amr simulations of the magneto-hydrodynamic richtmyer-meshkov instability
DESCRIPTION
AMR Simulations of the Magneto-hydrodynamic Richtmyer-Meshkov Instability. Ravi Samtaney Computational Plasma Physics Group Princeton Plasma Physics Laboratory Princeton University CPPG Seminar PPPL, 04/09/2003. Acknowledgement. Phillip Colella and Applied Numerical Algorithms Group, LBNL - PowerPoint PPT PresentationTRANSCRIPT
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AMR Simulations of the Magneto-hydrodynamic
Richtmyer-Meshkov Instability
Ravi SamtaneyComputational Plasma Physics Group
Princeton Plasma Physics Laboratory
Princeton University
CPPG SeminarPPPL, 04/09/2003
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Acknowledgement
• Phillip Colella and Applied Numerical Algorithms Group, LBNL
• Steve Jardin, PPPL• DOE SciDAC program
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Outline
• MHD primer (MHD with shocks)• Richtmyer-Meshkov Instability (RMI)• RMI suppression in MHD • Numerical Method
– Adaptive Mesh Refinement (AMR)– Unsplit upwinding method– div(B) issues
• Conclusion and future work
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Electromagnetic Coupling• Weakly coupled formulation
– Hydrodynamic quantities in conservative form, electrodynamic terms in source term
– Hydrodynamic conservation & jump conditions– One characteristic wave speed (ion-acoustic)
• Tightly coupled formulation– Fully conservative form– MHD conservation and jump conditions– Three characteristic wave speeds (slow, Alfvén, fast)– One degenerate eigenvalue/eigenvector
t
u
12u
2 1 1 p
uuu Ip
12 u
2 1 p u
T
T
0
10 jB0
Bt
uB
j1
0B
t
u
12 u
2 1 1 p
120B2
B
uuu p 1
20B2 I 1
0BB
12 u
2 1 p
12 0B2 u 1
0uB B
uB Bu
T
T
0
0
0
0
j¢ E
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Single-fluid resistive MHD Equations
• Equations in conservation formParabolic
Hyperbolic
Reynolds no.
Lundquist no.
Peclet no.
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MHD Discontinuities• MHD shock waves
– Fast shocks– Slow shocks– Intermediate shocks (unstable if =0)– Compound shocks (shock-rarefaction
structures do not occur in ideal MHD)– Alfvén shocks
• MHD contact discontinuities– No tangential jump in
velocity if normal component of B is present
– Shear flow discontinuities allowed if B.n=0
Figures from PhD thesis of Hans De Sterck, Katholieke Universiteit Leuven
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Richtmyer-Meshkov Instability (RMI)
• RMI – occurs at a fluid interface
accelerated by a shock-wave
– “impulsive Rayleigh-Taylor”
– Linear stability analysis by Richtmyer (1960)
– Experimental confirmation by Meshkov (1970)
h
l
g(t) = constant (RT)
= U (t) (RM)
Growth rate of perturbation:
Exponential (RT)
Linear (RM)
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Richtmyer-Meshkov Instability (RMI)• Occurs
– in inertial confinement fusion where it is the main inhibiting hydrodynamic mechanism
– in astrophysical situations
• Approx. 200 peer-reviewed publications + numerous proceedings in last 15 years.
X-ray images from Nova laser expts.Barnes et al. (LANL Progress Report 1997)
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Parameters and Initial/Boundary Conditions
• Parameters which completely define the posed Mathematical problem– Mach number of the incident shock, M– Density ratio (or Atwood number at the fluid interface),
, At=(-1)/(+1)– Perturbation – Magnetic field strength -1=B0
2/2p0
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Simulation Results
• Parameters which completely define the posed M=2, =45o, =3, -1=0 (B0=0) or-1=0.5 (B0=1) t=0.385
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Simulation Results
• t=1.86
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Simulation Results
• t=3.33 “Results! Why, man, I have gotten a lot of results. I know severalthousand things that won’t work”- Thomas Edison
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Simulation Results – Early Refraction Phase
• t=3.33
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Vortex Dynamical Interpretation
• During incident shock refraction on the density interface vorticity is generated due to misalignment of density and pressure gradients (Hawley and Zabusky, PRL 1989)
•
=0 in 2D
Baroclinic source
zero during incidentshock refraction phase
r p
r
r p
r
Light
HeavyHeavy
Light
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Why does B suppress the instability? – A mathematical explanation
• Consider jump conditions across a stationary discontinuity[ un] = 0[ un
2 + p + ½ B2 – ½ Bn2] = 0
[ un ut – Bn Bt ] = 0[Bn] = 0[Bt un – Bn ut] = 0[(e + p + ½ B2) un – Bn (u¢ B)] =0
• For un=0 (no flow through discontinuity) implies
– [ut] = 0 if Bn!=0 or
– [p+Bt2] = 0 and ut and Bt can jump if Bn=0
• For un!=0, ut jumps across the discontinuity.
