Lacuna-based Artificial Boundary Condition And Uncertainty Quantification of the Two-FluidPlasma Model

Eder Sousa1, Uri Shumlak1 and Guang Lin2

1 Computational Plasma Dynamics Lab, University of Washington, Seattle, WA2 Computational Sciences and Mathematics, Pacific Northwest National Laboratory, Richland, WA

Abstract

Modeling open boundaries is useful for truncating extended or infinite simulation domains to regions of greatest interest. However, artificial wave reflections at the boundaries can result for oblique wave intersections.

The lacuna-based artificial boundary condition (ABC) method is applied to numerical simulations of the two-fluid plasma model on unbounded domains to avoid unphysical reflections. The method is temporally

nonlocal and can handle arbitrary boundary shapes with no fitting needed nor accuracy loss. The algorithm is based on the presence of lacunae (aft fronts of the waves) in wave-type solutions in odd- dimensional

space. The method is applied to Maxwell’s equations of the two-fluid model. Placing error bounds on numerical simulations results is important for accurate comparisons, therefore, the multi-level Monte Carlo method

is used to quantify the uncertainty of the two-fluid plasma model as applied to the GEM magnetic reconnection problem to study the sensitivity of the problem to uncertainty on the mass ratio, speed of light to Alfven

speed ratio and the magnitude of the magnetic field initial perturbation.

Lacuna-based Artificial Boundary Conditions 3

INumerical simulation of wave phenomena on unbounded domains often produce unphysical reflections from the boundariesIConsequently, The original infinite domain has to be truncated and special artificial boundary conditions (ABCs) have to be developed

Lacunae are still regions present in wave-type solutions in odd-dimension

spaces.

Introduction

IThe key idea of using lacunae for computations isvery simple:I If the sources of waves are compactly supported in

space and time;I If the domain of interest has finite size;IThen it will completely fall inside the lacuna once a

certain time interval has elapsed since the inceptionof the sources.

IThe lacunae-based ABC is nonlocal in space andtime without loss of accuracy

IThe lacunae-based ABCs are not restrictedgeometrically to any particular shape of the externalartificial boundary

The computational domain and the auxiliary domain overlap by a couple gridcells where the transition multiplier, µ, varies from zero on the interior side to oneon the exterior one. The three steps of ABC implementation:ICalculate the exterior source, Ω(v) , from the interior solution;IReintegration of the exterior solution excluding earlier exterior sourcesICommunicate the exterior evolution with the interior problem ghost cells

3S.V.Tsynkov, ”On the Application of Lacunae-based Methods to Maxwell’s Equations”, JCP 199

(2004) 126-149

Auxiliary source generation

IThe computational domain is advanced using:∂q∂t

+∇ · F(q) = S(q)

IAuxiliary problem: ∂v∂t

+∇ · F(v) = S(v) + Ω(q)

IΩ(q) is the auxiliary source and v = µqIFor non-moving boundaries there is no time

dependent of µ(x), therefore:Ω(q) = µS(q)− S(µq) +∇ · F(µq)− µ∇ · F(q)

The boundary formulation is applied to the Maxwell equations using theWashington Approximate Riemann Plasma Solver (WARPX). The field aremodeled using the Perfectly Hyperbolic Maxwell4 formulation to account for thedivergence corrections,

∂~B∂t

+∇× ~E + γ∇Ψ = 0

1c2∂~E∂t−∇× ~B + χ∇Φ = −µo

∑s

qs

msρs~us

1χ

∂Φ

∂t+∇ · ~E =

∑s

qs

msρs

1γc2

∂Ψ

∂t+∇ · ~B = 0

Quantity being plotted is Bz.

– Interior domain boundary, – Auxiliary domain boundary

Problem setup: a quarter of spherical pulse propagating outwards, where the left

and the bottom boundary conditions are lacunae-based ABC’s. The wave front is

communicated to the auxiliary problems by the auxiliary sources in the transition

region (overlap region between the interior and the auxiliary regions).

Exterior domain re-integration

IThe auxiliary sources drive the problem in theauxiliary domain and guarantees both solutionsmatch in the exterior domain

IThe auxiliary problem is re-integrated every specifiedtime interval and early sources are removed fromcomputation

The following plot are slices of the previous figures at x=1.

