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Statement of the problem WaveEquationswith Numeric method MovingBoundaries Code tests Closing remarks NumericalSolutionand ApplicationtoCosmology SanjeevS.Seahra Departmentof Mathematics&Statistics

  1. Applications ■ classic Stefan problem: Statement of the problem ● Modified Stefan problem ◆ two phase thermal system where interface between the ● Applications ● Biofilms media evolves in time (i.e., melting) ● Braneworld models ● Braneworld IVP ■ free boundary problems arise in many other situations: ● Linearized braneworlds ● Master wave equations ◆ biology (biofilm growth) ● Separation of variables ◆ manufacturing (behaviour of steel during welding) Numeric method ◆ finance (American put option pricing) Code tests ◆ braneworld cosmology Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 5/29

  2. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  3. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks bacteria substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  4. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks anti-bacterial�agent bacteria substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  5. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks anti-bacterial�agent anti-bacterial�diffuses through�biofilm�and tries�to�kill�bacteria substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  6. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models bacteria�adapts�into ● Braneworld IVP ● Linearized braneworlds form�that�consumes ● Master wave equations ● Separation of variables anti-bacterial Numeric method Code tests Closing remarks anti-bacterial�diffuses through�biofilm�and tries�to�kill�bacteria substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  7. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models bacteria�adapts�into ● Braneworld IVP ● Linearized braneworlds form�that�consumes ● Master wave equations ● Separation of variables anti-bacterial biocide�action�causes Numeric method film�to�shrink,�cell Code tests adaption�slows�rate Closing remarks anti-bacterial�diffuses through�biofilm�and tries�to�kill�bacteria substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  8. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  9. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  10. Biofilms Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks substrate Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 6/29

  11. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  12. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  13. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  14. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  15. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations evolution�governed ● Separation of variables by�Isreal�junction Numeric method conditions Code tests Closing remarks evolution�governed by�Einstein�field equations Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  16. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms nasty�nonlinear�PDEs ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations evolution�governed ● Separation of variables by�Isreal�junction Numeric method conditions Code tests Closing remarks evolution�governed by�Einstein�field equations nasty�nonlinear�PDEs Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  17. Braneworld models braneworld�models�say Statement of the problem our�universe�is�the�4D ● Modified Stefan problem ● Applications boundary�of a�5D�bulk ● Biofilms ● Braneworld models ● Braneworld IVP brane�shape�evolves�in ● Linearized braneworlds ● Master wave equations reponse�to�bulk�gravity ● Separation of variables and�brane�matter Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 7/29

  18. Braneworld initial value problem principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  19. Braneworld initial value problem principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks initial�time�slice Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  20. Braneworld initial value problem principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks select�initial initial�time�slice bulk�geometry Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  21. Braneworld initial value problem principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks select�initial brane�shape select�initial initial�time�slice bulk�geometry Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  22. Braneworld initial value problem final�time�slice principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method evolution�of bulk Code tests geometry�given�by Closing remarks hyperbolic�PDEs subject�to�BCs�on brane select�initial brane�shape select�initial initial�time�slice bulk�geometry Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  23. Braneworld initial value problem final�time�slice principal�goal�in braneworld�models�is�to Statement of the problem ● Modified Stefan problem solve�the�initial�value ● Applications ● Biofilms problem ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method evolution�of brane evolution�of bulk Code tests shape�determined geometry�given�by Closing remarks by�bulk�geometry hyperbolic�PDEs subject�to�BCs�on brane select�initial brane�shape select�initial initial�time�slice bulk�geometry Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 8/29

  24. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  25. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  26. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  27. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables ■ observationally interesting to study linear fluctuations about Numeric method cosmological solutions Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  28. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables ■ observationally interesting to study linear fluctuations about Numeric method cosmological solutions Code tests Closing remarks ■ dynamical degrees of freedom in this case: Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  29. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables ■ observationally interesting to study linear fluctuations about Numeric method cosmological solutions Code tests Closing remarks ■ dynamical degrees of freedom in this case: ◆ bulk field ψ ⇒ gravitational potential perturbations Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  30. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables ■ observationally interesting to study linear fluctuations about Numeric method cosmological solutions Code tests Closing remarks ■ dynamical degrees of freedom in this case: ◆ bulk field ψ ⇒ gravitational potential perturbations ◆ brane field ∆ ⇒ matter density perturbations Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  31. Linearized braneworlds ■ in general, equations of motion (EOMs) for braneworlds are Statement of the problem ● Modified Stefan problem extremely difficult to deal with ● Applications ● Biofilms ■ can derive analytic solutions with high symmetry ● Braneworld models ● Braneworld IVP ◆ e.g. cosmology: three of the four spatial dimensions are ● Linearized braneworlds ● Master wave equations isotropic and homogeneous ● Separation of variables ■ observationally interesting to study linear fluctuations about Numeric method cosmological solutions Code tests Closing remarks ■ 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 Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 9/29