– In MHD shocks can support shear
• Jump conditions originally derived by Teller and do Hoffman (1952) Other sources: Friedrichs and Kranzer (1958), Anderson (1963)“Young man, in mathematics you don’t understand things,
you just get used to them” – J. Von Neumann
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Vortex sheets – Early Refraction Phase
• t=0.385• The shocked
interface is a vortex sheet (-1=0)
• In MHD shocks can support shear
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Vortex Dynamical Interpretation
• Baroclinic vorticity is taken away by slow shocks in the presence of a magnetic field. These shocks are stable (at least in our simulations)
• Net circulation in the domain
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Current Sheets – Early Refraction Phase
• t=0.385
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I don’t believe this! What if B is really small?
• In case the magnetic field is very small, the slow shocks move closer to the interface. The interface amplitude grows due to an “entrainment” effect.
t=1.91
t=3.31
RF
TF
TSRS
-1=0.05
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Results for < 1
• The same suppression effect is observed t=1.9
1
t=3.31
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Adaptive Mesh Refinement
• Adaptive mesh refinement provides a “numerical microscope”
• Provides resolution where we need it– Optimally should be based on a local truncation error analysis
• Mesh refined where local error exceeds a user defined threshold
– In practice, we refine the mesh where a threshold is exceeded (usually depends on gradients of flow quantities)
• LTE analysis is not very meaningful if discontinuities are present
– Usually leads to more efficient compuations
• Various AMR frameworks/packages– GrACE– SAMRAI– Paramesh– CHOMBO– AMRITA
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Why AMR?
• For the RMI case– Base mesh 256x32– Plus 3 levels of refinement (each with a factor of 4)– Effective uniform mesh of 16384x 2048– Estimated computational saving about a factor of 25– Each run took about 80 hours on NERSC– Uniform mesh run would have taken 25x80=2000 hours– % Area covered
• Finest mesh: 2.1%
• Level 2 : 6.5%
• Level 1 : 15.6%
“ It is unworthy of excellent mento lose hours, like slaves, in thelabors of calculation” -Gottfried Wilhelm Leibnitz
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Why AMR?
• Area covered – Finest mesh (321) : 7.6 %
– Level 2 (38): 18.5%
– Level 1 (5): 50 %
– Level 0 (2): 100 %
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AMR Basics
• Flag cells which need to be refined
• Using a clustering algorithm make patches
• Copy from previous overlapping fine mesh and interpolate from coarse mesh where new refinement occurs
• Surround each fine patch with a layer of ghost cells
• Interpolation must be consistent to the order of the method
Layer of ghost cells
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AMR Basics
• Update coarse mesh solution • Update fine mesh “r” times
– Coarse mesh solution interpolated in space and time to provide boundary conditions at CFI
• Synchronize at the end of each time step by flux refluxing to maintain strong conservation
• Average fine mesh solution replaces coarse solution where coarse/fine meshes overlap
Coarse at tn
Coarse at tn+1
Fine at tn
Fine at tn+1/2
Fine at tn+1
Adaptive time stepping
Synchronizeby replacingcoarse meshfluxes by fine mesh fluxes at CFI
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Adaptive Mesh Refinement with Chombo• Chombo is a collection of C++ libraries for implementing block-
structured adaptive mesh refinement (AMR) finite difference calculations (http://www.seesar.lbl.gov/ANAG/chombo) – (Chombo is an AMR developer’s toolkit)
• Mixed language model– C++ for higher-level data structures– FORTRAN for regular single grid calculations– C++ abstractions map to high-level mathematical description of AMR
algorithm components
• Reusable components. Component design based on mathematical abstractions to classes
• Based on public-domain standards– MPI, HDF5
• Chombovis: visualization package based on VTK, HDF5• Layered hierarchical properly nested meshes• Adaptivity in both space and time
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Chombo Layered Design• Chombo layers correspond to different levels of functionality in the AMR
algorithm space• Layer 1: basic data layout
– Multidimensional arrays and set calculus– data on unions of rectangles mapped onto distributed memory
