Adaptive time-space algorithms for plasma discharge simulation

Context and MotivationMulti-scale Simulation Algorithms

Numerical IllustrationPerspectives

Adaptive time-space algorithms for plasmadischarge simulation

Z. Bonaventura1,2 A. Bourdon2

M. Duarte2 M. Massot2

S. Candel S. Descombes T. DumontF. Laurent V. Louvet C. Tenaud

1Masaryk University - Czech Republic

2EM2C - Ecole Centrale Paris - France

ECOMOD - November 16th 2010

Bonaventura, Bourdon, Duarte, Massot Adaptive Time-Space Numerical Methods



Outline I1 Context and Motivation

Unsteady reactive phenomenaTime integration numerical strategiesOperator splitting and stiffness

2 Algorithms for multi-scale phenomena simulationSuitable Stiff IntegratorsSpace Adaptive Numerical MethodAdaptive Time-Space Numerical Method

3 Numerical IllustrationBZ ModelCombustion FlamesStroke ModelPlasma Applications

4 PerspectivesBonaventura, Bourdon, Duarte, Massot Adaptive Time-Space Numerical Methods



Unsteady reactive phenomenaNumerical StrategiesSplitting

Outline1 Context and Motivation




4 Perspectives





Application Background

Numerical simulation of unsteady reactive phenomena

Flames (Instabilities, dynamics, pollutants)

Chemical “waves” (spiral waves, scroll waves)Biochemical Engineering (migraines, Rolando’s region,strokes)

Dynamics involving many “species” and “reactions”

“Complex Chemistry”

Convection-diffusion coupled to chemistry

∂tU +∑

∂i(Φi(U, ∂xU)

)= Ω(U)







Flames (Instabilities, dynamics, pollutants)Chemical “waves” (spiral waves, scroll waves)

Biochemical Engineering (migraines, Rolando’s region,strokes)




∂tU +∑

∂i(Φi(U, ∂xU)

)= Ω(U)







Flames (Instabilities, dynamics, pollutants)Chemical “waves” (spiral waves, scroll waves)Biochemical Engineering (migraines, Rolando’s region,strokes)




∂tU +∑

∂i(Φi(U, ∂xU)

)= Ω(U)







Flames (Instabilities, dynamics, pollutants)Chemical “waves” (spiral waves, scroll waves)Biochemical Engineering (migraines, Rolando’s region,strokes)




∂tU +∑

∂i(Φi(U, ∂xU)

)= Ω(U)





Time integration numerical strategies

Resolving the large scale spectrum coupled

Explicit methods in time (high order in space)Fully implicit methods with adaptive time steppingSemi-implicit methods (IMEX, source/diffusion explicit intime)Method of lines coupled to a stiff ODE solver

The computational cost and memory requirements havesuggested the study of alternative methods : decoupling

Reduction of chemistry (large literature)Operator Splitting techniques





Time integration numerical strategies

Resolving the large scale spectrum coupled

Explicit methods in time (high order in space)Fully implicit methods with adaptive time steppingSemi-implicit methods (IMEX, source/diffusion explicit intime)Method of lines coupled to a stiff ODE solver

The computational cost and memory requirements havesuggested the study of alternative methods : decoupling

Reduction of chemistry (large literature)Operator Splitting techniques





Operator splitting techniques

Operator splitting : separate convection-diffusion and chemistry

High order methods existAllow the use of dedicated solver for each stepYield lower storage and optimization capability





Basis of operator splitting

Reaction-diffusion system to be solved (t : time interval)

U(t) = T tU0

∂tU −∆U = Ω(U)

U(0) = U0

Two elementary “blocks”.

V (t) = X tV0

∂tV −∆V = 0

V (0) = V0

W (t) = Y tW0

∂tW = Ω(W )

W (0) = W0






First order methods :

Lie Formulae.

Lt1 U0 = X t Y t U0 Lt

1U0 − T tU0 = O(t2),

Lt2 U0 = Y t X t U0 Lt

2U0 − T tU0 = O(t2),






Second order methods :

Strang Formulae.

