# Presentation: 509596

## Multiscale PIC modeling of beams and plasmas.

**Ec****o****le CEA-EDF-INRIA /***** CEA-EDF-INRIA School***** **
**Kinetic equations. Applications to plasma and beam physics.**
- September 19-22, 2005 - Rocquencourt, France

**J.-L. Vay **
**Heavy Ion Fusion-Virtual National Lab./Lawrence Berkeley National Lab.**
***In collaboration with:***** ****	**
**A. Friedman, D.P. Grote, R.H. Cohen - HIF-VNL/Lawrence Livermore Nat. Lab.**
**P. Colella, McCorquodale, D. Serafini - Lawrence Berkeley Nat. Lab.**** **
**J.-C. Adam, A. Héron - CPHT, Ecole Polytechnique, France**
**E. Sonnendrücker, Université de Strasbourg, France**

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## The U.S. Heavy Ion Fusion Program - Participation

**Lawrence Berkeley National Laboratory****	****	****MIT**
**Lawrence Livermore National Laboratory****	****	****Advanced Ceramics**
**Princeton Plasma Physics Laboratory****	****	****Allied Signal**
**Naval Research Laboratory ****	****	****	****National Arnold**
**Los Alamos National Laboratory****	****	****	****Hitachi**
**Sandia National Laboratory****	****	****	****Mission Research**
**University of Maryland ****	****	****	****	****Georgia Tech**
**University of Missouri****	**** ****	****	****	****General Atomic**
**Stanford Linear Accelerator Center ****	****	****MRTI ****	**
**Advanced Magnet Laboratory ****	****	****	****	**
**Idaho National Environmental and Engineering Lab ****	**
**University of California**
**	**** ****a.  Berkeley**
**	**** ****b.  Los Angeles **
**	**** ****c.  San Diego**
**Employees of LBNL, LLNL, and PPPL form the U.S. Virtual National Laboratory for Heavy Ion Fusion**

## Artist view of a Heavy Ion Fusion driver

**How near term goal is High-Energy Density Physics (HEDP).**

**Heavy Ion Inertial Fusion (HIF) ****goal is to develop an accelerator that can deliver beams to ignite an inertial fusion target**

**DT**

**Target requirements:**
**3-7 MJ x ~ 10 ns ******** *****~ 500 Terawatts***
**Ion Range: 0.02 - 0.2 g/cm****2**** ******** *****1-10 GeV***
**drive accelerator requirements:**
**A~200 ******** *****~10******16 ******ions, 100 beams, 1-4 kA/beam***** **

**Artist view of a Heavy Ion Fusion driver**

## Time and length scales in driver and chamber span a wide range 

_**Length scales:**_

**electron**

**cyclotron**

**in magnet**

**pulse**

**electron drift**

**out of magnet**

**beam**

**residence**

****

**pb**

****

**lattice**

**period**

**betatron**

**depressed**

**betatron**

****

**pe**

**transit**

**thru**

**fringe**

**fields**

**beam**

**residence**

**pulse**

**log of timescale**

**in seconds**

****

**pi**

****

**pb**

** ****electron gyroradius in magnet ~10 ********m**
** ********D,beam**** ~ 1 mm**
** ****beam radius ~ cm**
** ****machine length ~ km's**

_**Time scales:**_

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities.**** **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Modeling of a plasmas - classification (1)

**Collection of a large number of ****interacting**** charged ****particles**
**Particles**** mathematically described by**
** ****Lagrangian approach: evolution of singularities**
** ****Klimontovitch eq.**
** ****Eulerian approach: evolution of an incompressible fluid**
** ****in phase-space: Boltzmann/Fokker-Planck eq. (collisions), Vlasov eq. (no collisions)**
** ****in real space: fluid/MHD eq.**
**Interactions**** mathematically described by**
** ****Lagrangian approach: sum from all singularities, instantaneous or with ****	****	****	****      retardation**
** ****Eulerian approach: fields**
** ****instantaneous: Poisson**
** ****with retardation: Maxwell**

## In summary, the modeling of a plasma implies the modeling of 

interacting 

The numerical integration leads to further splitting
Partial differential equations: finite-differences/volumes/elements, Monte-Carlo, semi-Lagrangian, 
Time integration: explicit/implicit, 
Direct interaction: direct summation, multipole expansion (tree-codes),
...

**interacting **
**The numerical integration leads to further splitting**
**Partial differential equations: finite-differences/volumes/elements, Monte-Carlo, semi-Lagrangian, **
**Time integration: explicit/implicit, **
**Direct interaction: direct summation, multipole expansion (tree-codes),**
**...**

**Modeling of a plasmas - classification (2)**

**a collection of particles**

**fluid cells in phase-space**

**fluid cells in configuration space**

**or**

**or**

**directly**

**or through a field**

## Modeling of a plasmas - commonalities, speed-up

**All these methods have in common that they must update the status of N quantities (particle/fluid/field quantities) from time t to time t+********t**
**In order to minimize the computing time, N/********t should be minimized**
**grids: non-uniform, block-structured, high-order splines, filtering,**
**time steps: non-uniform, different for particle groups (species, velocity, ...), different for grid blocks (with different ********x), high-order integrators, averaging over smallest time scales,**
**particles: splitting/merging, high-order splines, filtering,**
**hybrids**
** ****Groups of particles modeled differently, according to species, velocities, momentum, charge state, ...**
** ****particle-particle-particle-mesh (pp-pm), **
** ****regions modeled differently (for example implicit in high-density parts, explicit in low-density parts)**
**...**
**The subject is very large. We will focus on a few recent developments.**

