Mathematical and numerical aspects on seismic imaging using … · 2013. 9. 20. · Mathematical...

Mathematical and numerical aspects on seismicimaging using high-order schemes

Helene BarucqInria Bordeaux Sud-Ouest, EPC Magique 3D

Laboratoire de Mathematiques et de leurs Applications,Universite de Pau et des Pays de l’Adour, UMR CNRS 5132

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Seismic imaging

Oil exploration using numerical seismic reflection:

Non invasive process to get an image of the subsurface thatgives information on a possible oil deposit in the region ofinterest.

Simulations are based on data provided by acquisitioncampains.

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What does the acquisition campain give rise to?

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An example of seismogram

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Numerical seismic imaging based on velocity models

1 A first step: Seismic Tomography. Use of data collectedduring the acquisition campain to provide a velocity model

2 Using the reflected waves recorded during the acquisitioncampain.

Solve two waveequations persource

(a) One reproduces the propagation of asource inside the velocity model

(b) One reproduces the propagation of thereflected waves inside the domain: itconsists in retro-propagating thereflected waves.

3 An image is performed by cross-correlating (a) and (b):Imaging condition

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Consider that the reflected waves which are known from theacquisition campain provide a record of the real propagationmedium

The record is available only at the receiver positions

The imaging condition is given by:

I (x , y , z) =n∑

i=1

∫ T

0U i

s(x , y , z , t)R i (x , y , z , t)

where U is is the wave generated by the source and R i is

generated by the reflected waves.

Observe that both U is and R i must be computed at each point

of the grid. Two wave equations must then be solved.

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Observe that for the retropropagation step, the source that isemployed is given by :

SRP(t, x) =Nr∑

j=1

(Rj (T − t, rj ) ∗ δxj (x))) (1)

where rj , j = 1,Nr represents a receiver, Nr is the number ofreceivers and R is the reflected wave that has been recorded andgenerated by one of the sources Sl , l = 1,NS , NS being thenumber of sources. δ denotes the Dirac distribution.

In practice, sources do not explode at the same time.Sometimes they are detonated simultaneously along a line.

Numerically, we reproduce the physical experience byconsidering only one source at a time.

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At first step, one get an image of the subsurface given by a velocity model

It is compared with the empirical velocity model

If it does not fit with the numerical one, a second step needsto be performed again after the numerical velocity model hasbeen modified

and so on...until convergence

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Toy problem

Seismogram obtained duringthe acquisition campain

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Toy problem

Seismogram obtained duringthe acquisition campain

=⇒

Foretold velocity model aftersimulations

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Toy problem

Deduced initial guess

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Toy problem

Deduced initial guess

=⇒

Image

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A two-dimensional example

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RTM provides a robust way for producing accurate images ofthe subsurface but it is computationally intensive

RTM is based on the solution of a collection of waveequations in heterogeneous media

Advanced numerical methods are needed: acceleratecomputations and reduce occupation of memory

Parallel computing is mandatory

Each algorithm must be designed by always keeping in mindthat the performance of existing software packages must bekept at least

Conclusion: Seismic imaging is a research topic that requires togather researchers with different background: geophysicists,mathematicians, computer scientists.

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Some issues

Design of high-order numerical schemes to improve theaccuracy of the space approximation

Design of high-order time schemes to decrease the numberof iterations while keeping the level of acccuracy andcompatible with parallel computing

Design of imaging conditions based on a limited number ofcomputations according to the memory capacity

Design of effective boundary conditions allowing totruncate the computational domain

Design of efficient solvers for harmonic wave equations

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Joint works with

Design of high-order numerical schemes to improve theaccuracy of the space approximation: Henri Calandra, JulienDiaz, Florent Ventimiglia

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Joint works with


Design of high-order time schemes to decrease the number ofiterations while keeping the level of acccuracy and compatiblewith parallel computing: Henri Calandra, Julien Diaz,Florent Ventimiglia

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Joint works with


Design of high-order time schemes to decrease the number ofiterations while keeping the level of acccuracy and compatiblewith parallel computing

Design of imaging conditions based on a limited number ofcomputations according to the memory capacity: HenriCalandra, Julien Diaz, Jerome Luquel

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Joint works with




Use of new computing architectures: Henri Calandra, JulienDiaz, Lionel Boillot, George Bosilca

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Joint works with




Use of new computing architectures

Design of efficient solvers for harmonic wave equations:Mohamed Amara, Marie Bonasse-Gahot, HenriCalandra, Theophile Chaumont-Frelet, Julien Diaz,Elodie Estecahandy, Rabia Djellouli, Christian Gout,Stephane Lanteri

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Focus on


Part of Florent Ventimiglia thesis

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Current numerical methods for the full wave equation

Regarding the space discretization:

Finite differences : the most popular technique

Fast computations;

Easy implementation;

Not adapted in case of highly varying topography and whenthe characteristics of the medium are highly oscillating.

