This work was funded by European Fusion Development Agreement

(EFDA) under WP13-SOL-01-01/CCFE/PS.

Greater than 10x Acceleration of fusion plasma edge simulations using the Parareal algorithm

Debasmita Samaddar1, David P. Coster2, Xavier Bonnin3,5, Christoph Bergmeister1, Eva Havlickova1, Wael R. Elwasif4, Lee A. Berry4, Donald B. Batchelor4 1 –CCFE, Culham Science Centre, UK , 2 - Max-Planck-Institut für Plasmaphysik, Germany, 3 – CNRS-LSPM, Université Paris 13, France, 4 – ORNL, USA, 5 – Now at ITER Organization, Route de Vinon sur Verdon, 13115 Saint Paul Lez Durance, France

Introduction :

Fusion power / Input power = Q ≥

10

Event based parareal implementation using the IPS framework

[4, 5] greatly improves resource utilization as well as gain.

•Relatively environmentally friendly : Waste with

radioactive toxicity < 100 years

•Virtually unlimited – Deuterium abundant in sea water. Tritium produced from lithium - also abundant in nature.

Temperature - Ti:

1-2108 K (10-20 keV)

~10 temperature of sun’s core

Density - ni:11020 m-3

~10-6 of atmospheric density

3.5 MeV

14 MeV

•Possible solution to our search for an alternate energy

source

Energy confinement time - E : 3-4s

Plasma pulse duration~1000s

Fusion :

ITER :

SOLPS :

Geometry:

- toroidal symmetry

- equations written in curvlinear

coordinates coinciding with

magnetic geometry

Physical plane Computational

plane

Package consists of 2 codes:

B2(plasma fluid transport) &

Eirene(neutral particle

transport).

Parallel & perpendicular

transport described in 2D

system.

Equations :

Results – Coarse model 1: Monte-Carlo model replaced by fluid neutrals model in G:

Results – Coarse model 2: Reduced grid size and bigger dt in G:

Current continuity

Energy conservation electrons (and ions)

Parallel momentum ions

Continuity

Parareal Algorithm :

Metric for

convergence:

i=0 to (N-1)

Check Convergence

k = k + 1

i = 0 to (N-1)

k = 0 to (N-1), F : a propagator evolving

the function (energy(t))

from initial time, t0, to a

later time .

G - faster but inaccurate

propagator

Electron density :

Coarse (G) solution Fine (F) solution

Parareal solution

Size of time slice solved per prpcessors =

10

Parareal convergence was sensitive to size of time slice (or number of

timesteps solved per processor).

Computational gain = 12.58 with 240

processors.

Coarse model 2 – Reduced grid model in G:

Fine grid: 150 X 36, Coarse: 150 X 18, dt_G =

50dt_F. Gain=12.53 for 32 processors

Results – Event based parareal significantly improves performance:

Understanding the physics of the edge plasma is crucial

for the design and operation of magnetic fusion devices

like ITER, JET, MAST, D|||-D. But, these simulations are

computationally intensive. This work explores the

parareal approach to temporally parallelize the SOLPS

code and demonstrates a computational gain > 10 which

is noteworthy for systems as complex as these.

References:

1) Lions J. L et al, C.R. Acad. Sci. Paris, Series I, 332 (2001), pp.661668 2) Samaddar D. et al, J. Comp Phys,18 (229) (2010)

3) Schneider, R. et al, Contrib. Plasma Phys. 46, No. 1-2,3-191 (2006)

4) Berry, L. A. et. al, Journal of Computational Physics 231(2012) 59455954

5) Elwasif W.R et al, 4th IEEE Workshop on Many-Task Computing on Grids and

Supercomputers, MTAGS 2011, (2011)

In this case, ‘coarse’ solution diverges away from ‘fine’ solution.

Coarse solution

G is faster than F by > 50 times. After

parareal convergence is achieved, the

parareal solution matches the serial

solution (or F solution).

Parareal fails for slice > 10

Parareal solution

Coarse model 1 – Monte-Carlo model replaced by

fluid neutrals model in G:

In F, neutral transport is modelled using Monte-Carlo treatment solving the Boltzman equation (Eirene

package). Replacing it by a fluid neutrals model for G

speeds up computations.

a) Fine grid 150X36 for MAST

plasma.

Coarse grids: 150X18, 76X36,

76X18 CFL condition allows bigger dt with reduced grid sizes.

b) Fine grid 96X36 for DIIID

plasma.

Coarse grids: 48X36, 32X36

Coarsening in space : 2 cases were explored for this purpose:

Fine grid: 150 X 36, Coarse: 150 X 18, dt_G =

64dt_F. Gain=32 for 64 processors

Fine grid: 96 X 36, Coarse: 48 X 36, dt_G =

30dt_F. Gain=21.8 for 96 processors

Size of time slice solved per processor

affects the gain (as also seen in

Coarse model 1).

For each processor:

NTIMF = Number of timesteps in Fine

NTIMG = Number of timesteps in

Coarse.

(DIIID case)

Performance may be improved by increasing NTIMF for the same NTIMG

MAST case: for NTIMG = 4 and 16 processors, the gain varies with

NTIMF :

Weak scaling: Gain may be maximized

by optimising NTIMF / NTIMG

Strong scaling: Gain will reduce for high

processor number as NTIMF & NTIMG

reduce significantly

Gain = T(serial) / T(parareal)

Actual 1st G 1st F 2nd G 2nd F 3rd G 3rd F

t0 t1 t2 t3 t4 t5 t6

ener

gy

(t)

New G Old G

Time

} } } } } } P0 P4 P3 P2 P1 P5