RungeKutta Discontinuous Galerkin Method for Numerical Simulations of Thermal Instability TI

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Runge-Kutta Discontinuous Galerkin Method for
Numerical Simulations of Thermal Instability (TI)
East Asian Numerical Astrophysics meeting 2004,
Nov 30--Dec 2, NAO, Japan

• Hiroshi Koyama (Kobe Univ.)
• Hiroaki Nishikawa (Dept. Aerospace Engineering,
  Univ. of Michigan)

• Outline
• Critical length scale of TI Field length
• RKDG method for Euler equations with source and
  diffusion terms
• Results Accurate on moderately resolved grids!!

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Radiative cooling hydrodynamics

• ISM, SNR, IGM, Solar atmosphere, etc.
• Optically thin cooling
• T10108K
• Thermal Instability
• Field (1965)

Unstable
Cooling function Wada Norman (2001)
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Thermal instability (TI)

• Field length
• ?F(KT/?2?)1/2
• Critical wavelength of the TI
• Without thermal conduction (TC)
• The most unstable wavelength is the smallest
  scale
• One Grid scale fluctuation develops
• Resolution dependent
• Inclusion of TC
• Necessary to attain the convergence the Field
  condition
• Koyama Inutsuka (2004) ApJ 602, L25
• TC makes smooth distribution of temperature
• Several grids are at least required to resolve
  Field length

Dispersion relation (Field 1965)
Without TC
Growth rate
Field length
Wave number
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Example two phase ISM

• An incorrect net source term leads to incorrect
  propagation speeds.
• Thermal conductivity (TC) sustains turbulent
  motions.
• MUSCL type 2nd order Godunov method (van Leer
  1979) for Euler equations

Without TC
With TC
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How to resolve conductive regions?

• Extremely find grids!!!
• We hope that numerical scheme is accurate when
  the Field length is unresolved (asymptotic
  preserving-AP).
• Lowrie Morel (2002) showed that Discontinuous
  Galerkin (DG) method is AP for a stiff relaxation
  problem.
• We apply the DG method for TI problems.

Convergence test
?x?F
By using MUSCL scheme Koyama Inutsuka (2004)
ApJ, 602, 25L
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RKDG method
DG method

• developed by Cockburn Shu (1989)
• widely used in the aeronautical community
• Discretize in space using the DG method with
  polynomial basis functions.
• (ODE) can be integrated in time with the TVD
  Runge-Kutta scheme
• Apply a limiter if it is necessary

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P2
P1

• RKDG
• A high-order extension of Godunov-type FVM
• Riemann solution can be used for surface
  integral,
• Arbitrary high-order
• Only information about U on grid and its edges is
  necessary
• Highly parallelizable (compact scheme)

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Test Entropy wave in 1D
v1
One period
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Test entropy wave in 2D
10 time periods
Initial condition 802 meshes
MUSCL (2nd order)
P1 RKDG (2nd order)
Evaluate points for surface and volume integrals
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RKDG for TI

• Euler equations (Hyperbolic system) source term
  diffusion term (viscosity conductivity)
• volume integral on grid for source term
• For diffusion term
• Important note!
• Simple replacement of numerical flux F by Ux make
  inconsistent solution which errors does not
  decrease with a mesh refinement!!
• Local DG method for diffusion operator (Cockburn
  Shu 1998)
• Recovery method (van Leer 17th AIAA CFD
  Conference 2005, submitted)
• Unsplit method

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Test TI in 1D
Density L1 error
Temperature distribution
?x?F
40 grids calculations are shown
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Test TI in 2D
Kinetic energy

• ?F ?L/(300 grids)
• P1 RKDG with 256 grids cannot resolve ?F but
  shows good agreement with the converged solution.

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Summary

• Accurate simulation involving thermal instability
  (TI) poses a severe demand to numerical methods.
• Extremely fine grids to resolve conductive
  regions
• We investigate a high resolution numerical
  method based on Runge-Kutta Discontinuous
  Galerkin (RKDG) method for TI simulations.
• arbitrary high order
• compact scheme
• RKDG methods are capable of accurately
  simulating TI on moderately resolved grids.

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references

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