Athena????GLM????

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Athena????GLM????

• ???
• 4?7?

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????

• ????
• ??(CESEHLL)
• Athena??????
• GLM-MHD????
• ????

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????

• ???CESE(in the near sun)and HLL(off-sun)?????,????
  ???
• ?off-sun??????Athena code????????
• ??????????????,?off-sun????6Rs???????????
• ???MHD?????GLM????,?????

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??(CESEHLL)

• ???????????,???????????????
• ??????,???????CESE??,???????????(AMR)???HLL??
• ????????????????????,???????????????AMR??????????
  ?????(far-field),???????????,??????????
• ????

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,?????????????
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Athena

• Athena
• The equations ideal MHD
• The numerical algorithms in Athna are based on
  directionally unsplit,higher order Godunov
  methods
• Discretization 1)mass,momentum,energyfinite
  volume
• 2)magnetic fieldbased on area rather than
  volumes averages

Volume-averaged
with
time- and area-averaged fluxes
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Athena
with
area-averaged
electromotive force averaged along the
appropriate line element

• advantages1) ideal for use AMR
• 2)Superior for shock capturing and evolving the
  contact and rotational discontinuties

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Athna
The chart for the steps in the 2D algorithm in
Athena
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Athena

• The algorithm for computing MHD interface
  statespiecewise contant(first-order)
  reconstruction,piecewise linear(second-order)resco
  nstruction,piecewise parabolic(third-order)
  reconstruction
• The algorithm for computing fluxesHLL
  solvers,Roes method
• Remarks1)the reconstruction used in Athena
  require characteristic variables and a
  characte-ristic evolution of the linearized
  systerm
• 2)The Godunov methods do not require expensive
  solvers based on complex characteristic
  decompositions

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Data reconstruction

• Piecewise constant reconstruction assume the
  primitive variables are piecewise constant within
  each cell
• Piecewise linear reconstruction assume the
  primitive variables vary linearly within each cell

• WENO reconstruction can achieve higher than
  second order
• Basic idealseveral cells can formulate a module
  (r denotes the number of cells formulated the
  module,k denotes the total number of
  modules,different modules have different
  interpolation polynomials

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Data reconstruction

• the total polynomial R(x) of reconstruction is a
  convex combination of the above polynomials Pj(x),

where is weight cofficient,
The WENO reconstruction can have (2k-1) order,
and is non-oscillatory,but the computation is
complex
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Godunov Fluxes

• First proposed by Godunov S.K. in 1959
• The basic idea at ,in each cell the
  primitive variables are constant. At the
  interface bewteen the neighbor cells
  , there is a initial discontinuity
• Godunov methods do require expensive solvers
  based on complex characteristic decompositions
  and capture high quality shock
• HLL-family solvers

then formulate a local Riemann problem bewteen
the neighbour cells.
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Roes method

• An useful linearization for the MHD equations
• Include all the characteristics of the
  systerm,and less diffusive and more accurate for
  intermediate waves
• Jacobian is evaluated using an average state(Roe
  average)
• where is the enthalpy
• the Roe fluxes are simply
• Disadvantage may return negative densities or
  pressures

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HLL

• assuming an average intermediate state between
  the fastest and slowest waves
• intermediate state
• the HLL fluxes

are the minimum signal speed and
the maximum signal speed
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HLL

• Remarks
• must be estimated appropriately
• Davis
• Einfeldt et al

• The solver is fast and do not need the
  characteristic decomposition
• too diffusive and cannot resolve isolated contact
  discontinuities very well

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HLLE

• Using a singal constant intermediate state
  computed from a conservative average
• Do not require a characteristic decomposition of
  MHD equations
• The HLLE flux at the interface

where
are the fluxes evaluated using the left and
right states of the conserved variables,and
If both (or
),the HLLE flux will be

• the HLLE can guarantee the pressure and
  density is positive,but in the multiple
  dimensions,it does not necessarily
  guarantee.Whats more,the HLLE neglects the
  Alfven,slow magnetosonic,and contact waves.

