1. Gradient Particle Magnetohydrodynamics and Adaptive Particle Refinement Astrophysical Fluid Dynamics Workshop Grand Challenge Problems in Computational Astrophysics Institute for Pure and Applied Mathematics UCLA 4-9 April 2005 Gregory G. Howes Department of Astronomy UC Berkeley

2. Collaborators Steve Cowley, Mark Morris UCLA Department of Physics and Astronomy Jim McWilliams UCLA Department of Atmospheric Science Jason Maron American Museum of Natural History

3. Outline 1. Gradient Particle Magnetohydrodynamics (GPM): a) Astrophysical Motivation b) Lagrangian Methods c) GPM Algorithm (see Maron and Howes (2003) ApJ, 595:564.) d) Convergence and Stability e) Magnetic Divergence f ) Test Results for Standard GPM g) Potential Weaknesses h) Error in Lagrangian Methods 2. Adaptive Particle Refinement: a) Strategy for Achieving General Adaptivity b) Preliminary Results 3. Disconnection Error and Computational Instability

4. Astrophysical Motivation Galactic Magnetic Field To understand the origin and evolution of the large- scale Galactic magnetic field Focus on the effect of the global geometry on the field Supernova explosion Vertical Field in Center B ~1mG Horizontal Field in Disc B ~ 3 μ G R ~ 10 kpc

5. Why Lagrangian? Advantages Disadvantages Inherently adaptive (but only in density) Easier implementation and setup No preferred direction Typically lower order of convergence More noisy than grid-based methods MHD is notoriously difficult

6. Lagrangian MHD Simulation Advancement in time requires knowledge of the values and gradients at the particle position

7. General Idea Behind GPM h The gradient is recovered by a local polynomial least squares fit. The local neighborhood is sampled by all particles within the a smoothing sphere, weighted by a smoothing function.

8. Gradient Particle Magnetohydrodynamics (GPM) Consider a fluid quantity with 1-D spatial profile Constructing the quantities Substituting for, we obtain a matrix equation for the mean, and the gradient. In 3-D, a 2nd-order fit yields a 10 x 10 matrix

9. Convergence Properties Fan and Gijbels (1996) review two decades of progress studying Local Polynomial Regression in Statistics For a th-order fit over smoothing radius, the local truncation error in th derivative is of order The optimal kernel (minimizing mean squared error) is the Epanechnikov kernel, The error for the 2nd order GPM algorithm for fitting gradients ( ) is thus.

10. Stability Properties No rigorous proof of computational stability exists Smoothing: In practice, periodic smoothing over the sphere is necessary to inhibit growth of noise and maintain stability. Particle Separation: Particle separation both improves stability and ensures better efficiency by preventing two particles from sampling the same point.

11. Magnetic Divergence The induction equation can be written in terms of vector potential with magnetic field calculated in an intermediate step This works for most applications, but requires two derivatives to yield the gradient of magnetic field. In some applications, the resulting error can swamp the gradual accumulation of magnetic field over long time periods. A better alternative exists...

12. Lagrange Multipliers for Add magnetic divergence constraint to minimization as a Lagrange multiplier, with no guarantee that Smoothing is necessary to maintain for particle values. The GPM algorithm minimizes the weighted least squares fit for each magnetic field component,

13. Sound Wave Dispersion

14. Polar Plot of MHD Waves

15. Magnetic Divergence

16. MHD Shocks

17. Kelvin-Helmholtz Instability

18. Potential Weaknesses of GPM Not conservative Divergence cleaning may be inadequate or cause excessive smoothing of magnetic field Smoothing periodically is required for stability Adaptivity is based only on density Robustness to unfavorable particle distributions

19. Error in Lagrangian Methods Truncation Error: Error in local least squares fit Adaptive methods can reduce this error Disconnection Error: Error arising from poor sampling within the smoothing sphere Adaptive methods adjusting to the error in the local fit do not resolve this problem Requires correction separate from adaptivity Two types of error plague Lagrangian methods h

20. Adaptive Particle Refinement

21. hh Adaptivity to Reduce Error

22. Adaptive Particle Refinement Error Estimation AMR: Richardson Extrapolation Repeat calculation at same order but different resolution APR: Repeat calculation at same resolution but different order For an order fit at position, denoted by Refinement/unrefinement is determined by the ratios,, and.

23. Refine, Unrefine, or Smooth? Depending on the quality of the fit, you must choose to refine, smooth, unrefine, or do nothing.

24. Refinement Typical values are: Two tests: If and if then refine (add a particle)

25. Unrefinement If then unrefine (combine two particles) Typical value is:

26. Other Adaptivity Parameters Three parameters should control adaptive scheme: Minimum/Maximum scales: and Sensitivity Threshold: Other parameters,, and should not control the overall performance.

27. 1-D Hydrodynamic Shock

28. 2-D Hydrodynamic Shock

29. 2-D Magnetized Vortex

30. Disconnection Error

31. Disconnection Error and Instability Disconnection can cause GPM to suffer both error and a virulent computational instability

32. Disconnection Instability

34. Correction for Disconnection Two measures prevent disconnection 1. If quadrant contains no neighbor, add particle in center of that quadrant 2. If the closest neighbor in each quadrant is not a mutual neighbor, add a particle midway in between

35. Summary Gradient Particle Magnetohydrodynamics (GPM): - New numerical method for Lagrangian particle simulation of astrophysical MHD systems Adaptive Particle Refinement (APR): - Strategy to introduce general adaptivity to Lagrangian particle methods - Preliminary results are promising Disconnection Error and Instability: - Disconnection can wreak havoc in Lagrangian particle codes - Ensuring mutual neighbors can reduce the error and prevent growth of the instability

36. References Fan, J. and Gijbels, I. (1996) Local Polynomial Modeling and Its Applications (New York: Chapman and Hall). Maron, J. L. and Howes, G. G. (2003) Gradient Particle Magnetohydrodynamics: A Lagrangian Particle Code for Astrophysical Magnetohydrodynamics, ApJ, 595:564.