Finite Elements

Accurate representation of the topographyFlexibilityImplementation not always obviousComputations are a priori more time consuming, in particularwhen the solution is not given explicitely at each iteration

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The model : First Order Formulation

Elastic wave equationρ(x)

∂v(x, t)

∂t+ ∇ · σ(x, t) = 0 in Ω× [0,T ]

∂σ(x, t)

∂t+ C (x)ε (v(x, t)) = 0 in Ω× [0,T ]

where ρ is the density, σ is the stress tensor, ε is the strain tensorand C is the elasticity tensor.

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Its acousic version

Acoustic wave equationρ(x)

∂v(x, t)

∂t+ ∇p(x, t) = 0 in Ω× [0,T ]

1

µ(x)

∂p(x, t)

∂t+ ∇ · (v(x, t)) = 0 in Ω× [0,T ]

where ρ is the density, µ is the compressibility modulus.

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Finite element/difference formulation

Mv

dV

dt+KpP = 0

MpdP

dt+Kv V = 0

Then the time discretization must be addressed:

The structure of the mass matrices impacts on the choice ofthe scheme

Implicit/Explicit scheme?

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Discontinuous Galerkin Method

Flexibility, Local Aspect and Multithread Oriented

Hybrid Method : Finite volumes and finite elementscharacteristics

The solution is defined into each element as a polynomialapproximationNumerical fluxes are used over interfaces

Large variety of meshes and elements

hp-adaptive method

Size of elements (h)Degrees of elements (p)

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Notations

n± is the outward unit normal vector to the element K±;

Jump notation :

[[u]] =

u+n+ + u−n− if F ∈ Fi\∂Ωu.n if F ∈ ∂Ω

Average notation :

u =

u+ + u−

2if F ∈ Fi\∂Ω

u if F ∈ ∂Ω

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Approximation by finite elements

The space approximation is defined by:

Vh = v ∈ L2(Ω), v|K ∈ Pp(K ), ∀K ∈ Th

Ω is a convex domain ;

Th : associated mesh to Ω ;

Pp : polynomial space of degree ≤ p;

S. Delcourte, L. Fezoui and N. Glinsky-Olivier (2009)

A high-order discontinuous Galerkin method for the seismic wave propagation,ESAIM.

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Variational formulation

∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γ[[p w]]

∑K

∫K

∂p

∂tq =

∑K

∫K

v · ∇p −∑Γint

∫Γ[[v q]]

For p and v regular “enough” [[p]] = 0, if F ∈ Fi\∂Ω

[[v]] = 0, if F ∈ Fi\∂Ω∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γp [[w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γv [[q]]

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∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γ[[p w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γ[[v q]]


[[v]] = 0, if F ∈ Fi\∂Ω

∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γp [[w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γv [[q]]

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∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γ[[p w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γ[[v q]]


[[v]] = 0, if F ∈ Fi\∂Ω∥∥∥∥∥∥∥∥∥∑

K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w−∑Γint

∫Γp [[w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γv [[q]]

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We then obtain the following matricial problem,Mv

dV

dt+KpP = 0

MpdP

dt+Kv V = 0

which becomes for instance with the widely-used second orderLeap-Frog Scheme

MvVn+1 − Vn

∆t+KpPn+ 1

2 = 0

MpPn+ 3

2 − Pn+ 12

∆t+Kv Vn = 0

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Some comments

1 Even if the Leap-Frog scheme is used, the solution is givenexplicitely only if the matrices Mv and Mp are diagonal

2 Most of the finite element methods do not lead accurately todiagonal mass matrices but:

The mass matrix of spectral elements is diagonalThe mass matrix of DG elements is block-diagonal

3 Spectral elements are not easy to use for industrial projects:they are based on hexahedra

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High-Order Schemes

High-Order explicit time schemes : No need for matrix inversionbut CFL condition.