  32. Master wave equations Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables in�linear�theory,�braneworld Numeric method cosmological�perturbation Code tests problem�reduces�to�the�following: Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  33. Master wave equations Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  34. Master wave equations Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  35. Master wave equations Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  36. Master wave equations Statement of the problem ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  37. Master wave equations what�about�the�notion�of a Statement of the problem ● Modified Stefan problem “free�boundary”? ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  38. Master wave equations what�about�the�notion�of a Statement of the problem ● Modified Stefan problem “free�boundary”? ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP in�linear�theory, ● Linearized braneworlds fluctuations�in�brane ● Master wave equations ● Separation of variables position�are�small Numeric method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  39. Master wave equations what�about�the�notion�of a Statement of the problem ● Modified Stefan problem “free�boundary”? ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP in�linear�theory, ● Linearized braneworlds fluctuations�in�brane ● Master wave equations ● Separation of variables position�are�small Numeric method Code tests brane�bending�mode Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  40. Master wave equations what�about�the�notion�of a Statement of the problem ● Modified Stefan problem “free�boundary”? ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP in�linear�theory, ● Linearized braneworlds fluctuations�in�brane ● Master wave equations ● Separation of variables position�are�small Numeric method Code tests brane�bending�mode Closing remarks via�a�coordinate�change,�one�can treat�the�brane�bending�mode�as a�scalar�function�on�the�brane Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  41. Master wave equations what�about�the�notion�of a Statement of the problem ● Modified Stefan problem “free�boundary”? ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP in�linear�theory, ● Linearized braneworlds fluctuations�in�brane ● Master wave equations ● Separation of variables position�are�small Numeric method Code tests brane�bending�mode Closing remarks via�a�coordinate�change,�one�can treat�the�brane�bending�mode�as a�scalar�function�on�the�brane in�this�picture,�the�brane�boundary is�fixed�to�its�background�pos’n Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 10/29

  42. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  43. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  44. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  45. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  46. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method n a Code tests r b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  47. Separation of variables even�though�brane�shape�is�fixed, Statement of the problem problem�is�still�complicated... ● Modified Stefan problem ● Applications ● Biofilms ● Braneworld models ● Braneworld IVP ● Linearized braneworlds ● Master wave equations ● Separation of variables e Numeric method Numerical n a Code tests r b Closing remarks Methods Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 11/29

  48. Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm Numeric method ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 12/29

  49. What others have done ■ other people have attacked similar problems: Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  50. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  51. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  52. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  53. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  54. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks ◆ mapping the brane to a stationary position and using ordinary finite differencing (Kobayashi and Tanaka 03) Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  55. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks ◆ mapping the brane to a stationary position and using ordinary finite differencing (Kobayashi and Tanaka 03) ■ works, but slow Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  56. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks ◆ mapping the brane to a stationary position and using ordinary finite differencing (Kobayashi and Tanaka 03) ■ works, but slow ■ doesn’t handle brane fields Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  57. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks ◆ mapping the brane to a stationary position and using ordinary finite differencing (Kobayashi and Tanaka 03) ■ works, but slow ■ doesn’t handle brane fields ◆ others . . . Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  58. What others have done ■ other people have attacked similar problems: Statement of the problem ◆ Fourier spectral decomposition with time dependent Numeric method coefficients (Koyama 02) ● What others have done ■ leads to integral equations that have to be solved ● Computational domain ● Discretization numerically (poor convergence) ● The algorithm ● Error budget ◆ decomposition of bulk field in terms of Tchebychev ● Diamond evolution ● Triangle evolution polynomials with time dependent coefficients (Hiramatsu ● “Nonlocal” terms ● Advantages of the method et al 03) Code tests ■ leads to (many) ODEs to solve (works, but slow) Closing remarks ◆ mapping the brane to a stationary position and using ordinary finite differencing (Kobayashi and Tanaka 03) ■ works, but slow ■ doesn’t handle brane fields ◆ others . . . ■ need a fast and accurate algorithm to facilitate comparison to observations Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 13/29