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## The Adaptive-Mesh-Refinement (AMR) method

** ****addresses the issue of wide range of space scales**
** ****well established method in fluid calculations**
** ****however, coupling to PIC/Vlasov/MHD methods has to be done with care**

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Mesh refinement implies a jump of resolution and some procedure for coupling the solutions at the interface. What kind of issues can we expect?

 loss of symmetry: self-force?

 conservation laws?

 waves (EM, plasma)?

We will look at some of these aspects using simple schemes in reduced dimensions.

-  loss of symmetry: self-force?
-  conservation laws?
-  waves (EM, plasma)?
- We will look at some of these aspects using simple schemes in reduced dimensions.

**Coupling of AMR to PIC/Vlasov/MHD: issues**

## Example: 2-D PIC-electrostatic

- Given a hierarchy of grids, there exists several ways to solve Poisson.
- Two considered here:
**‘****1-pass’**
- solve on coarse grid
- interpolate solution on fine grid boundary
- solve on fine grid
- different values on collocated nodes
**‘****N-pass’**
- interleave coarse and fine grid relaxations
- collocated nodes values reconciliation
- same values on collocated nodes

## Illustration of the spurious self-force effect

**Illustration of the spurious self-force effect**

-  1 grid with metallic boundary + 1 refinement patch

-  MR introduces spurious force,

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## Self-force amplitude map and mitigation

**Self-force amplitude map and mitigation**

-  Magnitude of self force decreases rapidly with distance from edge
-  with the 1-pass method, the coarse grid solution is free of self-force:
-  the self-force effect can be mitigated by defining a transition region surrounding the patch in which deposit charge and solve, but get field from underlying coarse patch
-  N-pass method: coarse grid solution has spurious self-force
-  no easy mitigation method

- patch

- main grid

- transition region

- Method

- Interpolation between patches

## Global error

-  global error larger with N-pass than 1-pass

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- Linear

- Quadratic

- 1-pass

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- N-pass

**Global error**

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## Global error

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-  N-pass: global error due to violation of Gauss’ law

**Global error**

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- Quadratic

- N-pass

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- N-pass

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## Electromagnetics: usual scheme

- Rc

- Rf

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- Rc: coarse resolution
- Rf: fine resolution

- P

-  the solution is computed as usual in the main grid and in the patch
-  interpolation is performed at the interface

## 1-D AMR-EM

**We consider 1d wave equation **
**staggered on a regular space time grid**
**We use finite-difference time-centered scheme**

## 1-D AMR-EM: space refinement only (factor 3)

**1-D AMR-EM: space refinement only (factor 3)**

## 1-D AMR-EM: space and time (factor 3)

## 1-D AMR-EM: illustration of instability*

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- Space only

- Space+Time

- o: E, x:B

- * J.-L. Vay, JCP (2001)

## Coupling of AMR to PIC/Vlasov/MHD: Issues (summary)

- Asymmetry of grid implies asymmetry
-     of field solution for one particle/marker/fluid cell
- spurious self-force
- Some implementations may violate Gauss’ Law
- total charge may not be conserved exactly
- EM: shortest wavelength resolved on fine grid not resolved on coarse grid may reflect at interface
- if reflection factor <=1, spurious waves
- if reflection factor >1, may cause instability by multiple reflections
- Remark: BTW, these are general and apply also to irregular griddings!

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Current HCX Configuration(High Brightness Beam Transport Campaign, 2005)

**INJECTOR**

**MATCHING**
**SECTION**

**ELECTROSTATIC**
**QUADRUPOLES**

**MAGNETIC**
**QUADRUPOLES**

**Focus of Current**
**Gas/Electron Experiments**

_**Additional Experiments:**_** Fill-Factor Measurements,                Head-Tail Correction, Wave Experiments**

**1 MeV, 0.18 A, t ≈ 5 ********s, **6x1012 K+/pulse

## We are using the accelerator PIC code WARP.

**Geometry: ****3D, (x,y), or (r,z)**
**Field solvers:**** FFT, capacity matrix, multigrid **
**Particle pusher****: Boris, subcycling**
**Boundaries:**** “cut-cell” --- no restriction to “Legos”**
**Bends:**** “warped” coordinates; no “reference orbit”**
**Lattice description:**** general; takes MAD input**
**solenoids, dipoles, quads, sextupoles, ...**
**arbitrary fields, acceleration**
**Beam injection: ****Child-Langmuir, and other models**
**Diagnostics:**** Extensive snapshots and histories**
**Parallel:**** MPI**
**Python and Fortran: ****“steerable,” input decks are programs **
**a GUI is also available**
**New advanced features:**
_**AMR**_**, **_**Electron mover with large time steps**_**, gas and electrons models, prototype Vlasov (soon)**

