[PPT] - Time-space adaptive numerical methods for multi-scale reaction waves PowerPoint Presentation

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time-space adaptive numerical methods for multi-scale reaction waves simulation

M. Duarte1
M. Massot1
S. Descombes2
T. Dumont3
V. Louvet3
C. Tenaud4

1EM2C - Ecole Centrale Paris - France 2Laboratoire J. A. Dieudonné - Nice - France 3ICJ - Université Claude Bernard Lyon 1 - France 4LIMSI - CNRS - France

In collaboration with S. Candel1and F. Laurent1.

SMAI 2011 - May 25th 2011

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 1 / 29

SLIDE 2

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Outline

1

Context and Motivation

2

Time/Space Adaptive Numerical Scheme

3

Numerical Illustration

4

Conclusions and Perspectives

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 2 / 29

SLIDE 3

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Context and Motivation

Predictive Simulations for industrial needs Strong evolution of Computer Power and Modeling Tools

source: CERFACS. source: R. Vicquelin, EM2C Lab.

Main Numerical Goals: Resolution of the dynamics of reaction fronts. Reliable accuracy control based on mathematical aspects. New Numerical Strategies for Time/Space Multi-scale Fronts

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 3 / 29

SLIDE 4

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Context and Motivation

Predictive Simulations for industrial needs Strong evolution of Computer Power and Modeling Tools

source: CERFACS. source: R. Vicquelin, EM2C Lab.

Main Numerical Goals: Resolution of the dynamics of reaction fronts. Reliable accuracy control based on mathematical aspects. New Numerical Strategies for Time/Space Multi-scale Fronts

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 3 / 29

SLIDE 5

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Application Background

Flames (dynamics,

pollutants, complex chemistry) source: Yale Univ.

Plasma (repetitive

discharges, streamers) source: A. Bourdon EM2C

Biochemical Engineering

(migraines, Rolando’s region, strokes)

Chemical waves

(spiral waves, scroll waves)

Time and Space Multi-scale Phenomena

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 4 / 29

SLIDE 6

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Numerical Strategies

Coupled resolution of large time scale spectrum Time explicit methods (high order in space) Implicit methods (adaptive time stepping)

SANDIA, ISTA/JAXA, CERFACS, ...

Alternative methods: decoupling time scale spectrum Partitioning methods

G. Warnecke et al, ...

Operator Splitting techniques

J.B. Bell et al, M.S. Day et al, H.N. Najm, O.M. Knio, ...

AMR techniques for flames, detonations, reactive flows

R. Deiterding, T. Ogawa et al, D.W. Schwendeman et al, J.W. Banks et al, S. Paolucci et al, K. Schneider et al, ...

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Numerical Strategies

Coupled resolution of large time scale spectrum Time explicit methods (high order in space) Implicit methods (adaptive time stepping)

SANDIA, ISTA/JAXA, CERFACS, ...

Alternative methods: decoupling time scale spectrum Partitioning methods

G. Warnecke et al, ...

Operator Splitting techniques

J.B. Bell et al, M.S. Day et al, H.N. Najm, O.M. Knio, ...

AMR techniques for flames, detonations, reactive flows

R. Deiterding, T. Ogawa et al, D.W. Schwendeman et al, J.W. Banks et al, S. Paolucci et al, K. Schneider et al, ...

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29

SLIDE 8

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Numerical Strategies

Coupled resolution of large time scale spectrum Time explicit methods (high order in space) Implicit methods (adaptive time stepping)

SANDIA, ISTA/JAXA, CERFACS, ...

Alternative methods: decoupling time scale spectrum Partitioning methods

G. Warnecke et al, ...

Operator Splitting techniques

J.B. Bell et al, M.S. Day et al, H.N. Najm, O.M. Knio, ...