The Leap-Frog scheme is currently used: easy to implement,limited storage due to a one-step strategy

High-order schemes: consistent with high-order spacediscretization, the numerical dispersion is controlled

Multistep methods : Runge-Kutta, Adams-Bashworth...Intermediate stages

To account for memory ties, we have decided to focus onLax-Wendroff procedures i.e. single step time integration

ADER process and a new scheme

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Historical Background: ADER method

M. Kaser, M. Dumbser (2000)

High-Order numerical scheme both in time and space

First order formulation

Can be extended to Local Time Stepping (2007).

M. Dumbser and M. aser (2000)

An arbitrary High-Order discontinuous Galerkin method for elastic waves on unstructured meshes. Thetwo-dimensional isotropic case with external source terms,Geophysical J. Int., Vol. 142, pp. 000-000.

M. Dumbser and M. aser (2007)

An arbitrary High-Order discontinuous Galerkin method for elastic waves on unstructured meshes. Localtime stepping and p-adaptivity,Geophysical J. Int., Vol. 171, pp. 695-717.

M. A. Dablain (1986)

The application of high order differencing for the scalar wave equation,Geophysics, Vol. 1, pp. 51:54-56.

S. Delcourte, L. Fezoui and N. Glinsky-Olivier (1987)

A modified equation apporach to construct a fourth-orders methods for acoustic wave porpagation,SIAM J. Sci. Statist. Comput. 8:135-151.

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The method: Space Discretization

By rewritting the system asdV

dt= ApP

dP

dt= Av V

with Ap = −M−1v ×Mp and Av = −M−1

p ×Mv .

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The method: Time Discretization

Taylor expansion :

V (tn + ∆t)− Vn(tn)

∆t= ∂tV

(tn +

∆t

2

)+

∆t2

24∂3

t V

(tn +

∆t

2

)+ O(∆t4)

P(tn + 3∆t

2

)− Pn

(tn + ∆t

2

)∆t

= ∂tP (tn + ∆t) +∆t2

24∂3

t P (tn + ∆t) + O(∆t4)

∂tV = ApP

∂tP = Av V=⇒

∂3t V = ApAvApP

∂3t P = AvApAv V

Vn+1 − Vn

∆t= ApPn+ 1

2 + ∆t2

24ApAvApPn+ 1

2

Pn+ 32 − Pn+ 1

2

∆t= Av Vn +

∆t2

24AvApAv Vn

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Algorithmic Aspects

Algorithmic Aspects

Computation of

Q = ApPn+ 12

W = AvQ

Vn+1 = Vn + ∆t

(Q +

∆t2

24ApW

)

Computation of

W = Av Vn+1

Q = ApW

Pn+ 32 = Pn+ 1

2 + ∆t

(W +

∆t2

24AvQ

)

Two additional unknowns and three times more operations.

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Algorithmic Aspects

Algorithmic Aspects

Computation of

Q = ApPn+ 12

W = AvQ

Vn+1 = Vn + ∆t

(Q +

∆t2

24ApW

)Computation of

W = Av Vn+1

Q = ApW

Pn+ 32 = Pn+ 1

2 + ∆t

(W +

∆t2

24AvQ

)

Two additional unknowns and three times more operations.

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Algorithmic Aspects

Algorithmic Aspects

Computation of

Q = ApPn+ 12

W = AvQ

Vn+1 = Vn + ∆t

(Q +

∆t2

24ApW

)Computation of

W = Av Vn+1

Q = ApW

Pn+ 32 = Pn+ 1

2 + ∆t

(W +

∆t2

24AvQ

)Two additional unknowns and three times more operations.

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Computational Cost

4th order in time:

3 times more costly

The Leap-Frog CFL is multiplied by 2.8

Leap-Frog vs ADER-4th

2.8

3= 0.93

6th order in time:

5 times more costlyThe Leap-Frog CFL is multiplied by 3.7


3.7

5= 0.74

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Computational Cost

4th order in time:

3 times more costlyThe Leap-Frog CFL is multiplied by 2.8Leap-Frog vs ADER-4th

2.8

3= 0.93

6th order in time:

5 times more costlyThe Leap-Frog CFL is multiplied by 3.7


3.7

5= 0.74

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Computational Cost

4th order in time:


2.8

3= 0.93

6th order in time:

5 times more costly

The Leap-Frog CFL is multiplied by 3.7


3.7

5= 0.74

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Computational Cost

4th order in time:


2.8

3= 0.93

6th order in time:


3.7

5= 0.74

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Historical Background : ∇p method

C. Agut, J. Diaz and A. Ezziani (2011)

High order numerical schemes both in space and in time usingIPDG method.