## 3-D WARP simulation of High-Current Experiment (HCX)

**In the following slides, we will follow the story of the why and how we implemented mesh refinement to get to numerical convergence.**

- (from a WARP movie;
- see http://hif.lbl.gov/theory/simulation_movies.html)

## 3-D WARP simulation of HCX shows beam head scrapping 

- Rise-time  = 800 ns
- beam head particle loss < 0.1%

**z (m)**

**z (m)**

**x (m)**

**x (m)**

- Rise-time  = 400 ns
- zero beam head particle loss

** ****Head cleaner with shorter voltage rise-time. **
** ****Questions:**
** ****what is the optimal rise-time?**
** ****can we produce and model very-fast rise-time?**

## Test: 1-D time-dependent modeling of ion diode

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**Applied voltage for Heavyside **
**current history?**

**time**

**current**

**Analytic solution from Lampel-Tiefenback**

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## Test: 1D time-dependent modeling of ion diode (algo 1)

_**Injection algorithm**_

- Child-Langmuir solution* + voltage drop between emitter and virtual surface determines current to inject.

- *1-D; => JI (J=I/S, S=1)

## Unphysical oscillation related to Nb particles injected/time step (Ni)

- I(A)

- Time (s)

**N = 160**
******t = 1ns**
**d = 0.4m**

## Cure: derive voltage history numerically

_**Injection algorithm**_

- We apply Lampel-Tiefenback method at the discrete level

- solve for V using linearity of Poisson
**       ****(V-V****i****) = (V-V****i****)****V=0**** + (V-V****i****)********=0**

## Cure #2: apply irregular gridded patch around emitter.

- V (volts)

**Initial potential **
**V****0**** given by**
**V****0****/V****max****=10****-2**** **
**means d****i****/d=10****-6****! **

-  Apply irregular gridded patch covering di
-  Mesh sizes such that number of  particles per cell is a constant in patch assuming Child-Langmuir solution for (z)
-  Apply same injection algorithm as before in patch

**Cure #2: apply irregular gridded patch around emitter.**

- Time (s)

## Cure #3: apply regularly gridded patch following front.

**An Adaptive-Mesh-Refinement patch**
**Follows the front**

**Cure #3: apply regularly gridded patch following front.**

## Summary

- I (A)

- Time (s)

**N = 160**
******t = 1ns**
**d = 0.4m**

-  Discrete voltage solution or MR patch suppressed long wavelength oscillation
-  AMR patch suppressed front peak

**Regular grid**

**Discrete voltage solution**

**Emitter MR patch**

**Summary**

**Emitter MR patch + **
**AMR patch**

## Without MR, WARP predicts overshoot

Run with MR predicts very sharp risetime (not square due to erosion)

- T (s)

- V (kV)

**“****Optimized” Voltage**

- Without MR, WARP predicts overshoot
- Run with MR predicts very sharp risetime (not square due to erosion)

**Extension to three dimensions**

**Current at Z=0.62m**

- X (m)

- Z (m)

**Ion source diode**

- MR patch

- side    face

- Specialized 1-D patch implemented in 3-D injection routine, as a 2-D array of 1-D patches.

- Extended Lampel-Tiefenback technique to 3-D,  and implemented in WARP
-  predicts a voltage waveform which extracts a nearly flat current at emitter

## Test of MR patch on modeling of STS500 Experiment.

- Z (m)

- Current history (Z=0.62m)

- Current history (Z=0.62m)

**Test of MR patch on modeling of STS500 Experiment.**

- * J.-L. Vay et al, PoP (2003)

- MR OFF

- MR ON

##  Modeling of source critical - determines initial shape of beam.

** ****Modeling of source critical - determines initial shape of beam.**

**WARP-RZ (axi-symmetric)  simulations show that a fairly high resolution is needed to reach convergence**

| Run | Grid size | Nb particles |
| --- | --- | --- |
| Low res. | 56x640 | ~1M |
| Medium res. | 112x1280 | ~4M |
| High res. | 224x2560 | ~16M |
| Very High res. | 448x5120 | ~64M |

## First MR attempt - 1 MR block surrounding emitter.

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| Run | Grid size | Nb particles |
| --- | --- | --- |
| Low res. | 56x640 | ~1M |
| Medium res. | 112x1280 | ~4M |
| High res. | 224x2560 | ~16M |
| Medium res. + MR | 112x1280 | ~4M |

**Refining around the emitter area is enough to recover emittance from converged high-resolution  case.**

## First MR attempt - 1 MR block surrounding emitter (2).

- Low res.

- Medium res.

- High res.

- Medium res. + MR

- R’ (rad.)

**However, it is not enough for recovering details of distribution.**

- R (m)

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- R (m)

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- (plots from data at z=0.4m)

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- R (m)

- R (m)

- Rho (C/m3)

- Rho (C/m3)

- Rho (C/m3)

- Rho (C/m3)

## Second attempt - 1 MR block with adaptive excavation.

**Second attempt - 1 MR block with adaptive excavation.**

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- 4 nRMS  (mm.mrad)

| Run | Grid size | Nb particles |
| --- | --- | --- |
| Low res. | 56x640 | ~1M |
| Medium res. | 112x1280 | ~4M |
| High res. | 224x2560 | ~16M |
| Medium res. + MR | 112x1280 | ~4M |

**Emittance recovered, again.**

## Second attempt - 1 MR block with adaptive excavation (2).

**Second attempt - 1 MR block with adaptive excavation (2).**

- Low res.