AMR techniques for flames, detonations, reactive flows

R. Deiterding, T. Ogawa et al, D.W. Schwendeman et al, J.W. Banks et al, S. Paolucci et al, K. Schneider et al, ...

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29

SLIDE 9

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time/Space Adaptive Numerical Strategy

Time integration method ⇓ Strang Operator Splitting ֒ → based on Numerical Analysis for stiff PDEs ֒ → splitting time steps larger than fastest scales

Descombes & Massot 04, Descombes et al 07-11

Adaptive splitting time step technique ֒ → dynamic accuracy control based on NA

Descombes et al 11

֒ → applied to instationary problems

Duarte et al 11

Space adaptive multiresolution technique ֒ → based on wavelet transform and NA

Harten 95, Cohen et al 01

֒ → applied to stiff PDEs

Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 6 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time/Space Adaptive Numerical Strategy

Time integration method ⇓ Strang Operator Splitting ֒ → based on Numerical Analysis for stiff PDEs ֒ → splitting time steps larger than fastest scales

Descombes & Massot 04, Descombes et al 07-11

Adaptive splitting time step technique ֒ → dynamic accuracy control based on NA

Descombes et al 11

֒ → applied to instationary problems

Duarte et al 11

Space adaptive multiresolution technique ֒ → based on wavelet transform and NA

Harten 95, Cohen et al 01

֒ → applied to stiff PDEs

Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 6 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Strang Operator Splitting

∂tu − ∂x · F(u) − ∂x · (D(u) ∂xu) = Ω(u) ∂tu = Ω(u) ∂tu = ∂x · (D(u) ∂xu) ∂tu = ∂x · F(u) S∆tu0 = R∆t/2D∆t/2C∆tD∆t/2R∆t/2u0 R∆tR → Radau5 (Hairer & Wanner 91) D∆tD → ROCK4 (Abdulle 02) C∆tC → OSMP3 (Daru & Tenaud 04)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 7 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Strang Operator Splitting

∂tu − ∂x · F(u) − ∂x · (D(u) ∂xu) = Ω(u) ∂tu = Ω(u) ∂tu = ∂x · (D(u) ∂xu) ∂tu = ∂x · F(u) S∆tu0 = R∆t/2D∆t/2C∆tD∆t/2R∆t/2u0 R∆tR → Radau5 (Hairer & Wanner 91) D∆tD → ROCK4 (Abdulle 02) C∆tC → OSMP3 (Daru & Tenaud 04)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 7 / 29

SLIDE 13

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time Adaptive Numerical Strategy

Time integration method ⇓ Strang Operator Splitting ֒ → based on Numerical Analysis for stiff PDEs ֒ → splitting time steps larger than fastest scales

Descombes & Massot 04, Descombes et al 07-11

Adaptive splitting time step technique ֒ → dynamic accuracy control based on NA

Descombes et al 11

֒ → applied to instationary problems

Duarte et al 11

Space adaptive multiresolution technique ֒ → based on wavelet transform and NA

Harten 95, Cohen et al 01

֒ → applied to stiff PDEs

Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 8 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Time Step

We define two time integration solvers: S∆tu0 − T ∆tu0 = O(∆t3) = ⇒ Strang formula

S∆tu0 − T ∆tu0 = O(∆t2)

= ⇒ embedded Strang formula and considering

S∆tu0 −

S∆tu0

≈ O(∆t2) < η

yields ∆tnew = ∆t

η
S∆tu0 −

S∆tu0

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud

Time-Space Adaptive Numerical Methods 9 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Time Step

We define two time integration solvers: S∆tu0 − T ∆tu0 = O(∆t3) = ⇒ Strang formula

S∆tu0 − T ∆tu0 = O(∆t2)

= ⇒ embedded Strang formula and considering

S∆tu0 −

S∆tu0

≈ O(∆t2) < η

yields ∆tnew = ∆t

η
S∆tu0 −

S∆tu0

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud

Time-Space Adaptive Numerical Methods 9 / 29

SLIDE 16

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Space Adaptive Multiresolution