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HLLC

• the intermediate states in the Riemann fan are
  separated into two intermediate states by a
  contact discontinuity can resolve isolated
  contact discontinuities exactly

be evaluated from HLLaverage

• the numerical flux of HLLC

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HLLC

• Positively conservative
• HLLC can dramatically improve the results of the
  HLL solver, and has much less computational time
  than the HLLEM

athe soud speed
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HLLD

• Five-wave Riemann solver for MHD,HLL-Discontinuiti
  es solver
• Composed of four intermediate states

indicate the speeds of the fast
magnetosonic waves, Alfvén waves, and entropy
wave
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HLLD

• The numerical flux vector of the HLLD Riemann
  solver for MHD equations

• Positively conservative

the fast magnetosonic speed
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???????

• ?off-sun????6Rs???????????Cartesian??,????????????
  ?Parker??6Rs???
• Call parker(r6rs,ur,gamma0,T0)
• ????????????,???????????,????????
• subroutine nearpoint(r)

??Parker?????????????,??????
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GLM-MHD

• GLM(generalized Lagrange multiplier)
• The form of equations
• Solver for GLM-MHD

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GLM

• Coupling the divergence constraint by introducing
  a generalized Lagrange multiplier
• the divergence errors are transported to the
  domain boundaries with the maximal admissible
  speed and are damped at the same time
• Magnetic induction equations are replaced

Different choices for the linear operator D
Elliptic correction
Parabolic correction
Hyperbolic correction
Combination of parabolic and hyperbolic ansatz
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• MHD?????

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• The MHD equations can be symmetrized by adding
  some hyperbollic terms on the right-hand side
• The changed eqations

Remarks 1)call the equations the extended
GLM(EGLM) formulation of MHD equations 2)
significantly depends on the grid size and the
scheme used, is a function of
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??????

• The eigenvalues of the GLM-MHD coinside with the
  ordinary MHD waves plus two additional modes
  ,for a total of nine characteristic waves
• For one dimensional ,x direction

remarks1)show that the system is hyperbolic
2)only the waves traveling with speeds
can carry a change in or

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The solver of GLM-MHD

• Solver for the GLM-MHD without additional source
• Treat the linear system given by the B and
  from the other ordinary 7-wave MHD equations
  in an operator-split fashion

where S and A are the advection and source step
operators separately
1)Advection step based on the corner transport
upwind(CTU) method, second order accurate
discretization
Where F,G,H are the numerical fluxes computed by
solving a Riemann problem between suitable
time-centered left and right states
R(, ) denotes the flux obtained by means of a
Riemann solver, are computed
via a Taylor expansion in the direction normal to
a given interface
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The solver for GLM-MHD

• 2)Source step solver the initial value problem
  without the term

can be integrated exactly for a time increment

• Remarks
• numerical experiments indicate that the
  divergence errors are mininized when the lies
  in the range 0,1
• is an unphysical variable,the initial
  condition given by the output of the most
  recent step
• Boundary condition for assume that the
  behavior of and at the boundary is
  identical,use a homogeneous Dirichlet condition
  ,nonreflecting boundary condition

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????

• Xueshang Feng,Shaohua Zhang,Changqing
  Xiang,Liping Yang,Chaowei Jiang,A Hybrid Solar
  Wind Model of CESEHLL Method with Yin-Yang
  Overset Grid and AMR Grid
• Takahiro Miyoshi,Naoki Terada,The HLLD
  Approximate Riemann Solver for Magnetospheric
  Simulation
• Takahiro Miyoshi,Kanya Kusano,A multi-state HLL
  approximate solver for ideal magnetohydrodynamics
  
• A.Mignone,G.Bodo,PLUTOA NUMERICAL CODE FOR
  COMPUTATIONAL ASTROPHYSICS
• Shengtai Li,An HLLC Riemann solver for
  magneto-hydrodynamics
• James M.Stone,Thomas A.Gardiner,ATHENAA NEW
  CODE FOR ASTROPHYSICAL MHD
• A.Dedner,F.Kemm,Hyperbolic Divergence Cleaning
  for the MHD Equations
• Andrea Mignone,Petros Tzeferacos,Gianluigi
  Bodo,High-order conservative finite difference
  GLM-MHD schemes for cell-centered MHD

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????

• Andrea Mignone,Petros Tzeferacos,A Second-Order
  Unsplit Godunov Scheme for Cell-Centeres
  MHDCTU-GLM scheme
• Shengtai Li,Hui Li,A Modern Code for Solving
  Magneto-hydrodynamics or Hydro-
• dynamics Equations
• Dinshaw S.Balsara,Multidimensional HLLE Riemann
  SolverApplication to Euler and
  Magnetohydrodynamic Flows

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