- Medium res.

- High res.

- Medium res. + MR

- R’ (rad.)

**Refining emission are AND beam edge sufficient for recovering details of distribution.**

- R (m)

- R’ (rad.)

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- R (m)

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- (plots from data at z=0.4m)

- R (m)

- R (m)

- R (m)

- R (m)

- Rho (C/m3)

- Rho (C/m3)

- Rho (C/m3)

- Rho (C/m3)

## Full AMR implementation: speedup ~10.5

**Full AMR implementation: speedup ~10.5**

| Run | Grid size | Nb particles |
| --- | --- | --- |
| Low res. | 56x640 | ~1M |
| Medium res. | 112x1280 | ~4M |
| High res. | 224x2560 | ~16M |
| Low res. + AMR | 56x640 | ~1M |

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## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Study of e-cloud in LHC FODO cell

**The problem:****  **
**	****Simulate “multibunch, multiturn” passage of beam through FODO cell (~100 m):**
**	****	****	****   ****	****dipoles**
**	****	****	****	****quadrupoles**
**	****	****	****	****drifts**
**	****Electrons  ********  synchrotron radiation, secondary emission**
**Study:**
**	****Electron accumulation and trapping in quads**
**	****Power deposition from electrons**
**First try with one bunch in periodic FODO cell.**
**   **

## Frame 2nd passage of bunch through cell - 2

-  We use actual LHC pipe shape: beam size << pipe radius
-  Mesh Refinement provides speedup of **x20,000** on field solve

**beam**

**electrons**

## Frame 2nd passage of bunch through cell - 1

**(particles colored according to radius)**

**beam (scaled 10x)**

**electrons**

**1 LHC FODO cell**

**F B  B   B D B   B   B**

**T~0.5********s**

**left over from 1st passage**

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## A cartoon of fast ignition.

## Laser-plasma interaction in the context of fast ignition

**A laser impinges on a cylindrical target which density is far greater than the critical density. **
**The center of the plasma is artificially cooled to simulate a cold high-density core.**
**Patch boundary surrounds plasma. Laser launched outside the patch.**

** ****Implemented new MR technique in EM PIC code Emi2d (E. Polytech.)**

## PIC electromagnetic 2D, cubic splines (- noise, + stable), Esirkepov exact current deposition scheme
Boundary conditions: open system
particles
ions leave the box freely
electrons reflected until an ion exit (overall charge conserved)
EM fields: absorbing layer (“Asymmetric PML”*) + incoming wave 

* Vay, JCP (2002)

**Boundary conditions: open system**
**particles**
**ions leave the box freely**
**electrons reflected until an ion exit (overall charge conserved)**
**EM fields: absorbing layer (“Asymmetric PML”*) + incoming wave **
*** Vay, *****JCP***** (2002)**

**EMI2D code**

## New MR method implemented in EM PIC code Emi2d

- coarse

**PMLs**

- M

- Outside patch:
- F = FM

- Inside patch:
- F = FM-FC+FF

- F

- C

**Mesh refinement by substitution***

- * J.-L. Vay, J.-C. Adam, A. Heron, CPC (2004)

## Comparison patch on/off

**Comparison patch on/off**

-      MR off

-    MR on

-    MR on

## Partial cancellation due to numerical dispersion

**Partial cancellation due to numerical dispersion**

- Bz

- Bz main grid

- Bz fine patch

- Bz coarse patch

- main grid: laser + plasma response + residual
- patches: plasma response at two slightly different velocities

## Use less dispersive Maxwell solver

Inject residual of waves on main grid at patch interface

Do not use coarse patch and solve on fine patch with source term J as a correction to J

Go back to usual scheme with a hole in the main grid
put PML inside hole and on fine patch border 
couple using clean cross-injections

**Inject residual of waves on main grid at patch interface**
**Do not use coarse patch and solve on fine patch with source term ********J as a correction to J**
**Go back to usual scheme with a hole in the main grid**
**put PML inside hole and on fine patch border **
**couple using clean cross-injections**

**Possible paths for better scheme**

- coarse

**PML**

- M

- fine

- Outside patch: J

- Inside patch: J

- F

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## 	4D Vlasov testbed (with constant focusing) showed structure of the halo in a density-mismatched beam

**Solution of Vlasov equation on a grid in phase space offers low noise, large dynamic range for beam halo studies**

## New ideas: moving grid to model time-dependent applied field, AMR-Vlasov to resolve fine structures

- moving phase-space grid, based on non-split
- semi-Lagrangian advance    [E. Sonnendrucker,          F. Filbet, A. Friedman,     E. Oudet, J.-L. Vay, *CPC*, 2004]
- 

- 
- adaptive mesh [M. Gutnic, M. Haefele, I. Paun , E Sonnendrucker, CPC 2004]

