A BSTRACT B UBBLE V IBRATION M ODEL A PPLICATIONOFAN AMR TECHNIQUETOAN

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A PPLICATION OF AN AMR TECHNIQUE TO AN A BSTRACT B UBBLE V IBRATION M ODEL

Yohan PENEL¹,2, Anouar Mekkas¹^,³, St ´ephane Dellacherie¹, Juliet Ryan²^,³, M. Borrel³

1DEN/DANS/DM2S/SFME/LETR, CEA Saclay, France

2LAGA, Institut Galil ´ee, University of Paris 13, France

3DTIM/CHP, ONERA Ch ˆatillon, France

19THAIAA COMPUTATIONALFLUIDDYNAMICSCONFERENCE San Antonio, Texas June, 23rd, 2009

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I NTRODUCTION

Modeling of the evolution of bubblesin a nuclear reactor at the scale of bubbles (DNS)

GOAL

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I NTRODUCTION

GOAL

Navier-Stokes equationsfor a diphasic compressible im- miscible flow

Mach Number supposed to be small (still compressible flow but without acoustic wave effects)

Diphasic Low Mach Number system (DLMN)

Simplification but keeping a similar mathematical structure

Abstract Bubble Vibration model (ABV)

Numerical representation of bubble interfaces

Numerical scheme for a transport equation

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I NTRODUCTION

GOAL

AMR

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I NTRODUCTION

GOAL

AMR

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O UTLINE

1 Derivation of the model

2 Theoretical results

3 Interfaces: Numerical resolution of a transport equation

4 Numerical Results for the ABV model

5 Conclusion

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P ARTIAL O UTLINE

5 Conclusion

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C OMPRESSIBLE D IPHASIC N AVIER -S TOKES S YSTEM

Starting point

DIPHASIC COMPRESSIBLE NAVIER-STOKES 









∂t(ρY₁) +∇·(ρY₁u) =0,

∂tρ+∇·(ρu) =0,

∂t(ρu) +∇·(ρu⊗u) =−∇P+∇·σ+ρg,

∂t(ρE) +∇·(ρuE) =−∇·(Pu) +∇·(κ∇T) +∇·(σu) +ρg·u,

together with transmission and boundary conditions.

Nomenclature

• Y₁: mass fraction of Fluid 1

• T : temperature

• P: pressure

• u: global velocity

• E: total energy

• ρ : density

• g: gravity field

• σ : Cauchy stress tensor

• κ : thermal conductivity

• θ= (Y₁,T,P)

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M ODELING BUBBLES

Initial Condition

Y1(t=0,x) =Y⁰(x) =

(1, six∈Ω1(0), 0, six∈Ω₂(0).

AsY₁is solution of the equation∂t(ρY₁) +∇·(ρY₁u) =0⇐⇒∂tY₁+u·∇^Y1=0,the interface of the bubble coincides with its discontinuity. The resolution of this equation with that initial condition amounts to determining for a domainΩ1(t).

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H YPOTHESES AND T OOLS

PHYSICS MATHEMATICS

Bounded domain (reactor) Ω = [−1,1]^d,d∈ {2,3}

No void ρ>0

Linear elasticity Linearized Cauchy tensor Common physical properties for both fluids Single nondimensioned system

Low Mach Number Asymptotic expansion w.r.t.M∗1 Based on earlier Majda’s & Embid’s works on combustion (’84).

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DLMN S YSTEM

Diphasic Low Mach Number System: At order 0 in the asymptotic expansion, the system reads:











∂tY₁+u·∇Y₁=0,

∇·u=G_θ,

ρ(∂tu+ (u·∇)u) =−∇Π +2∇·[µD(u)] +ρg, ρ^cp(∂tT+u·∇^T) =α^TP⁰(t) +∇·(κ∇^T), P⁰(t) =H_θ(t),

wherePis the thermodynamic pressure,Πthe dynamic pressure and:

G_θ(t,x) :=−^D^tρ ρ =−¹

Γ P⁰(t)

P(t) +β∇·(κ∇T) P(t) ,

H_θ(t) :=

Z

Ω

β(θ)∇·

κ(θ)∇T dx Z

Ω

1 Γ(θ)dx

.