St1 U0 = Y t/2 X t Y t/2 U0 St

1U0 − T tU0 = O(t3),

St2 U0 = X t/2 Y t X t/2 U0 St

2U0 − T tU0 = O(t3),





Stiffness comes into play

Various origins of stiffnessLarge spectrum of temp. scales in chemical source(Descombes & Massot Numerische Mathematik 04)

Large spatial gradients of the solutions(Descombes et al. International Journal of ComputerMathematics 07)

ConsequencesOrder reduction due to fast scales

Lie RD order 0 fast variable onlyStrang DRD order 0 fast variable only

To summarize : Important to use Strang RDR




Suitable Stiff IntegratorsAdaptive MultiresolutionAdaptive Time-Space Strategy





4 Perspectives





Stability + Accuracy

We are particularly looking for:

A-stable methodsHigh order methodsL-stable methods

Type of Methods :

Explicit one-step methodsExplicit/Implicit Adams Multistep MethodsBDF Multistep Methods

=⇒ Needs of very stable high order one-step methods








Type of Methods :










Type of Methods :







Implicit Runge Kutta Methods

Based on Ehle’s Methods of type II: RadauIIA

Order: p = 2s − 1 (s: stage number)A-stable

L-stable

RADAU5(Hairer & Wanner Springer-Verlag 91)

Based on RadauIIA with s = 3 and p = 5

Simplified Newton Method =⇒ Linear Algebra tools

Adaptive time integration step





Explicit Runge-Kutta methods

We want to solve the discrete heat equation

u = Au,

with an explicit s-stage Runge-Kutta method.

Because of the properties of the matrix A, we need to find astable Runge-Kutta method for the simple problem

u = λu,

with λ real, negative and as big as possible...





ROCK4

(Abdulle SIAM J. Sci. Comput. 02)

Extended Stability Domain (along R−)

Order 4

Adaptive time integration step

Explicit Methods =⇒ NO Linear Algebra problems

Low Memory Demand





Explicit/Implicit Operator Splitting

Numerical Strategy:

∂tU − ∆U︸︷︷︸ROCK4

= Ω(U)︸︷︷︸Radau5

Reduction in Computational Time

Reduction in Memory Requirement

Same previous accuracy established by Splitting Scheme





Adaptive Multiresolution

(Cohen et al. Mathematics of Computation 01)Principles of the MR:

Represent a set of function data as values on a coarsergrid plus a series of differences at different levels of nestedgrids.The information at consecutive scale levels are related byinter-level transformations: projection and predictionoperators.The wavelet coefficients are defined as prediction errors,and they retain the detail information when going from acoarse to a finer grid.

Main idea: use the decay of the wavelet coefficients to obtaininformation on local regularity of the solution.





Inter-level Transformations

Projection Operator P jj−1: Maps Uj to Uj−1.

We obtain the exact averages at the coarser level by

uγ = |Ωγ |−1∑

|µ|=|γ|+1,Ωµ⊂Ωγ

|Ωµ|uµ

where |γ| := j if γ ∈ Sj .





Inter-level Transformations

Prediction Operator P j−1j : Maps Uj−1 to an approximation Uj

of Uj .

There is an infinite number of choice for definition of P j−1j , but

we impose at least two basic constraints:The prediction is local, i.e., uµ depends on the values uγon a finite stencil Rµ surrounding Ωµ.The prediction is consistent with the projection in the sensethat

|Ωγ |uγ =∑

|µ|=|γ|+1,Ωµ⊂Ωγ

|Ωµ|uµ

i.e., P jj−1P j−1

j = Id .





Multiresolution Transformation

We define the prediction error at level j as the differencesbetween the exact and predicted values, i.e.,

dµ := uµ − uµ.

From the consistency assumption:

∑|µ|=|γ|+1,Ωµ⊂Ωγ

|Ωµ|dµ = 0.