## 3.3 - Development of AMR library for PIC at LBNL

## Researchers from AFRD (PIC) and ANAG (AMR-Phil Colella’s group) collaborate to provide a library of tools that will give AMR capability to existing PIC codes (on serial and parallel computers)

The base is the existing ANAG’s AMR library Chombo (http://seesar.lbl.gov/anag/chombo) 

The way it works

WARP is test PIC code but library will be usable by any PIC code

**Researchers from AFRD (PIC) and ANAG (AMR-Phil Colella’s group) collaborate to provide a library of tools that will give AMR capability to existing PIC codes (on serial and parallel computers)**
**The base is the existing ANAG’s AMR library Chombo (http://seesar.lbl.gov/anag/chombo) **
**The way it works**
**WARP is test PIC code but library will be usable by any PIC code**

**Effort to develop AMR library for PIC at LBNL**

- WARP/MLI

- Chombo

## Examples of pre-AMR-PIC simulations using Chombo

- * P. McCorquodale, P. Colella, D.P. Grote, J.-L. Vay, JCP (2004)

**WARP-Chombo injector field calculation***

**MLI-Chombo beam field calculation**

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Motivation

- Our historical motivation: e-clouds in induction accelerators for HIF
- Need to follow electron orbits both in magnets (strongly magnetized) and in between (unmagnetized).
- Analytic integration of orbits in B field impossible because beam potential known only numerically and can’t be considered as impulsive.
- Need a way to accurately calculate electron orbits without having to take timesteps small compared to cyclotron period
- Note:
- above considerations apply to:
-  other kinds of accelerators
-  plasmas with both strong and weak magnetic fields
- Magnetic-fusion
- Inertial confinement fusion
- Space plasmas

## Statement of the problem

- Historical inspiration: Parker & Birdsall (JCP ’91)
- showed that standard Boris mover at large ct produces correct ExB and magnetic drifts
- Price: anomalously large “gyro” radius (~  ct) and anomalously low “gyro” frequency (particle orbit advances by almost in gyrophase per timestep; precesses at frequency ~ 1/ ct2)
- For our applications, low “gyro” freq. OK but large “gyroradius” is not

- Problem: Electron gyro timescale
-      << other timescales of interest
- 	 brute-force integration _very_ slow due to small t

**Magnetic quadrupole**

**Sample electron motion in a quad**

## We have developed an interpolation technique that allows us to skip over electron-cyclotron timescale

- Our solution: interpolation between full-electron dynamics (Boris mover) and drift kinetics (motion along B plus drifts).
- Choice 1/[1+(ct/2)2]1/2 gives, at both small and large ct,
- physically correct “gyro” radius
- correct drift velocity
- Correct parallel dynamics.
- Incorrect “gyration frequency” at large ct (same as pure Boris mover)
- Time step constraint set by next longer time scale -- typically electron cross-beam transit time.

## Test problem: drit orbits in quadrupole field with a specified beam space-charge potential

- Compare full orbit (ct ~ 0.25) to interpolated mover (ct ~ 2.5).
- Single orbit comparisons of some regular and nonadiabatic (chaotic) orbits:
- Chaotic orbits: ones launched on field lines that pass very close to field null.
- Good agreement on drift & bounce velocity, orbit size for regular orbits
- Expected non-agreement for chaotic orbits (expect similar statistics, but not tested).

- chaotic

- Average slope gives drift, frequency of stairsteps is bounce frequency

## Ion-ion two-stream instability test shows good agreement in time histories

**Ion-ion two-stream instability test shows good agreement in time histories**

- Test problem:
- Uniform B field; counter-streaming proton beams along B, 10 i across
- c/p = 48; vb/vth = 0.1; L/ i  60
- Compare: small t (c t = 0.6), large t=20 t (c t = 12) with interpolation; t with Boris mover (Parker-Birdsall)
- Finite beam-size effect: comparison with 20 i beam

- 20 i

- Small t
- t=20t interpolated
- t=20t Boris
- 20 i

- random seed => small time shift of (b)

- Small t
- t=20t interpolated
- t=20t Boris
- 20 i

- (c)

- (c)

## Ion-ion two-stream instability test shows good agreement in z-vz, x-y   phase space plots	

- small t

- interpolated mover

- Boris/Parker-Birdsall

- t=20t

- t=20t

- Reference run

## Coupled electron-ion test problem: electrons desorbed at end plate upon ion bombardment 

**Simulates experiment performed in High-Current Experiment (HCX) at LBNL**
**Ion beam allowed to hit end plate**
**Copious electrons produced**
**Here: calculate electron cloud produced in fourth magnet**
**We have also calculated the electron cloud in all 4 magnets and the resultant change in the ion phase space, and compared with experiment.**

## Comparison simulations in 4th magnet are another demo that the long-timestep electron mover works

**Comparison simulations in 4th magnet are another demo that the long-timestep electron mover works**

- Reference run

- interpolated mover

- Boris/Parker-Birdsall

- ions

- small t

- t=20t

- t=20t

- pipe wall

- pipe wall

- pipe wall

**Instability**

## New mover: summary and plans

- Interpolation between Boris algorithm and drift kinetics enables particle simulation with large ct that preserves physically correct gyroradius, drifts, and motion parallel to B
- Several tests demonstrate validity of particle mover in situations where simple application of Boris at large ct fails.
- Enables simulation on next-longer time scale -- electron bounce motion for the accelerator examples; wave period for the instability example
- Future directions:
- Bounce orbit averaging or projective integration to jump over electron bounce scales
- Combine with implicitness and collisions for applications to high-density plasmas