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DLMN S YSTEM

Diphasic Low Mach Number System: At order 0 in the asymptotic expansion, the system reads:











∂tY₁+u·∇Y₁=0,

∇·u=G_θ, =⁶⁼⇒⁰ compressibility, elliptic contribution

ρ(∂tu+ (u·∇)u) =−∇Π +2∇·[µD(u)] +ρg, ρ^cp(∂tT+u·∇^T) =α^TP⁰(t) +∇·(κ∇^T), P⁰(t) =H_θ(t),

wherePis the thermodynamic pressure,Πthe dynamic pressure and:

G_θ(t,x) :=−^D^tρ ρ =−¹

Γ P⁰(t)

P(t) +β∇·(κ∇T) P(t) ,

H_θ(t) :=

Z

Ω

β(θ)∇·

κ(θ)∇T dx Z

Ω

1 Γ(θ)dx

.

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D ERIVED S YSTEMS FOR PRELIMARY STUDIES

Hodge Decomposition:u=∇φ+wwith∇·w=0 and boundary conditions.

Potential DLMN System w→0











∂tY1+∇φ·∇Y1=0,

∆φ=G_θ,

ρc_p(∂tT+∇φ·∇T) =αTP⁰(t) +∇·(κ∇T), P⁰(t) =H_θ(t).

Abstract Bubble Vibration Model G_θ→G_Y







∂tY1+∇φ·∇Y1=0,

∆φ(t,x) =ψ(t)

Y₁(t,x)− ¹

|Ω|

Z

Ω

Y₁(t,y)dy

.

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P ARTIAL O UTLINE

5 Conclusion

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E XISTENCE AND U NIQUENESS FOR DLMN

DIPHASICLOWMACHNUMBER











∂tY1+u·∇Y1=0,

∇·u=G_θ,

ρ(∂tu+ (u·∇)u) =−∇Π +2∇·[µD(u)] +ρg, ρcp(∂tT+u·∇T) =αTP⁰(t) +∇·(κ∇T), P⁰(t) =H_θ(t).

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E XISTENCE AND U NIQUENESS FOR DLMN











∂tY1+u·∇Y1=0,

∇·u=G_θ,

ρ(∂tu+ (u·∇)u) =−∇Π +2∇·[µD(u)] +ρg, ρcp(∂tT+u·∇T) =αTP⁰(t) +∇·(κ∇T), P⁰(t) =H_θ(t). THEOREM(PENEL, ’09)

Assumings>d 2

+4,θ0∈H^ssuch that, for allx∈T^d^,θ0(x)∈G0 withG0⊂Θ, andu0∈H^s−1, there existsT =T(kθ0ks,ku0ks)>0 such that there exists a unique classical solution(θ,u,∇π)to the DLMN system:

• T,P∈Xs,T(T^d),Y1∈Xs−1,T(T^d),∂ θ

∂t ∈Xs−2,T(T^d);

• u∈Xs−1,T(T^d),∂u

∂t ∈Xs−3,T(T^d);

• ∇π∈Xs−3,T(T^d).

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E XISTENCE AND U NIQUENESS FOR DLMN











∂tY1+u·∇Y1=0,

∇·u=G_θ,

ρ(∂tu+ (u·∇)u) =−∇Π +2∇·[µD(u)] +ρg, ρcp(∂tT+u·∇T) =αTP⁰(t) +∇·(κ∇T), P⁰(t) =H_θ(t). THEOREM(PENEL, ’09)

Assumings>d 2

+4,θ0∈H^ssuch that, for allx∈T^d^,θ0(x)∈G0 withG0⊂Θ, andu0∈H^s−1, there existsT =T(kθ0ks,ku0ks)>0 such that there exists a unique classical solution(θ,u,∇π)to the DLMN system:

• T,P∈Xs,T(T^d),Y1∈Xs−1,T(T^d),∂ θ

∂t ∈Xs−2,T(T^d);

• u∈Xs−1,T(T^d),∂u

∂t ∈Xs−3,T(T^d);

• ∇π∈Xs−3,T(T^d). Xs,T(T^d) =C⁰ [0,T],L²(T^d)

∩L^∞ [0,T],H^s(T^d)

∩L² [0,T],H^s+1(T^d)

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E XISTENCE AND U NIQUENESS FOR ABV

ABSTRACTBUBBLEVIBRATIONMODEL











∂tY1+∇φ·∇Y1=0, Y1(0,x) =Y⁰(x),

∆φ(t,x) =ψ(t)

Y1(t,x)− ¹

|Ω|

Z Ω

Y1(t,y)dy

,

∇φ·n_|∂_Ω=0.

