need to generalize the code to deal with a genuinely
dynamic boundary
◆ work in progress
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
◆ extra dimensional scenario that explains late time
acceleration of the universe
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
◆ extra dimensional scenario that explains late time
acceleration of the universe
■ the so-called “dark energy” problem
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
◆ extra dimensional scenario that explains late time
acceleration of the universe
■ the so-called “dark energy” problem
■ need quantitative predictions for the behaviour of linear
cosmological perturbations
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
◆ extra dimensional scenario that explains late time
acceleration of the universe
■ the so-called “dark energy” problem
■ need quantitative predictions for the behaviour of linear
cosmological perturbations
◆ compare with observed distribution of galaxies
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 28/29
■ in cosmology, the code is important for testing of the DGP
braneworld model
◆ extra dimensional scenario that explains late time
acceleration of the universe
■ the so-called “dark energy” problem
■ need quantitative predictions for the behaviour of linear
cosmological perturbations
◆ compare with observed distribution of galaxies ■ future telescopes with be able to confirm or refute DGP
based on these calculations
Statement of the problem Numeric method Code tests Closing remarks
Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 29/29