Time integration method ⇓ Strang Operator Splitting ֒ → based on Numerical Analysis for stiff PDEs ֒ → splitting time steps larger than fastest scales

Descombes & Massot 04, Descombes et al 07-11

Adaptive splitting time step technique ֒ → dynamic accuracy control based on NA

Descombes et al 11

֒ → applied to instationary problems

Duarte et al 11

Space adaptive multiresolution technique ֒ → based on wavelet transform and NA

Harten 95, Cohen et al 01

֒ → applied to stiff PDEs

Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 10 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Space Adaptive Multiresolution

V J(x, y) = V 0(x, y) + J

j=1 Dj(x, y)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 11 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Space Adaptive Multiresolution

V J(x, y) = V 0(x, y) + J

j=1 Dj(x, y)

V J

ε (x, y)

V J(x, y) − V J

ε (x, y) ≤ ε

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 11 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time/Space Adaptive Numerical Scheme

code MBARETE (Duarte et al.) Time integration method ⇓ Strang Operator Splitting ֒ → High order dedicated methods Adaptive splitting time step technique ֒ → Independent of stability issues Space adaptive multiresolution technique

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 12 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Application Framework

Combustion flames interacting with vortex structures ∂tYi + v · ∂xYi = D ∂2

xYi + νiWi

ρ ˙ ω ∂tT + v · ∂xT = D ∂2

xT + νFWFQ

ρcp ˙ ω v(x, t) = ⇒ 2D vortex configuration vθ(r, t) = Γ 2πr

1 − exp
− r 2

4νt

Single step chemistry

⇓ Arrhenius law Thermodiffusive approach ⇓ Laminar flames with constant density

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 13 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Application Framework

Combustion flames interacting with vortex structures ∂tYi + v · ∂xYi = D ∂2

xYi + νiWi

ρ ˙ ω ∂tT + v · ∂xT = D ∂2

xT + νFWFQ

ρcp ˙ ω v(x, t) = ⇒ 2D vortex configuration vθ(r, t) = Γ 2πr

1 − exp
− r 2

4νt

Single step chemistry

⇓ Arrhenius law Thermodiffusive approach ⇓ Laminar flames with constant density

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 13 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Premixed Flame with Vortex (Laverdant & Candel J. Propulsion 87)

Introducing progress variable c(x, y, t) c = T − To Tb − To = YFo − YF YFo c = 1 → burnt gases c = 0 → fresh gases ∂c ∂t⋆ +u⋆ ∂c ∂x⋆ +v⋆ ∂c ∂y⋆ = ∂2c ∂x2

⋆

+∂2c ∂y2

⋆

+Da(1−c) exp

−

Ta To(1 + τc)

vθ⋆(r⋆, t⋆) = Re Sc

r⋆

1 − exp
−

r 2

⋆

4 Sc t⋆

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud

Time-Space Adaptive Numerical Methods 14 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Premixed Flame with Two Vortices (Re = 1000)

Finest Grid = 10242 CPU time ∼ 40h15 Active Grid Finest Grid 9% CPU time ∼ 0h57 η = 10−3 2.7 GHz AMD Shanghai RAM memory 32 GB

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 15 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Dynamic Adaptive Grid

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 16 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Time Evolution of Data Compression

time Active Grid Finest Grid [%] ∼ 9%

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 17 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Multiple Time Steps

time time steps ∆t ∼ O(10−5) ∆tD ∼ O(10−6) ∆tR ∼ [O(10−7), O(10−5)] ∆tC ∼ O(10−7)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 18 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Premixed Flame Toroidal Vortex (Re = 1000)

Finest Grid = 2563 CPU time (out of reach) Active Grid Finest Grid 17% CPU time ∼ 14h40 η = 10−3 2.7 GHz AMD Shanghai RAM memory 32 GB

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 19 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Dynamic Adaptive Grid

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 20 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Ignition of Diffusion Flame with Vortex (Thévenin & Candel Phys. Fluids 95)