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E XISTENCE AND U NIQUENESS FOR ABV











∂tY1+∇φ·∇Y1=0, Y1(0,x) =Y⁰(x),

∆φ(t,x) =ψ(t)

Y1(t,x)− ¹

|Ω|

Z Ω

Y1(t,y)dy

,

∇φ·n_|∂_Ω=0. THEOREM(LAFITTE& DELLACHERIE, ’05) Assumings>_d

2

+2,Y⁰∈H^s. There existsT =T(kY⁰ks)>0 such that there exists a unique classical solution(Y1,∇φ)to the ABV system, withY1∈C⁰ [0,T],L²(T^d)

∩L^∞ [0,T],H^s(T^d) .

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E XISTENCE AND U NIQUENESS FOR ABV











∂tY1+∇φ·∇Y1=0, Y1(0,x) =Y⁰(x),

∆φ(t,x) =ψ(t)

Y1(t,x)− ¹

|Ω|

Z Ω

Y1(t,y)dy

,

∇φ·n_|∂_Ω=0.

LEMMA

Assuming there exists a weak solution of the formY1(t,x) =1_Ω

1(t)(x) whereΩ1(t)is a smooth open set inΩ, andψ∈C⁰(0,+∞), then the volume of the bubble is given by:

|Ω1(t)|=|Ω|

|Ω1(0)|exp Zt

0

ψ(τ)dτ

|Ω2(0)|+|Ω1(0)|exp Zt

0

ψ(τ)dτ . THEOREM(LAFITTE& DELLACHERIE, ’05) Assumings>_d

2

+2,Y⁰∈H^s. There existsT =T(kY⁰ks)>0 such that there exists a unique classical solution(Y1,∇φ)to the ABV system, withY1∈C⁰ [0,T],L²(T^d)

∩L^∞ [0,T],H^s(T^d) .

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P ARTIAL O UTLINE

3 Interfaces: Numerical resolution of a transport equation Representation of interfaces

AMR

Numerical results

5 Conclusion

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I NTERFACES

^DL

Numerical handling of interfaces

Interface Capturing

Front Tracking

Level Set

Volume Of Fluid

Antidiffusive schemes

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I NTERFACES

^DL

Antidiffusive schemes Resolution of the transport equation∂tY₁+u·∇Y₁=0

by means of theDespr ´es-Lagouti `ere Scheme

B. Despr ´es, F. Lagouti `ere, Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics.J. of Sc. Comp., ’01.

Idea: combining stability properties of the upwind scheme and non-diffusive properties of the downwind scheme.

Properties:

L^∞-stable under CFL TVD

uniform control of the number of diffusion cells

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F UNDAMENTAL TEST (K OTHE & R IDER )

^Data

Resolution of∂tY₁+u·∇Y₁=0with a periodic rotational prescribed velocity in 2D and 3D cases. The velocity is such that one should recover the initial condition after one period.

Improving accuracy requires additional techniques likeAMR.

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S OME REFERENCES

General concept of AMR(Grouping-Clustering Algorithm)

M. Berger and J. Oliger,Adaptive methods for hyperbolic partial differential equations. JCP, ’84.

M. Berger and P. Colella,Local adaptive mesh refinement for shock hydrodynamics. JCP, ’89.

M. Berger and I. Rigoutsos,An Algorithm for Point Clustrering and Grid Generation. IEEE, ’91.

AMR techniques for the transport equation

J. Quirk,An Adaptive Grid Algorithm for Computational Shock Hydrodynamics.

Ph.D. Thesis, Cranfield Univ., ’91.

J.-C. Jouhaud,M ´ethode d’Adaptation de Maillages Structur ´es par Enrichissement.

Ph.D. Thesis, Bordeaux Univ., ’97.

J. Ryan and M. Borrel,Adaptive Mesh Refinement: a coupling Framework for Direct Numerical

Simulation of reacting Gas Flow. ICFD, ’04.

Local Defect Correction Methods for the Laplace equation

W. Hackbush,Local defect correction and domain decomposition techniques.

Defect Corr. Meth., ’84.

M. Anthonissen,Local Defect Correction Techniques: Analysis and Application to Combustion.

Ph.D. Thesis, Eindhoven Univ., ’01.