## Outline

**Who we are. Our interest in multiscale modeling.**
**Modeling of plasmas: generalities. **
**AMR**
**issues**
**Electrostatic**
**modeling of the High-Current Experiment (HCX)**
**modeling of the Large Hadron Collider (LHC)**
**Electromagnetic**
**modeling of laser-plasma interaction**** **
**Vlasov**
**New particle mover for large time steps in magnetic fields**
**Toward multiscale modeling of plasmas: some methods**
**Conclusion**

## Toward multiscale modeling of plasmas

- A multiscale approach to the modeling of plasmas should
- advance particles, or group of particles, according to pre-established criteria of accuracy, on an adaptive basis
- integrate methods that solve on different time scales in one single coherent scheme
- We will look at five methods:
- Implicit Multiscale PIC (Friedman et al)
- Discrete Event Simulation PIC (Karimabadi et al)
- Relaxed Iterative Methods for Coupling Disparate Scales (Shestakov et al)
- Equation Free Projective Integration (Shay et al)
- Implicit/explicit solvers coupling (Adam et al)

## Implicit MultiScale PIC - IMSPIC

- Advance each particle using a timestep that resolves the local field variations (assumed to be at scale of the grid spacing)
- Implicitness to:
- Afford stability with t > plasma and x > Debye
- in *selected* regions of phase space where that physics is deemed unimportant
- ... requires judgment on part of user, and/or smart controls
- afford a time-centered, second-order-accurate scheme
- Particle push is a variant of “d1” scheme, which allows time step adaptivity
- Poisson equation includes an *“implicit susceptibility”* (x)

## Timestep sizes are all multiples of some smallest “micro” step size; field-solve is done every micro-step

## A timeline shows the procedure for both active and inactive blocks

**A timeline shows the procedure for both active and inactive blocks**

## Timestep size control is an “art” as much as a “science”

- Seek to control truncation error
- Static control associates *ab initio* a step size  with each location in phase space
- Dynamic control sets  based on evolving gradients, etc.
- In the sheath application (see next slides), particle travel through the sheath (*x*E), rather than the time-dependent variation of E, is most limiting
- Would like to limit *kv**t** **** ****1*, where *k ~ **x*E/E .  However,
- 	if E and *x*E are fluctuating about zero (as is often the case),
- 	then where E ~ 0 there may be spuriously large values of *k*
- It’s somewhat easier to limit 2trap *t*2  (*q*/*m*)*x*E*t*2* **** ****2*
- 	by computing*x*E on the grid
- For our sheath work we used static control

## Application to the modeling of a sheath near a “floating” wall

- ions fixed,
- electrons cutoff Maxwellian

- particles are trapped

- particles hit the wall & are absorbed

- *	S. E. Parker, A. Friedman, S. L. Ray, and C. K.  Birdsall, “Bounded Multi-Scale Plasma Simulation: Application to Sheath Problems,” *J. Comp. Phys*. **107**, 388 (1993).

## Application to sheath showed effectiveness of method

- Potential vs. x at t = 100 (p = 1)

- Multi-
- group,
- max
- timestep
- ratio
- G = 64
- Single-
- group

-  vs. x at t = 100

- CPU time vs. max timestep ratio G
- (speedup of 4 at optimum G = 32)

- 	Another series of runs examined propagation of an ion acoustic shock front toward a conducting absorbing plate; see paper by Parker, *et al.*

## Motivation: 
Electron temperature in the accelerating region of a Hall thruster (40 eV) is higher than the temperature of charge saturation of SEE in Maxwellian plasma (17 eV).           [Staack, Raitses, Fisch, Appl. Phys. Lett. 84, 3028 (2004).]
Objective: 
The investigation of modification of electron velocity distribution function by SEE effects.

Simulation requirements: 
Both the sheath and the bulk plasma must be resolved.

PIC code: 
Electrostatic implicit multi-scale with non-uniform grid constant in time. [Friedman, Parker, Ray,  Birdsall, J. Comput. Phys. 96, 54 (1991).] The external fields Bx and Ez are assumed constant. 

- Motivation:
- Electron temperature in the accelerating region of a Hall thruster (40 eV) is higher than the temperature of charge saturation of SEE in Maxwellian plasma (17 eV).           [Staack, Raitses, Fisch, *Appl. Phys. Lett.* 84, 3028 (2004).]
- Objective:
- The investigation of modification of electron velocity distribution function by SEE effects.
- Simulation requirements:
- Both the sheath and the bulk plasma must be resolved.
- PIC code:
- Electrostatic implicit multi-scale with non-uniform grid constant in time. [Friedman, Parker, Ray,  Birdsall, *J. Comput. Phys.* 96, 54 (1991).] The external fields Bx and Ez are assumed constant.