Then, we define the detail vector as Dj = (dµ)µ∈∇j , where theset ∇j ⊂ Sj is obtained by removing for each γ ∈ Sj−1 oneµ ∈ Sj such that Ωµ ⊂ Ωγ .





Multiresolution Transformation

Finally, there is a one-to-one correspondence

Uj ←→ (Uj−1,Dj),

which can be implemented using the operators P jj−1 and P j−1

j .

By iteration of this decomposition, we obtain a multiscalerepresentation of UJ in terms of MJ = (U0,D1,D2, · · · ,DJ).Using the local structure of the projection and predictionoperators, we can implement the multiscale transformation

M : UJ 7−→ MJ .





Compression

Given a set Λ ⊂ ∇J of indices λ, we define a truncationoperator TΛ, that leaves unchanged the component dλ if λ ∈ Λand replaces it by 0, otherwise.

In practice, we are typically interested in sets Λ obtained bythresholding: given a set of level-dependent threshold(ε0, ε1, · · · , εJ), we set

Λ = Λ(ε0, ε1, · · · , εJ) := λs.t .|dλ| ≥ ε|λ|.

Applying TΛ on the multiscale decomposition of UJ amounts to

building an approximation AΛUJ , where the operator AΛ isgiven by

AΛ :=M−1TΛM.





Adaptive Time-Space Numerical Strategy

Refinement Precautionary measure to account for possibletranslation or presence of finer scales in thesolution.

Time Evolution Explicit/Implicit Strang RDR Operator SplittingMethod.

Thresholding Wavelet Thresholding Operation:

εj = 2d2 (j−J)ε, j ∈ [0, J],

in order to ensure a thresholding error ofprescribed order ε (d = spatial dimension).

If S∆t denotes the Strang splitting time integration operator

Un+1 = S∆t (M−1RTΛnεMUn)





Algorithm

(unλ)λ∈L(Λn

ε)M−→ (un

λ0, dnλ)λ∈Λn

ε

(unλ0, dnλ)λ∈Λn

ε

TΛnε−→ (un

λ0, dnλ)λ∈Λn

ε

(unλ0, dnλ)λ∈Λn

ε

R−→ (unλ0, dnλ)λ∈Λn+1

ε

(unλ0, dnλ)λ∈Λn+1

ε

M−1−→ (un

λ)λ∈L(Λn+1ε )

(unλ)λ∈L(Λn+1

ε )S∆t−→ (un+1

λ )λ∈L(Λn+1ε )




BZ ModelCombustion FlamesStroke ModelPlasma Applications





4 Perspectives





BZ Model

Belousov-Zhabotinsky system of equations

∂a∂τ− Da∆a =

1µ

(−qa− ab + fc),

∂b∂τ− Db∆b =

1ε

(qa− ab + b(1− b)) ,

∂c∂τ− Dc∆c = b− c,

ε = 10−2 µ = 10−5 f = 1,6 q = 2.10−3

Da = 2,5.10−3 Db = 2,5.10−3 Dc = 1,5.10−3





3D Configuration

(Duarte et al. Submitted 10)

Time Domain : T = [0,2]

Spatial Domain : Ω = [0,1]× [0,1]× [0,1]

Integration Time Step : ∆t = 2/256Tolerances of Time Integrators: tol = 1.10−5

Finest Grid Level : J = 9Number of cells at Grid J:2d×J = 134217728 = 512× 512× 512MR Tolerance : ε = 1.10−1