Defining Z(x, y, t) and θ(x, y, t) 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 − Z φτ + 1 χ(Z − θ) Z + τ χ(Z − θ)

exp
−

τa 1 + τθ

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud

Time-Space Adaptive Numerical Methods 21 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

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

Finest Grid = 10242 CPU time ∼ 5h24 Active Grid Finest Grid 6% CPU time ∼ 0h08 η = 10−3 2.7 GHz AMD Shanghai RAM memory 32 GB

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 22 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Time Step

time time steps ∆t ∼ O(10−5) REJECTION! ∆t ∼ O(10−7)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 23 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Time Step

time time steps ∆t ∼ O(10−5) REJECTION! ∆t ∼ O(10−7) ∆tC ∼ [O(10−8), O(10−7)]

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 24 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Some Remarks and Conclusions

Concerning implementation aspects: Important savings in computing time and memory requirements. Straightforward parallelization in shared memory computing environments. Needs of dedicated data structures and optimized routines for MR. Main contribution: Efficient time/space adaptive strategy for multi-scale phenomena. Time/space dynamic accuracy control. Mathematical background.

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 25 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Some Remarks and Conclusions

Concerning implementation aspects: Important savings in computing time and memory requirements. Straightforward parallelization in shared memory computing environments. Needs of dedicated data structures and optimized routines for MR. Main contribution: Efficient time/space adaptive strategy for multi-scale phenomena. Time/space dynamic accuracy control. Mathematical background.

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 25 / 29

SLIDE 35

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Complex Chemistry

Finest Grid = 10242 49 species - 299 reactions

(Lindstedt et al 98)

Active Grid Finest Grid 4% CPU time ∼ 30h30 with 12 cores η = 10−2 2.7 GHz AMD Shanghai RAM memory 48 GB

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 26 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Complex Chemistry

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 27 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Funds

Ph.D. Thesis grant from CNRS (Maths and Engineering Institutes). INCA Label. DIGITEO RTRA MUSE, 2010-2014. Coordinator

M. Massot

ANR Blanche SECHELLES, 2009-2013. Coordinator

S. Descombes

PEPS from CNRS MIPAC, 2009-2010. Coordinator

V. Louvet

PEPS from CNRS, 2007-2008. Coordinators F . Laurent and A. Bourdon

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 28 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

References

M. Duarte, M. Massot and S. Descombes. Parareal Operator Splitting Techniques for

Multi-Scale Reaction Waves: Numerical Analysis and Strategies. M2AN, 45:825-852, 2011.

M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvet and F.
Laurent. New resolution strategy for multi-scale reaction waves using time operator

splitting, space adaptive multiresolution and dedicated high order implicit/explicit time

integrators. Submitted to SIAM, available on HAL (2010)
T. Dumont, M. Duarte, S. Descombes, M.A. Dronne, M. Massot and V. Louvet.

Simulation of human ischemic stroke in realistic 3D geometry: A numerical strategy. Submitted to Bulletin of Mathematical Biology, available on HAL (2010)

M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvet and F.
Laurent. New resolution strategy for multi-scale reaction waves using time operator

splitting space adaptive multiresolution: Application to human ischemic stroke. Accepted to ESAIM Proceedings (2011)

S. Descombes, M. Duarte, T. Dumont, V. Louvet and M. Massot. Adaptive time splitting

method for multi-scale evolutionary PDEs. Accepted to Confluentes Mathematici (2011)

M. Duarte, Z. Bonaventura, M. Massot, A. Bourdon, S. Descombes and T. Dumont. A

new numerical strategy with space-time adaptivity and error control for multi-scale gas discharge simulations. Submitted to J. of Comp. Physics, special issue on "Computational Plasma Physics" coordinated by Barry Koren and Ute Ebert (2011)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 29 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Detailed Chemistry

RD 21 variables model MR on simple geometry

(Duarte et al. 11)