- Hall thruster, cylindrical geometry:

- 1D3V PIC simulations, plane geometry, approximation of accelerating region of a Hall thruster:

**Application of IMSPIC to secondary electron emission (SEE) effects in a plasma slab in crossed electric and magnetic fields ****[Sydorenko, Smolyakov, 46****th**** APS DPP, Savannah GA, 2004, NM2B.008]**

## The results of the single-scale and multi-scale simulations are close to each other and reproduce the results of Schwager.
The multi-scale simulation is 8 times faster than the single scale simulation.

- The multi-scale simulation is _8 times faster_ than the single scale simulation.

- The code was applied to simulations of the _region between the Maxwellian plasma source (x=0) and the wall with SEE (x=L)_.  No collisions, zero external fields.
- Such a problem was considered by Schwager [*Phys. Fluids* B 5, 631 (1993)]

- Snapshots of profile of potential.
- The insert figure zooms into the
- potential dip near the emitting wall.
- Blue arrows – Schwager’s data.
- Black curves – uniform grid,
- Red curves – nonuniform grid,

**Application of IMSPIC to secondary electron emission ...****Benchmarking of the multi-scale code**

## Discrete Event Simulation is an alternative approach

- DES PIC has similar goals to Implicit Multi-Scale PIC but differs fundamentally
- Event-driven, not time-driven
- Particle timesteps fully independent, asynchronous
- Not (necessarily) implicit
- Builds on established discrete-event methodology
- Incremental field solution may be a challenge
- Successfully applied to spacecraft charging in 1D spherical geometry*:

- *H. Karimabadi, J. Driscoll, Y. A. Omelchenko, and N. Omidi,to be publ. in *JCP*

## Relaxed Iterative Methods for Coupling Disparate Scales (RIC)

- Fluxes

- From: S. Jardin

- From: Waltz/Candy

**Motivation:**
-  evolution of T, n, v in toroidal MFE devices governed by **transport** which is usually dominated by fluxes driven by plasma **turbulence**** **
-  Significant spread of scales (especially time)

- Transport (nominally 1-D) sets
- the profiles (P(r), v(r), n(r), etc)
- whose gradients drive turbulence

- Turbulence (3-D, anisotropic)
- produce the fluxes that drive
- transport

## RIC: Step 1, split the equations

- Consider nonlinear equations containing two timescales, of form:
- 	u = density, temperature, etc
- Define average & fluctuating parts u = u + u, u = 0
- Split equations into averaged (transport) and fluctuating (turbulence) parts:
- Notes:
-    denotes average over ensemble, spatial dimension, or time.
- Method applies to systems where short and long timescales not derivable from single set of eqs.
- Next step (2) is predicated on disparity of   and ~ timescales

- ~

- ~

- “transport”
- “turbulence”

## RIC: Step 2, solve coupled system via relaxed functional iteration -- fully implicit and Jacobian-free

** ****t**** and ****∆t**** = ****turbulence**** and ****transport**** code timesteps**
**For each iteration j:**

- ~

- In the turbulence code:
- set input profile **u** = **u**n+1,j
- take turb. code timestep
**u**j+1  ****j+1  ****j+1

- j

- tturb

- n

- ttransp

- ∆t

- j

- **u**

****

## Slide 87

## RIC summary

- RIC is method which allows running a turbulence code on transport timescale and thus obtain transport profiles self-consistent with turbulent fluxes
- It can be interpreted as an integration of a Delta f and a f solver, which follows both f (transport) and f (turbulence)
- Fully implicit transport timestepping -- no stability limit on transport timestep
- one transport timestep (including t=∞) costs ~ saturated turbulence code run with fixed profiles
- implies time savings ~ (turbulence timescale/transport timescale)
- A coupling that works for local and (with extensions) non-local transport
- demonstrated solutions for cases where flux locally runs up-hill

## Equation Free Projective Integration

- Projective Integration
- A method for using a combination of a few small time step integrations to cover large time steps.
- Restriction and Lifting
- Mappings between a high dimensional representation (microscopic or fine) and a low dimensional representation (macroscopic or coarse), for example:
- Microscopic - a collection of particles in Monte Carlo simulations to a low-dimensional description
- Macroscopic - finite element approximation to a *distribution* of the particles
- Projection done on macroscopic representation
- “Experiment” (kinetic code) evaluated on microscopic representation.

- Kevrekidis et al., 2002

## Projective Integration - a sequence of outer integration steps:

- Need to study the accuracy and stability of these methods

**Equation Free Projective Integration**

## Equation Free Projective Integration cycle

## Projecting forward in time

## Test: ion acoustic mode

**Test: ion acoustic mode**

**Wave propagation speed matches exactly.**
**P****i**** diverges first. **
**So far, x12 speed-up**

- efree full particle

- ni`

- Vix`

- Ex

- X

- X

- X

- Pi

- X

- Parameters
- nxcrs = 32nxmicro = 512lx = 1.2Nextrap   = 20Nproj     = 200

## Coupling of explicit/implicit solvers

- Motivation: high/low density region better handled by implicit/explicit solver.
- J.-C. Adam and A. Héron have proposed to extend the new AMR techniques developed by J.-L. Vay to the coupling of explicit and implicit solvers.
- The left system would be terminated on the right by absorbing boundary conditions that will suppress the outgoing wave of the explicit part, and vice-versa for the right system.
- Particles move freely through the boundary and give the correct source terms in both regions.
- Because mesh size can be different on both sides of the boundary M, mesh refinement is de facto built into the method.