3D Configuration





Adaptive Grid

Compression −→ 4.46%





Adaptive Grid






Adaptive Grid






Adaptive Grid






Adaptive Grid






Memory requirements

Radau5 −→ L1 = 4×W1 ×W1 + 12×W1 + 20

Rock4 −→ L2 = 8×W2

W = 3× 512× 512× 512 ≈ 4.03× 108 −→ 24 Gb

W1 W2 L = L1 + L2

Quasi-exact W 0 6.5× 1017 −→ 36 Eb

Splitting 3 W 3.2× 109 −→ 191 Gb

MR/Splitting 1.10−1 3 0.13W 4.2× 108 −→ 25 Gb





Memory requirements

Radau5 −→ L1 = 4×W1 ×W1 + 12×W1 + 20

Rock4 −→ L2 = 8×W2

W = 3× 512× 512× 512 ≈ 4.03× 108 −→ 24 Gb

W1 W2 L = L1 + L2

Quasi-exact W 0 6.5× 1017 −→ 36 Eb

Splitting 3 W 3.2× 109 −→ 191 Gb

MR/Splitting 1.10−1 3 0.13W 4.2× 108 −→ 25 Gb





Memory requirements

Radau5 −→ L1 = 4×W1 ×W1 + 12×W1 + 20

Rock4 −→ L2 = 8×W2

W = 3× 512× 512× 512 ≈ 4.03× 108 −→ 24 Gb

W1 W2 L = L1 + L2

Quasi-exact W 0 6.5× 1017 −→ 36 Eb

Splitting 3 W 3.2× 109 −→ 191 Gb

MR/Splitting 1.10−1 3 0.13W 4.2× 108 −→ 25 Gb





Single Step Chemistry

∂Yi

∂t+ v · ∇Yi = D∆Yi +

νiWi

ρw

∂T∂t

+ v · ∇T = D∆T +νF WF Qρcp

w

with νiWi = −νF WF , −νOWO, νPWP , 0 for i = F , O, P, N

v(x, t) =⇒ 2D vortex configuration

vθ(r , t) =Γ

2πr

(1− exp

(− r2

4νt

))Bonaventura, Bourdon, Duarte, Massot Adaptive Time-Space Numerical Methods




Premixed Flame with Vortex

(Laverdant & Candel J. Propulsion 87)Introducing progress variable c(x , y , t)

c =T − To

Tb − To=

YFo − YF

YFo

With dimensionless variables:

x? =xL, y? =

yL, u? =

uU, v? =

vU, t? =

tτd

∂c∂t?

+u?∂c∂x?

+v?∂c∂y?

=∂2c∂x2

?

+∂2c∂y2

?

+Da(1−c) exp(− Ta

To(1 + τc)

)





Premixed Flame with Two Vortices (Re = 1000)





Premixed Flame with Two Vortices (Re = 1000)





Data Compression





Splitting Time Step





Splitting Time Step





Ignition of Diffusion Flame with Vortex

Defining Z (x , y , t) and θ(x , y , t) (Thévenin & Candel Phys. Fluids 95)

θ = (T − TO0)/(TF0 − TO0), Z =χYF/YF0 + τθ

χ+ τ

With dimensionless variables:∂Z∂t?

+ u?∂Z∂x?

+ v?∂Z∂y?

=∂2Z∂x2

?

+∂2Z∂y2

?

∂θ

∂t?+ u?

∂θ

∂x?+ v?

∂θ

∂y?=

∂2θ

∂x2?

+∂2θ

∂y2?

+ F (Z , θ)

F (Z , θ) = DaφχYO0

[1− Zφτ

+1χ

(Z − θ)

] [Z +

τ

χ(Z − θ)

]exp

(− τa

1 + τθ

)Bonaventura, Bourdon, Duarte, Massot Adaptive Time-Space Numerical Methods




Ignition of Diffusion Flame with Vortex(TO0 = 1000 - Re = 1000)





Data Compression





Splitting Time Step





Splitting Time Step





0-d model

System of 19 ODEs.Unknowns :

ions: (Na+, K +, Ca2+, Cl−, glu−) inside the neurons,glial cells and extracell space,fn, fa,Vn et Va, potentials inside the neurons and glial cells.





In 3D, system of reaction-diffusion

diffusion of ions in the astrocytes and in the extracellspace,no diffusion of ions in neurons,no diffusion for fn, fa, Vn et Va.

System of reaction-diffusion:

dui

dt− εi ∆ ui = Fi(u1, ...,u19).

Homogeneous Neumann boundary conditions.