• Layer 2: operators that couple different levels– conservative prolongation and restriction– averaging between AMR levels– interpolation of boundary conditions at coarse-fine interfaces– refluxing to maintain conservation at coarse-fine interfaces
• Layer 3: implementation of multilevel control structures– Berger-Oliger time stepping– multigrid iteration
• Layer 4: complete PDE solvers– Godunov methods for gas dynamics– Ideal and single-fluid resistive MHD– elliptic AMR solvers
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Numerical Method: Upwind Differencing
• The “one-way wave equation” propagating to the right:
• When the wave equation is discretized “upwind” (i.e. using data at the old time level that comes from the left the wave equations becomes:
• Advantages:– Physical: The numerical scheme “knows” where the information is coming from
– Robustness: The new value is a linear interpolation between two old values and therefore no new extrema are introduced
t
ax
0 a 0
in1 i
n atx
i 1n i
n 1 in i 1
n 1
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Symmetrizable MHD Equations• The symmetrizable MHD equations can be written in a near-
conservative form (Godunov, Numerical Methods for Mechanics of Continuum Media, 1, 1972, Powell et al., J. Comput. Phys., vol 154, 1999):
• Deviation from total conservative form is of the order of B truncation errors
• The symmetrizable MHD equations lead to the 8-wave method. The eigenvalues are
– The fluid velocity advects both the entropy and div(B)
in the 8-wave formulation
t
u
12 u
2 1 1 p
120B2
B
uuu p 1
20B2 I 1
0BB
12 u
2 1 p
12 0B2 u 1
0uB B
uB Bu
T
T
B
01 0B
1 0uB u
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• Conservative (divergence) form of conservation laws:
• Volume integral for computational cell:
• Fluxes of mass, momentum, energy and magnetic field entering from one cell to another through cell interfaces are the essence of finite volume schemes. This is a Riemann problem.
dU
dtF S
Numerical Method: Finite Volume Approach
dU i, j ,k
dt AF
faces S i, j ,k
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Numerical method: Riemann Solver
• Discontinuous initial condition– Interaction between two states– Transport of mass, momentum, energy
and magnetic flux through the interface due to waves propagating in the two media
• Riemann solver calculates interface fluxes from left and right states
• The eigenvalues and eigenvectors of the Jacobian, dF/dU are at the heart of the Riemann solver:
– Each wave is treated in an upwind manner
– The interface flux function is constructed from the individual upwind waves
– For each wave the artificial dissipation (necessary for stability) is proportional to the corresponding wave speed
FLR 1
2FL FR 1
2Rk
k1
8
k L k UR UL
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Unsplit method – Basic concept• Original idea by P. Colella (Colella, J. Comput. Phys., Vol 87, 1990)• Consider a two dimensional scalar advection equation
• Tracing back characteristics at t+ t
• Expressed as predictor-corrector
• Second order in space and time• Accounts for information propagating across corners of zone
-U tA3 A1
A4 A2
Corner coupling
I II
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Unsplit method: Hyperbolic conservation laws
• Hyperbolic conservation laws
• Expressed in “primitive” variables
• Require a second order estimate of fluxes
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Unsplit method: Hyperbolic conservation laws
• Compute the effect of normal derivative terms and source term on the extrapolation in space and time from cell centers to cell faces
• Compute estimates of Fd for computing 1D Flux derivatives Fd / xd
- I +
I
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Unsplit method: Hyperbolic conservation laws• Compute final correction to WI,§,d due to final transverse derivatives
• Compute final estimate of fluxes
• Update the conserved quantities
• Procedure described for D=2. For D=3, we need additional corrections to account for (1,1,1) diagonal couplingD=2 requires 4 Rieman solves per time stepD=3 requires 12 Riemann solves per time step
II
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The r¢ B=0 Problem• Conservation of B =0:
– Analytically: if B =0 at t=0 than it remains zero at all times– Numerically: In upwinding schemes the curl and div operators do not
commute
• Approaches:– Purist: Maxwell’s equations demand B =0 exactly, so B must
be zero numerically– Modeler: There is truncation error in components of B, so what is
special in a particular discretized form of B?