Initially the interior problem pulse propagates

through the interior domain.

As the re-integration is preformed the earlier

sources are removed from the auxiliary

domain as they no longer affect the interior

solution.

As the interior pulse enters the transition

region, the auxiliary source is applied to the

auxiliary domain.

The pulse is reintegrated out of the exterior

domain an no reflection are present in the

interior problem.

Conclusion

The lacuna-based ABC’s can correctly simulated unbounded

domains accurately while removing artificial reflections that

otherwise would be present.

4Munz et. al., ”Divergence Correction Techniques for Maxwell Solvers Based on Hyperbolic

Model”, JCP 161, 484-511 (2000)

Uncertainty Quantification

Motivation

IDetermining the region of acceptable results forexperimental and computational is not only desiredbut required

IUncertainty quantification will allow for errorbars to beput into computational results

IThere are numerous sources of uncertainty in theTwo-Fluid plasma model

ITreating all the inputs as stochastic is computationallyexpensive

Introduction

IThe mean square error (MSE)e(PM)2 = V [PMC

ml,N] + (E [Pml − P])2

ITo achieve root MSE less than εI the variance V [PMC

ml,N] = N−1V [Pml] has to be less

than ε2/2 meaning N ≥ ε−2 for the first term (largenumber of samples)

IHigh discretization level: ml ≥ ε−1/α where α is thediscretization convergence rate for the second term

Multi-level Monte Carlo (MMC)5,6

IThe method is based on the multi-grid method as thesolution is obtained from different solutions at differentgrid refinement levels

IThe estimator comes from the same random sample,N, but at different refinement levels, L

PMLml

=L∑

l=0

1Nl

Nl∑i=1

(P iml− P i

ml−1)

IThe multilevel variance V [Yl] = V [Pml − Pml−1]→ 0 asl →∞ which implies that Nl → 1 as l →∞

I It is less costly to achieve an overall RMSE of ε for themultilevel than the standard Monte Carlo

Probabilistic Collocation Method (PCM)

IThe PCM approach based on selecting the samplingpoints and corresponding weights.

ICollocation points in probability space of randomparameters as independent random inputs based ona quadrature formula

IThe solution statistics is estimated using thecorresponding quadrature rule

5M.B.Giles, ”Multilevel Monte Carlo Path Simulation”, Operations Research, 56, 981-986, 2008

6K.A.Cliffe, et. al. ”Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random

Coefficients.” Submitted, to appear in Numerische Mathematik, 2011

Results

The methods are applied to the two-fluid magnetic reconnection problem for

the cases where the electron-to-ion mass ratio, the speed of light and the

initial B-field perturbation are considered stochastic.

LogM of the variance and mean for Pl and

Pl − Pl−1 (velocity at x=0) for the Euler

equations with dispersive source term. The

variance and the mean converge as the

refinement level is increased.

Mean (top) and variance (bottom) of the

reconnected flux for varying speed of light to

Alfven speed ratio ranging from 10 to 20.

Reconnected flux for different plasma models

used in the GEM challenge7 compared to

two-fluid model (red). The error bars are the

standard deviation caused by varying the

c/vA ratio, calculated using the MMC method.

Mean and variance of the reconnected flux

for varying electron-to-ion mass ratio ranging

from 25 to 100.

Mean and variance of the reconnected flux

for varying amplitude of the B-field

perturbation ranging from 8.5% to 11.5% of

the background field.

Computational Cost

The following are actual

computational cost for the case of

the uncertainty in the initial B-field

perturbation.

CPU-hoursMC 6496PCM 4660MMC 4075

ConclusionThe MMC method produced the same accuracy as the standard MC and the

PCM. A 37.3% cost saving was calculated for the MMC method over the MC

and a 14.4% saving over the PCM. MMC allows for an easy and inexpensive

way to determine the error of computational plasma codes.

7 ”Geospace Environmental Modeling (GEM) Magnetic Reconnection Challenge,” Journal of

Geophysical Research, vol. 106, pp. 3715-3719, March 2001.

Computational Plasma Dynamics Lab - Aeronautics and Astronautics Department - University of Washington - Seattle, WA sousae@uw.edu http://www.aa.washington.edu/research/cfdlab