Grid = 5123 Active Grid 5% CPU time ∼ 37h24 with 8 cores - parallel gain ratio ∼ 7 Unfeasible with fixed grid and same computing resource

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 30 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Detailed Chemistry

RD 21 variables model MR on simple geometry

(Duarte et al. 11)

Grid = 5123 Active Grid 5% CPU time ∼ 37h24 with 8 cores - parallel gain ratio ∼ 7 Unfeasible with fixed grid and same computing resource

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 30 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Detailed Chemistry

RD 21 variables model MR on simple geometry

(Duarte et al. 11)

Grid = 5123 Active Grid 5% CPU time ∼ 37h24 with 8 cores - parallel gain ratio ∼ 7 Unfeasible with fixed grid and same computing resource

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 30 / 29

SLIDE 42

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Taking into Account Detailed Chemistry

RD 21 variables model MR on simple geometry

(Duarte et al. 11)

Grid = 5123 Active Grid 5% CPU time ∼ 37h24 with 8 cores - parallel gain ratio ∼ 7 Unfeasible with fixed grid and same computing resource

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 30 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

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

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 31 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of operator splitting

First order methods : Lie Formulae. Lt

1 U0 = X t Y t U0

Lt

1 U0 − T t U0 = O(t2),

Lt

2 U0 = Y t X t U0

Lt

2 U0 − T t U0 = O(t2),

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 32 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of operator splitting

Second order methods : Strang Formulae. St

1 U0 = Y t/2 X t Y t/2 U0

St

1 U0 − T t U0 = O(t3),

St

2 U0 = X t/2 Y t X t/2 U0

St

2 U0 − T t U0 = O(t3),

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 33 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Suitable Stiff Integrators

Numerical Strategy: ∂tU + ∇ · F(U)

OSMP3

− ∆U

ROCK4

= Ω(U)

Radau5

(Hairer & Wanner Springer-Verlag 91)

A-stable and L-stable Based on RadauIIA Implicit RK with order 5 Simplified Newton Method = ⇒ Linear Algebra tools Adaptive time integration step

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 34 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Suitable Stiff Integrators

Numerical Strategy: ∂tU + ∇ · F(U)

OSMP3

− ∆U

ROCK4

= Ω(U)

Radau5

(Abdulle SIAM J. Sci. Comput. 02)

Extended Stability Domain (along R−) Order 4 Adaptive time integration step Explicit RK Method = ⇒ NO Linear Algebra problems Low Memory Demand

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 34 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Suitable Stiff Integrators

Numerical Strategy: ∂tU + ∇ · F(U)

OSMP3

− ∆U

ROCK4

= Ω(U)

Radau5

(Daru & Tenaud JCP 04)

One-step monotonicity preserving scheme Order 3 at least Explicit Method = ⇒ standard CFL constraint Low Memory Demand

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 34 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of Splitting Numerical Analysis

Considering Lt

2 U0 = Y t X t U0

Lt

2 U0 − T t U0 = O(t2)

and introducing Lie formalism DFG(u) = G′(u)F(u) yield Y t U0 =

k≥0

tk k!

Dk

YId(U)

t=0

= etDY IdU0

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 35 / 29

SLIDE 50

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of Splitting Numerical Analysis

Considering Lt

2 U0 = Y t X t U0

Lt

2 U0 − T t U0 = O(t2)

and introducing Lie formalism DFG(u) = G′(u)F(u) yield Y t U0 =

k≥0

tk k!