  59. Computational domain to�solve�problem�numerically,�we�need Statement of the problem to�specify�computational�domain Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  60. Computational domain to�solve�problem�numerically,�we�need Statement of the problem to�specify�computational�domain Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  61. Computational domain to�solve�problem�numerically,�we�need Statement of the problem to�specify�computational�domain Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution good�idea:�design�domain ● Triangle evolution ● “Nonlocal” terms e based�on�the�causal�properties n ● Advantages of the method a of the�wave�equation r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  62. Computational domain Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  63. Computational domain Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  64. Computational domain Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  65. Computational domain Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  66. Computational domain most�applications�only�care Statement of the problem about�value�of fields�on�brane between�an�intial�and�final�time Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  67. Computational domain most�applications�only�care Statement of the problem about�value�of fields�on�brane between�an�intial�and�final�time Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  68. Computational domain most�applications�only�care Statement of the problem about�value�of fields�on�brane between�an�intial�and�final�time Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm e ● Error budget v s o ● Diamond evolution p l u e ● Triangle evolution t c i o i ● “Nonlocal” terms e f n i e � n o ● Advantages of the method d f � a b f i y r e � Code tests b l i n d i s t � i o a Closing remarks n l � � d b a r t a a n � o e n � c � o o n m e p � o l e f t t e h l e y s e Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  69. Computational domain most�applications�only�care Statement of the problem about�value�of fields�on�brane between�an�intial�and�final�time Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks our�choice:�specify initial�data�here Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  70. Computational domain most�applications�only�care Statement of the problem about�value�of fields�on�brane between�an�intial�and�final�time Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks our�choice:�specify initial�data�here Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  71. Computational domain note:�a�more Statement of the problem traditional�choice�of Numeric method domain�may�have ● What others have done looked�like�this ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  72. Computational domain note:�a�more Statement of the problem traditional�choice�of Numeric method domain�may�have ● What others have done looked�like�this ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms e n ● Advantages of the method a r Code tests b our�choice�is�better: Closing remarks we�only�calculate the�bulk�field�where we�need�it Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 14/29

  73. Discretization now�need�to�partition domain�into�finite�segments Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

  74. Discretization now�need�to�partition domain�into�finite�segments Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget break�brane ● Diamond evolution ● Triangle evolution up�into�small ● “Nonlocal” terms ● Advantages of the method segments Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

  75. Discretization now�need�to�partition domain�into�finite�segments Statement of the problem Numeric method ● What others have done construct�grid�by ● Computational domain ● Discretization drawing�future�and�past ● The algorithm null�rays�from�brane ● Error budget break�brane ● Diamond evolution ● Triangle evolution up�into�small ● “Nonlocal” terms ● Advantages of the method segments Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

  76. Discretization now�need�to�partition domain�into�finite�segments Statement of the problem Numeric method ● What others have done construct�grid�by ● Computational domain ● Discretization drawing�future�and�past ● The algorithm null�rays�from�brane ● Error budget break�brane ● Diamond evolution ● Triangle evolution up�into�small ● “Nonlocal” terms bulk�“diamond”�cell ● Advantages of the method segments Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

  77. Discretization now�need�to�partition domain�into�finite�segments Statement of the problem Numeric method ● What others have done construct�grid�by ● Computational domain ● Discretization drawing�future�and�past ● The algorithm null�rays�from�brane ● Error budget break�brane ● Diamond evolution ● Triangle evolution up�into�small ● “Nonlocal” terms bulk�“diamond”�cell ● Advantages of the method segments Code tests Closing remarks brane “triangle” cell Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

  78. Discretization some�“real”�computational�grids�used�in�applications: Statement of the problem Numeric method ● What others have done ● Computational domain ● Discretization ● The algorithm ● Error budget ● Diamond evolution ● Triangle evolution ● “Nonlocal” terms ● Advantages of the method Code tests Closing remarks Sanjeev S. Seahra; 26 August, 2008 Numeric solution of wave equations with moving boundaries - p. 15/29

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