Electric potential inside the glial cells





3D Configuration

(Dumont et al. Submitted 10)

Time Domain : T = [0,3600]

Spatial Domain : Ω = [0,50000]× [0,50000]× [0,50000]

Integration Time Step : ∆t = 3600/360 = 10Tolerances of Time Integrators: tol = 1.10−5

Finest Grid Level : J = 8Number of cells at Grid J:2d×J = 16777215 = 256× 256× 256MR Tolerance : ε = 1.10−2





Adaptive Grid





Adaptive Grid





Adaptive Grid





Adaptive Grid





Data Compression





Computing Time





Simplified RD Model

∂ne

∂t−∇ · neve −∇ · (De∇ne) = S+

e − S−e + Sph,

∂np

∂t+∇ · npvp −∇ · (Dp∇np) = S+

p − S−p + Sph,

∂nn

∂t−∇ · nnvn −∇ · (Dn∇nn) = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,

No advectionConstant diffusion coefficientsNo photoionizationConstant localized electric field





Simplified RD Model

∂ne

∂t−∇ · (De∇ne) = S+

e − S−e + Sph,

∂np

∂t−∇ · (Dp∇np) = S+

p − S−p + Sph,

∂nn

∂t−∇ · (Dn∇nn) = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,






Simplified RD Model

∂ne

∂t− D∆ne = S+

e − S−e + Sph,

∂np

∂t− D∆np = S+

p − S−p + Sph,

∂nn

∂t− D∆nn = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,






Simplified RD Model

∂ne

∂t− D∆ne = S+

e − S−e ,

∂np

∂t− D∆np = S+

p − S−p ,

∂nn

∂t− D∆nn = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,






Simplified RD Model

∂ne

∂t− D∆ne = S+

e − S−e ,

∂np

∂t− D∆np = S+

p − S−p ,

∂nn

∂t− D∆nn = S+

n − S−n ,






High Frequency Repetitive Discharges

Pulse period: 1µs Pulse length: 10ns





Adaptive splitting time step strategy

Adaptive splitting time-step





Adaptive splitting time step strategy

Splitting Error Control





Streamer Discharge

Global system

∂ne

∂t−∇ · neve −∇ · (De∇ne) = S+

e − S−e + Sph,

∂np

∂t+∇ · npvp −∇ · (Dp∇np) = S+

p − S−p + Sph,

∂nn

∂t−∇ · nnvn −∇ · (Dn∇nn) = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,





Streamer Discharge

Considered system

∂ne

∂t−∇ · neve −∇ · (De∇ne) = S+

e − S−e ,

∂np

∂t+∇ · npvp −∇ · (Dp∇np) = S+

p − S−p ,

∂nn

∂t−∇ · nnvn −∇ · (Dn∇nn) = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,





Streamer Discharge

EF/Transport+Source splitting

∂ne

∂t−∇ · neve −∇ · (De∇ne) = S+

e − S−e ,

∂np

∂t+∇ · npvp −∇ · (Dp∇np) = S+

p − S−p ,

∂nn

∂t−∇ · nnvn −∇ · (Dn∇nn) = S+

n − S−n ,

ε0∇2V = −qe(np − nn − ne), E = −∇V ,





Transport Computation

Operator splitting

∂ne

∂t−∇ · neve︸︷︷︸

OSMP31

−∇ · (De∇ne)︸︷︷︸ROCK 4

= S+e − S−e︸︷︷︸Radau5

Time integration method =⇒ Strang scheme order 2

→ ne(∆T ) = R∆T/2 D∆T/2 A∆T D∆T/2 R∆T/2(ne(0))

1Daru & Tenaud JCP 04Bonaventura, Bourdon, Duarte, Massot Adaptive Time-Space Numerical Methods