• Purposes to control B numerically:– To improve accuracy– To improve robustness– To avoid unphysical effects (Parallel Lorentz force)
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Approaches to address the r¢ B=0 constraint • 8-wave formulation: r¢ B = O(h) (Powell et al, JCP 1999; Brackbill and Barnes, JCP 1980)
• Constrained Transport (Balsara & Spicer JCP 1999, Dai & Woodward JCP 1998, Evans & Hawley Astro. J. 1988)
– Field Interpolated/Flux Interpolated Constrained Transport – Require a staggered representation of B– Satisfy r¢ B=0 at cell centers using face values of B
• Constrained Transport/Central Difference (Toth JCP 2000)
– Flux Interpolated/Field Interpolated– Satisfy r¢ B=0 at cell centers using cell centered B
• Projection Method• Vector Potential (Claim: CT/CD schemes can be cast as an “underlying”
vector potential. Evans and Hawley, Astro. J. 1988)• Require ad-hoc corrections to total energy• May lead to numerical instability (e.g. negative pressure – ad-hoc fix
based on switching between total energy and entropy formulation by Balsara)
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r¢ B=0 by Projection• Compute the estimates to the fluxes Fn+1/2
i+1/2,j using the unsplit formulation
• Use face-centered values of B to compute r¢ B. Solve the Poisson equation r2 = r ¢ B
• Correct B at faces: B=B-r• Correct the fluxes Fn+1/2
i+1/2,j with projected values of B
• Update conservative variables using the fluxes – The non-conservative source term S(U) r¢ B has been algebraically removed
• On uniform Cartesian grids, projection provides the smallest correction to remove the divergence of B. (Toth, JCP 2000)
• Does the nature of the equations change? – Hyperbolicity implies finite signal speed– Do corrections to B via r2=r¢ B violate hyperbolicity?
• Conservation implies that single isolated monopoles cannot occur. Numerical evidence suggests these occur in pairs which are spatially close.
– Corrections to B behave as 1/r2 in 2D and 1/r3 in 3D
• Projection does not alter the order of accuracy of the upwinding scheme and is consistent
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Unsplit + Projection AMR Implementation
• Implemented the Unsplit method using CHOMBO• Solenoidal B is achieved via projection, solving the elliptic equation r2=r¢
B– Solved using Multgrid on each level (union of rectangular meshes)– Coarser level provides Dirichlet boundary condition for
• Requires O(h3) interpolation of coarser mesh on boundary of fine level
– a “bottom smoother” (conjugate gradient solver) is invoked when mesh cannot be coarsened
– Physical boundary conditions are Neumann d/dn=0 (Reflecting) or Dirichlet
• Multigrid convergence is sensitive to block size• Flux corrections at coarse-fine boundaries to maintain
conservation– A consequence of this step: r¢ B=0 is violated on coarse meshes in
cells adjacent to fine meshes.
• Code is parallel• Second order accurate in space and time
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Conclusion
• Numerical evidence presented that the RMI is suppressed in the presence of a magnetic field– Mathematical explanation– Physical explanation still lacking
• Conjectures: Main result will remain essentially same– For non-ideal MHD– For Hall MHD– In three dimensions
• A conservative solenoidal B AMR MHD code was developed– Unsplit upwinding method for hyperbolic fluxes– r¢ B=0 achieved via projection
(These, gentlemen, are the opinions upon which I will base my facts” – Winston Churchill)
(“If the facts don’t fit the theory, change the facts”- Albert Einstein)
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Related AMR Work
• High resolution parallel 2D magnetic reconnection runs.
• Implicit treatment of viscous/conductivity terms• Pellet injection AMR simulations