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G ENERATING A HIERARCHICAL STRUCTURE OF GRIDS

Steps

tagging grid regions that need a higher resolution clustering tagged cells into subgrids (patches) refining patches

Balance between general constraints

as few patches as possible to reduce computation

patches as small as possible to avoid unrelevant refined regions

Grouping-ClusteringAlgorithm coarse level grid:G0

moving patchesG₁_,_nadapting to the solution of eq.∂tY₁+u·∇Y₁=0

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G ENERATING A HIERARCHICAL STRUCTURE OF GRIDS

G0 G1,n

Grouping-ClusteringAlgorithm coarse level grid:G₀

moving patchesG1,nadapting to the solution of eq.∂tY1+u·∇Y1=0

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S KETCH OF RESOLUTION

Y_Gⁿ

1,n

Y_Gⁿ

0

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0 ¹ Yⁿ_G⁺₀¹

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1’ 1,n 1

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

1’

1 2

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

Y_Gⁿ⁺¹

0∩G_1,n

1’

1

2’

2

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S KETCH OF RESOLUTION

G0

G₁_,_n

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

Y_Gⁿ⁺¹

0∩G_1,n

Y_Gⁿ⁺¹

0

1’

1

2’

2

3 3

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S KETCH OF RESOLUTION

G0

G₁_,_n

G₁_,_n₊₁

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

Y_Gⁿ⁺¹

0∩G_1,n

Y_Gⁿ⁺¹

0

G₁_,_n₊₁

1’

1

2’

2

3 3

4

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S KETCH OF RESOLUTION

G0

G₁_,_n

G₁_,_n₊₁

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

Y_Gⁿ⁺¹

0∩G_1,n

Y_Gⁿ⁺¹

0

G₁_,_n₊₁

Yⁿ_G⁺¹

1,n+1

1’

1

2’

2

3 3

4

5 5

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S KETCH OF RESOLUTION

G0

G₁_,_n

G₁_,_n₊₁

Y_Gⁿ

1,n

Y_Gⁿ

0 Yⁿ_G⁺₀¹

Y_Gⁿ⁺¹

1,n

Y_Gⁿ⁺¹

0\G_1,n

Y_Gⁿ⁺¹

0∩G_1,n

Y_Gⁿ⁺¹

0

G₁_,_n₊₁

Yⁿ_G⁺¹

1,n+1

Y_Gⁿ⁺¹

1,n+1

1’

1

2’

2

3 3

4

5 5

6

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F UNDAMENTAL TEST ( WITH AMR)

^Data

Refinement rate equal to 10

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P ARTIAL O UTLINE

5 Conclusion

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

Y_Gⁿ

0,G1,n,Y_Gⁿ

1,n,φⁿ

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

Y_Gⁿ

0,G1,n,Y_Gⁿ

1,n,φⁿ

DL-Scheme with AMR for one time iteration of the resolution of the transport equation with discrete velocity∇φⁿ

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

Y_Gⁿ

0,G1,n,Y_Gⁿ

1,n,φⁿ

Y_Gⁿ⁺¹

0 ,G₁_,_n+1,Y_Gⁿ⁺¹

1,n+1

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

Y_Gⁿ

0,G1,n,Y_Gⁿ

1,n,φⁿ

Y_Gⁿ⁺¹

0 ,G₁_,_n+1,Y_Gⁿ⁺¹

1,n+1

LDC method (using fine grid built in the hyperbolic step) for the resolution of the Laplace equation with rhsP(Yⁿ⁺¹)

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C ODE STRUCTURE

^LDC

(

∂tY₁+∇φ·∇Y₁=0,

∆φ=P(Y₁).

Y_Gⁿ

0,G1,n,Y_Gⁿ

1,n,φⁿ

Y_Gⁿ⁺¹

0 ,G₁_,_n+1,Y_Gⁿ⁺¹

1,n+1

LDC method (using fine grid built in the hyperbolic step) for the resolution of the Laplace equation with rhsP(Yⁿ⁺¹)

Y_Gⁿ⁺¹

0 ,G₁_,_n+1,Y_Gⁿ⁺¹

1,n+1,G₁_,_n+1

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N UMERICAL TEST

ABV model:











∂tY₁+∇φ·∇Y₁=0, Y₁(0,x) =Y⁰(x),

∆φ=Y1−¹ 4

ZZ

[−1,1]²

Y1(t,y)dy,

∇φ·n_|∂_Ω=0.

ψ≡1 implies constant growth.

The initial conditionY⁰is defined by two circles with different radii.