Dk

YId(U)

t=0

= etDY IdU0

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 35 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of Splitting Numerical Analysis

Hence, Lt

2 U0 = Y t X t U0 = etDX etDY IdU0

And considering BHC formula etDX etDY = eL(t) L(t) = t (DX + DY) + t2 2 [DX, DY] + O(t2) yield T t U0 − Y t X t U0 = t2 2 [DY, DX]IdU0 + O(t3)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 36 / 29

SLIDE 52

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of Splitting Numerical Analysis

Hence, Lt

2 U0 = Y t X t U0 = etDX etDY IdU0

And considering BHC formula etDX etDY = eL(t) L(t) = t (DX + DY) + t2 2 [DX, DY] + O(t2) yield T t U0 − Y t X t U0 = t2 2 [DY, DX]IdU0 + O(t3)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 36 / 29

SLIDE 53

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Basis of Splitting Numerical Analysis

Hence, Lt

2 U0 = Y t X t U0 = etDX etDY IdU0

And considering BHC formula etDX etDY = eL(t) L(t) = t (DX + DY) + t2 2 [DX, DY] + O(t2) yield T t U0 − Y t X t U0 = t2 2 [DY, DX]IdU0 + O(t3)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 36 / 29

SLIDE 54

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Considering now St

1 U0 = Y t/2 X t Y t/2 U0

St

1 U0 − T t U0 = O(t3)

it follows St

1 U0 = Y t/2 X t Y t/2 U0 = e

t 2 DY etDX e t 2 DY IdU0

and T t U0 − Y t/2 X t Y t/2 U0 = t3 24

DY, [DY, DX]
IdU0

− t3 12

DX, [DX, DY]
IdU0 + O(t4)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 37 / 29

SLIDE 55

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Considering now St

1 U0 = Y t/2 X t Y t/2 U0

St

1 U0 − T t U0 = O(t3)

it follows St

1 U0 = Y t/2 X t Y t/2 U0 = e

t 2 DY etDX e t 2DY IdU0

and T t U0 − Y t/2 X t Y t/2 U0 = t3 24

DY, [DY, DX]
IdU0

− t3 12

DX, [DX, DY]
IdU0 + O(t4)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 37 / 29

SLIDE 56

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Considering now St

1 U0 = Y t/2 X t Y t/2 U0

St

1 U0 − T t U0 = O(t3)

it follows St

1 U0 = Y t/2 X t Y t/2 U0 = e

t 2 DY etDX e t 2DY IdU0

and T t U0 − Y t/2 X t Y t/2 U0 = t3 24

DY, [DY, DX]
IdU0

− t3 12

DX, [DX, DY]
IdU0 + O(t4)

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 37 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Defining

St

1U0 = Y( 1

2 +ε)t X t Y( 1 2 −ε)t U0

it follows

St

1 U0 = e( 1

2 −ε)tDY etDX e( 1 2 +ε)tDY IdU0

and T tU0−Y( 1

2+ε)t X t Y( 1 2 −ε)t U0 = εt2[DY, DX]IdU0+O(εt3)+O(t3)

T t U0 − St

1 U0 = T t U0 − St 1 U0

O(t3)

+ St

1 U0 −

St

1 U0

O(εt2)

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SLIDE 58

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Defining

St

1U0 = Y( 1

2 +ε)t X t Y( 1 2 −ε)t U0

it follows

St

1 U0 = e( 1

2 −ε)tDY etDX e( 1 2 +ε)tDY IdU0

and T tU0−Y( 1

2 +ε)t X t Y( 1 2 −ε)t U0 = εt2[DY, DX]IdU0+O(εt3)+O(t3)

T t U0 − St

1 U0 = T t U0 − St 1 U0

O(t3)

+ St

1 U0 −

St

1 U0

O(εt2)

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SLIDE 59

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Adaptive Splitting Numerical Analysis

Defining

St

1U0 = Y( 1

2 +ε)t X t Y( 1 2 −ε)t U0

it follows

St

1 U0 = e( 1

2 −ε)tDY etDX e( 1 2 +ε)tDY IdU0

and T tU0−Y( 1

2 +ε)t X t Y( 1 2 −ε)t U0 = εt2[DY, DX]IdU0+O(εt3)+O(t3)

T t U0 − St

1 U0 = T t U0 − St 1 U0

O(t3)

+ St

1 U0 −

St

1 U0

O(εt2)

