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The Effects of Divergence B Errors on MHD Magnetosphere Simulations |
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The Problem:
- Why is it difficult to maintain
numerically?
- In what numerical representation should be
?
- How can we achieve this in a parallel, adaptive mesh, shock-capturing
code like BATS-R-US?
- Which scheme is most accurate, robust and efficient for magnetospheric
simulations?
- How efficient is local time stepping vs. time accurate simulation?
- How efficient and accurate is adaptive vs. uniform grid?
The Methods:
- 8-wave formulation (Powell 1999)
- Allow truncation error in
- Rederive MHD equations with
- Discretize this form to improve stability and accuracy
- Constrained Transport (Evans, Hawley 1988, Balsara, Spicer 1999)
- Interpolate F*Bfluxes of the
base scheme to obtain a time and edge centered electric field
En+1/2
- Obtain Bn+1 = Bn -
x En+1/2 with simple finite differences
- Conserve
to machine accuracy in a particular discretization
- Projection scheme (Brackbill, Barnes 1980)
- Calculate B* with the base scheme
- Solve Poisson equation
- Obtain divergence free
The Simulations:
- Steady state magnetosphere with northward and southward IMF
- Simulation box is 256 x 128 x 128 R3e
- Base scheme: 2nd order Rusanov
- 8-wave and projection schemes
- Block-adaptive grid: smallest cell size is 1Re
with ~ 256,000 cells and .25 Re with ~340,000
cells
- local timestepping: 6000 or 12000 time steps
- Constrained transport scheme
- Uniform grid: cell size is 2Re with ~460,000
cells
- time-accurate simulation: 90,000 time steps
- Earth is represented by a conducting sphere with an embedded non-tilted,
non-rotating magnetic dipole.
- Inner simulation boundary is an Earth centered sphere with R
= 3 Re where temperature and density are set to
10000 K and 1/ccm.
- Solar wind parameters are n=5/ccm, T=180,000K, v=-400
km/s
- Upstream Bz = +5nT and -5nT for northward and
southward IMF
Results:
- Definition of Normalized Difference:
where Vi is the volume of cell i, and U
is some variable
- Normalized Difference between 8-wave and Projection Schemes
| Variable |
Northward IMF |
Southward IMF |
 |
0.035 |
0.011 |
| vx |
0.033 |
0.011 |
| vy |
0.146 |
0.043 |
| vz |
0.155 |
0.043 |
| p |
0.059 |
0.021 |
| Bx |
0.134 |
0.081 |
| By |
0.110 |
0.094 |
| Bz |
0.025 |
0.015 |
- Figures:
Conclusions:
- Algorithms
- There is no unique discretization of
- Unphysical effects (Lorentz force || B) cannot be fully avoided
in shock-capturing codes
- The projection scheme gives correct weak solutions
- Local time stepping is not possible for constrained transport
- Comparison of numerical results
- The qualitative agreement among the three schemes is excellent
for both northward and southward IMF cases
- Quantitative differences between 8-wave and projection schemes
are of order ~10% for 250,000 cells and ~5% for 340,000 cells
- Comparison of efficiency
- Adaptive grid gives superior accuracy for given computational
cost
- Local time stepping greatly accelerates convergence
- Projection scheme is more expensive (x2 - x4) than 8-wave scheme
because it does not scale well for many blocks and many processors
- Constrained transport scheme is expensive for steady state calculation
due to lack of local time stepping (x10 - x20)
- For steady state calculations the 8-wave scheme is the most efficient
(accuracy/computational cost)
- Future
- For time-accurate simulations the flux interpolated constrained
transport scheme in combination with a shock-capturing base scheme
on an adaptive grid is a promising alternative to the 8-wave scheme
- For steady state calculations the 8-wave and projection schemes
can be combined: the converged steady state of the 8-wave scheme
can be further improved with the projection scheme for little extra
cost
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