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N UMERICAL TEST

100×100 grid with a refinement rate equal to 6

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V OLUMES

^Lemma

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P ARTIAL O UTLINE

5 Conclusion

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C ONCLUSION AND P ROSPECTS

Done

Implementationof an AMR technique for a transport equation Couplingbetween hyperbolic and elliptic

Couplingbetween three algorithms (DL scheme, AMR, LDC)

To do

Application to theDLMN System

Enrichment of the model withphysical aspects Theoretical studies fornon-smooth initial conditions

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T HANK YOU FOR YOUR ATTENTION

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P ARTIAL O UTLINE

6 Despr ´es-Lagouti `ere Scheme

7 LDC

8 Data

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D EFINITION OF THE SCHEME

Set:











bⁿ_i(u) :=^Y

n

i −M_i₋₁_/₂

u∆t/∆x +M_i₋₁_/₂, Bⁿ_i(u) := ^Y

n

i −m_i₋₁_/₂

u∆t/∆x +m_i₋₁_/₂, withm_i₋₁_/₂=min(Y_iⁿ₋₁,Y_iⁿ)andM_i₋₁_/₂=max(Y_iⁿ₋₁,Y_iⁿ).

Then the fluxes are defined by:

Y_iⁿ₊₁_/₂=







b_iⁿ(u) ifY_iⁿ₊₁≤bⁿ_i(u),

Y_iⁿ₊₁ ifbⁿ_i(u)<Y_iⁿ₊₁<B_iⁿ(u), Bⁿ_i(u) ifB_iⁿ(u)≤Y_iⁿ₊₁.

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C OMPARISON BETWEEN DL AND UPWIND SCHEMES

Sch éma Despr és - Lagouti ère Sch éma Upwind

Data u(x,y) =





−y x



 Ω = [−1,1]² ∆x= ∆y= ¹

200

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P ARTIAL O UTLINE

7 LDC

8 Data

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1D LDC METHOD FOR −∆Φ = f , WITH N EUMANN BC

Step 0

Coarse grid resolution (modified conjugate gradient)







Φ_H_,₀(x_i₊₁)−2Φ_H_,₀(x_i) + Φ_H_,₀(x_i₋₁)

H² =f_H(x_i), i={1, . . . ,N−1},

Φ⁰_H_,₀(x₀) =0, Φ⁰_H_,₀(x_N) =0;

Fine grid resolution with updated Dirichlet BC (CG)







Φ_h_,₀(x_i₊₁)−2Φ_h_,₀(x_i) + Φ_h_,₀(x_i₋₁)

h² =f_h(x_i), i={L₁, . . . ,L₂},

Φh,0(x_L₁) = ΦH,0(x_L₁), Φh,0(x_L₂) = ΦH,0(x_L₂).

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1D LDC METHOD FOR −∆Φ = f , WITH N EUMANN BC (2)

Stepk≥1

Coarse grid resolution with updated rhs







Φ_H_,_k(x_i+1)−2Φ_H_,_k(x_i) + Φ_H_,_k(x_i−1)

H² =f_H(x_i) +R_H_,_k₋₁(x_i), Φ⁰_H_,_k(x₀) =0, Φ⁰_H_,_k(x_N) =0,

with

R_H_,_s(x_i) =







Φ_h_,_s(x_i₊₁)−2Φ_h_,_s(x_i) + Φ_h_,_s(x_i₋₁)

H² −f_H(x_i), i={L₁, . . . ,L₂},

0, i={1, . . . ,L₁−1} ∪ {L₂+1, . . . ,N};

Fine grid resolution







Φ_h_,_k(x_i₊₁)−2Φ_h_,_k(x_i) + Φ_h_,_k(x_i₋₁)

h² =f_h(x_i), Φh,k(x_L₁) = ΦH,k(x_L₁), Φh,k(x_L₂) = ΦH,k(x_L₂).

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P ARTIAL O UTLINE

7 LDC

8 Data

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D ATA FROM K OTHE - R IDER TESTS

^off ^on

u(t,x,y) =2 cos ₂

πt T

sin(πx)sin(πy)×





−sin(πx)cos(πy) sin(πy)cos(πx)





Ω = [0,1]x[0,1]

∆x= ∆y=₁₂₈¹ T=14

u(t,x,y,z) =cos ₂

πt T

×







2 sin²(πx)sin(2πy)sin(2πz)

−sin²(πy)sin(2πx)sin(2πz)

−sin(2πx)sin(2πy)sin²(πz)







Ω = [0,1]x[0,1]x[0,1]

∆x= ∆y= ∆z=₆₄¹,T=6