EF Computation

Computation in 1.5D geometry

E(x if ) =

nx∑j=1

sρjwj

2ε0

1−wj + 2hi,j√

h2i,j + R2

s +

√(hi,j + wj

)2+ R2

s

,





EF Computation

Computation in 1.5D geometry

E(x if ) =

nx∑j=1

sρjwj

2ε0

1−wj + 2hi,j√

h2i,j + R2

s +

√(hi,j + wj

)2+ R2

s

,





Numerical results

Electric Field





Numerical results

Electron density





Numerical results

Electron density





Numerical results

Adapted Grid








4 Perspectives




Work in progress

Computation of EF on adapted grid

Poisson solver + MR

Multidimensional configurations

Higher order time integration schemes

Complex chemistry

Complex geometry




Funds

Thesis grant BDI from CNRS (departments ST2I et MPPU)

ANR CIS PITAC, 2006-2010. Coordinator Y. MadayANR Blanche SECHELLES, 2009-2013. CoordinatorS. DescombesPEPS from CNRS, 2007-2008. Coordinators F. Laurentand A. BourdonPEPS from CNRS MIPAC, 2009-2010. CoordinatorV. Louvet




References I

E. Hairer and G. Wanner.Solving ordinary differential equations II. Stiff and differential-algebraicproblems.Springer-Verlag, Berlin, 1991.

A. Abdulle.Fourth order Chebyshev methods with recurrence relation.SIAM J. Sci. Comput., 23, pp. 2041-2054, 2002.

A. Cohen, S.M. Kaber, S. Müller and M. Postel.Fully Adaptive Multiresolution Finite Volume Schemes for ConservationLaws.Mathematics of Computation, 72, pp. 183-225, 2001.

A. Harten.Multiresolution algorithms for the numerical solution of hyperbolicconservation laws.Comm. Pure and Applied Math., 48, pp. 1305-1342, 1995.




References II

V. Daru and C. Tenaud.High order one-step monotonicity-preserving schemes for unsteadycompressible flow calculations.J. of Comp. Physics, 193, pp. 563-594, 2004.

A. Laverdant and S. Candel.Computation of diffusion and premixed flames rolled up in vortexstructures.J. Propulsion and Power, 5, pp. 134-143, 1989.

D. Thévenin and S. Candel.Ignition dynamics of a diffusion flame rolled up in a vortex.Phys. Fluids, 7, pp. 434-445, 1995.




References III

S. Descombes, T. Dumont and M. MassotOperator splitting for nonlinear reaction-diffusion systems with anentropic structure : singular perturbation, order reduction andapplication to spiral wavesProceeding of the Workshop “Patterns and waves : theory andapplications”, Saint-Petersbourg (2003)

S. Descombes and M. MassotOperator splitting for nonlinear reaction-diffusion systems with anentropic structure : singular perturbation and order reductionNumerische Mathematik (2004)

S. Descombes, T. Dumont, V. Louvet and M. MassotOn the local and global errors of splitting approximations ofreaction-diffusion equations with high spatial gradientsInternational Journal of Computer Mathematics (2007)




References IV

S. Descombes, T. Dumont, V. Louvet, M. Massot, F. Laurent and J.BeaulaurierOperator splitting techniques for multi-scale reacting waves andapplication to Low Mach number flames with complex chemistry:Theoretical and numerical aspects.Submitted to SIAM, available on HAL (2010)

T. Dumont, M. Duarte, S. Descombes, M.A. Dronne, M. Massot and V.LouvetSimulation of human ischemic stroke in realistic 3D geometry: Anumerical strategySubmitted to Bulletin of Mathematical Biology, available on HAL (2010)




References V

M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvetand F. LaurentNew resolution strategy for multi-scale reaction waves using timeoperator splitting, space adaptive multiresolution and dedicatedhigh order implicit/explicit time integrators.Submitted to SIAM, available on HAL (2010)

M. Duarte, Z. Bonaventura, M. Massot, A. Bourdon, S. Descombes andT. Dumont.A new numerical strategy based on operator splitting with timeadaptivity for plasma discharges.In preparation, invited paper for J. of Comp. Physics special issue on"Computational Plasma Physics" coordinated by Barry Koren and UteEbert (2010)


Documents

Adaptive time-space algorithms for plasma discharge simulation