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SLIDE 60

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Wavelet Representation

Considering J embedded grids j ∈ [0, J] and the box function φ(x) = χ[0,1)(x), we define for u(x) ∈ L2 Pj(u) :=

2j−1

k=0

u, φj,kφj,k with φj,k = 2j/2φ(2j · −k) Thus, defining ψj,k = 2j/2ψ(2j · −k) yields

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SLIDE 61

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Wavelet Representation

Considering J embedded grids j ∈ [0, J] and the box function φ(x) = χ[0,1)(x), we define for u(x) ∈ L2 Pj(u) :=

2j−1

k=0

u, φj,kφj,k with φj,k = 2j/2φ(2j · −k) Thus, defining ψj,k = 2j/2ψ(2j · −k) yields

2j+1−1

k=0

u, φj+1,kφj+1,k =

2j−1

k=0

u, φj,kφj,k +

2j−1

k=0

u, ψj,kψj,k

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SLIDE 62

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Wavelet Representation

Considering J embedded grids j ∈ [0, J] and the box function φ(x) = χ[0,1)(x), we define for u(x) ∈ L2 Pj(u) :=

2j−1

k=0

u, φj,kφj,k with φj,k = 2j/2φ(2j · −k) Thus, defining ψj,k = 2j/2ψ(2j · −k) yields (Pj+1 − Pj)u =

2j−1

k=0

dj,k(u)ψj,k, dj,k(u) := u, ψj,k

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 39 / 29

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Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Multiresolution Transformation

(Cohen et al. Mathematics of Computation 01)

There is a one-to-one correspondence Uj ← → (Uj−1, Dj), which defines by iteration a multiscale representation of UJ: M : UJ − → MJ, MJ = (U0, D1, D2, · · · , DJ) And thus, thresholding is performed if |dj,k| < εj = ⇒ dj,k = 0, εj = 2

d 2 (j−J)ε,

which implies Un

J − V n J L2 ∝ nε

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 40 / 29

SLIDE 64

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Multiresolution Transformation

(Cohen et al. Mathematics of Computation 01)

There is a one-to-one correspondence Uj ← → (Uj−1, Dj), which defines by iteration a multiscale representation of UJ: M : UJ − → MJ, MJ = (U0, D1, D2, · · · , DJ) And thus, thresholding is performed if |dj,k| < εj = ⇒ dj,k = 0, εj = 2

d 2 (j−J)ε,

which implies Un

J − V n J L2 ∝ nε

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 40 / 29

SLIDE 65

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Multiresolution Transformation

(Cohen et al. Mathematics of Computation 01)

There is a one-to-one correspondence Uj ← → (Uj−1, Dj), which defines by iteration a multiscale representation of UJ: M : UJ − → MJ, MJ = (U0, D1, D2, · · · , DJ) And thus, thresholding is performed if |dj,k| < εj = ⇒ dj,k = 0, εj = 2

d 2 (j−J)ε,

which implies Un

J − V n J L2 ∝ nε

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 40 / 29

SLIDE 66

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Wavelet Representation

If the ψλ have m vanishing polynomial moments, P, ψj,kSj,k = 0, P ∈ Pm−1 then, |dj,k(u)| = |u, ψj,k| = inf

P∈Pm−1

|u − P, ψj,k| Thus, in practice, we compute dµ := uµ − ˆ uµ. uµ ⇒ Projection: exact values computed from finer grids ˆ uµ ⇒ Prediction: approximated values from coarser grids

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 41 / 29

SLIDE 67

Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives

Wavelet Representation

If the ψλ have m vanishing polynomial moments, P, ψj,kSj,k = 0, P ∈ Pm−1 then, |dj,k(u)| = |u, ψj,k| = inf

P∈Pm−1

|u − P, ψj,k| Thus, in practice, we compute dµ := uµ − ˆ uµ. uµ ⇒ Projection: exact values computed from finer grids ˆ uµ ⇒ Prediction: approximated values from coarser grids

Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 41 / 29