## A lot of effort has been/is devoted to develop techniques to address multiscale issues in plasma modeling.

AMR can be of great help for PIC/Vlasov multiscale plasma simulations but scheme must be derived with care (spurious self-force, conservation of charge, reflection of waves, non-cancellations due to numerical errors (dispersion), ...)

in electrostatic, ‘problem solved’ to some extend but cutoff of plasma modes at interface remains to be studied,

in electromagnetic, existing schemes can be successfully applied to some problems but more research is needed to get better scheme(s),

with irregular geometries, AMR on regular cartesian grids may not be enough: sometimes need to apply irregularly gridded patch which maps to conductors, field line, ...,

We have developed a new solver that allows to jump over the cyclotron period.

This is a very active field with several promising emerging methods.

**AMR can be of great help for PIC/Vlasov multiscale plasma simulations but scheme must be derived with care (spurious self-force, conservation of charge, reflection of waves, non-cancellations due to numerical errors (dispersion), ...)**
**in electrostatic, ‘problem solved’ to some extend but cutoff of plasma modes at interface remains to be studied,**
**in electromagnetic, existing schemes can be successfully applied to some problems but more research is needed to get better scheme(s),**
**with irregular geometries, AMR on regular cartesian grids may not be enough: sometimes need to apply irregularly gridded patch which maps to conductors, field line, ...,**
**We have developed a new solver that allows to jump over the cyclotron period.**
**This is a very active field with several promising emerging methods.**

**Conclusion**

## References

- _	__PIC-AMR_
- J.L. Vay, “An extended FDTD scheme for the wave equation: application to multiscale electromagnetic simulation.”, *Journal of Computational Physics* **167, **72 (2001)
- J.-L. Vay, J.-C. Adam, A. Héron, “Asymmetric PML for the absorption of waves. Application to mesh refinement in electromagnetic particle-in-cell plasma simulations”, *Computer Physics Communication* **164***, *171 (2004)
- J.-L. Vay , P. Colella, J. W. Kwan, P. McCorquodale, D. B. Serafini, A. Friedman, D. P. Grote, G. Westenskow, J.-C. Adam, A. Héron, I. Haber, “Application of adaptive mesh refinement to particle-in-cell simulations of plasmas and beams”, *Physics of Plasmas ***11**, 2928 (2004)
- P. McCorquodale, P. Colella, D. P. Grote, J.-L. Vay, “A node-centered local refinement algorithm for Poisson's equation in complex geometries.”, *Journal of Computational Physics ***201**, 34 (2004)
- J.-L. Vay , P. Colella, J. W. Kwan, P. McCorquodale, D. B. Serafini, A. Friedman, D. P. Grote, G. Westenskow, J.-C. Adam, A. Héron, I. Haber, “Application of Adaptive Mesh Refinement to PIC Simulations in Inertial Fusion”, *Nuclear Inst. and Methods in Physics Research A* **544**, 347 (2005)
- _	__Vlasov with moving grid/AMR_
- E. Sonnendrucker, F. Filbet, A. Friedman, E. Oudet, J.-L. Vay,** “**Vlasov simulations of beams with a moving grid” ”, *Computer Physics Communications* **164**, 390
- M. Gutnic, M. Haefele, I. Paun , E Sonnendrucker, “Vlasov simulations on an adaptive phase-space grid”, *Computer Physics Communications* **164**, 214
- _	__Drift-kinetic / Newton “blend” mover_
- Cohen, R. H.; Friedman, A.; Kireeff Covo, M.; Lund, S. M.; Molvik, A.W.; Bieniosek, F. M.; Seidl, P. A.; Vay, J-L.; Stoltz, P.; Veitzer, S., “Simulating Electron Clouds in Heavy-Ion Accelerators”, *Phys. Plasmas* **12**, 56718 (2005).  A *J. Comp. Phys*. manuscript is in preparation.

## References

- _	__Implicit Multiscale PIC_
- A. Friedman, S. E. Parker, S. L. Ray, and C. K. Birdsall, “Multi-Scale Particle-in-Cell Plasma Simulation,” *J. Comp. Phys*. **96**, 54 (1991).
- S. E. Parker, A. Friedman, S. L. Ray, and C. K.  Birdsall, “Bounded Multi-Scale Plasma Simulation: Application to Sheath Problems,” *J. Comp. Phys*. **107**, 388 (1993).
- _	__Discrete Event Simulations_
- H. Karimabadi, J. Driscoll, Y. A. Omelchenko, and N. Omidi, *J. Comp. Phys*., to be published
- _	__Relaxed Iterative Method for Coupling Disparate Scales_
- A. I. Shestakov, R. H. Cohen, J. A. Crotinger, L. L. LoDestro, A. Tarditi, and X. Q. Xu, “Self-Consistent Modeling of Turbulence and Transport”, JCP March 2003
- _	_
- _	__Equation Free Projective Integration_
- http://www.cscamm.umd.edu/cmpd/projective_integration.htm