Second order Formulation

C. Agut, J. Diaz and A. Ezziani (2012)

High-Order Schemes Combining the modified Equation Approach and Discontinuous GalerkinApproximations for the Wave Equation,Commun. Comput. Phys., 11 (2012), pp. 691-708.

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The method : time discretization

By rewritting the problem as∂v(x, t)

∂t= − ∇p(x, t) in Ω× [0,T ]

p(x, t)

∂t= − ∇ · (v(x, t)) in Ω× [0,T ]

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The method : time discretization

Using Taylor expansion on the continuous variables :

v(x, tn + ∆t)− v(x, tn)

∆t= ∂tv

(x, tn +

∆t

2

)+

∆t2

24∂3

t v

(x, tn +

∆t

2

)+ O(∆t4)

p(x, tn + 3∆t

2

)− p

(x, tn + ∆t

2

)∆t

= ∂tp (x, tn + ∆t) +∆t2

24∂3

t p (x, tn + ∆t) + O(∆t4)

∂tv = −∇p

∂tp = −∇ · v=⇒

∂3t v = −∇∇ ·∇p

∂3t p = −∇ ·∇∇ · v

vn+1 − vn

∆t= −∇pn+ 1

2 − ∆t2

24∇∇ ·∇pn+ 1

2

pn+ 32 − pn+ 1

2

∆t= −∇ · vn+1 −∇ ·∇∇ · vn+1

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Space discretization

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∑K

∫K

vn+1 − vn

∆t·w =

∑K

∫Kpn+ 1

2∇ ·w

− ∆2

24

∑K

∫Kpn+ 1

2∇ ·∇∇ ·w

−∑Γint

∫Γ[[p w]]

− ∆t2

24

∑Γint

∫Γ[[∇ ·∇p w]]

+∆t2

24

∑Γint

∫Γ[[∇p∇w]]− ∆t2

24

∑Γint

∫Γ[[p∇ · ∇w]]

2nd order terms.

4thorder terms.

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∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∑K

∫K

vn+1 − vn

∆t·w =

∑K

∫Kpn+ 1

2∇ ·w− ∆2

24

∑K

∫Kpn+ 1

2∇ ·∇∇ ·w

−∑Γint

∫Γ[[p w]]− ∆t2

24

∑Γint

∫Γ[[∇ ·∇p w]]

+∆t2

24

∑Γint

∫Γ[[∇p∇w]]− ∆t2

24

∑Γint

∫Γ[[p∇ · ∇w]]

2nd order terms.

4thorder terms.

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For p regular “enough”[[p]] = 0, if F ∈ Fi\∂Ω

[[∇ · p]] = 0, if F ∈ Fi\∂Ω

[[∇∇ · p]] = 0, if F ∈ Fi\∂Ω

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∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∑K

∫K

vn+1 − vn

∆t·w =

∑K

∫Kpn+ 1

2∇ ·w−∆2

24

∑K

∫Kpn+ 1

2∇ ·∇∇ ·w

−∑Γint

∫Γp [[w]]−∆t2

24

∑Γint

∫Γ∇ ·∇p [[w]]

+∆t2

24

∑Γint

∫Γ∇p [[∇w]]−

∑Γint

∫Γp [[∇ · ∇w]]

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We then obtain the following matricial system,Mv

dV

dt+ BpP = 0

MpdP

dt+ Bv V = 0

No additional cost providing the storage of B at thebeginning

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Numerical results: 1D

The length of the domain is 6 m with 60 s of simulation time.

The coarsest space step is 0.2 m and the finest is 0.05 m

We consider periodic boundary conditions

Initial data is such that

U(x , t) = (x − x0− t)e(−

(4π2

r0(x−x0−t)2

)

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P3 in Space and 4th-order in time : Convergence study

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P3 in Space and 4th-order in time : Number of operations

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P3 in Space and 4th-order in time : Number of unknowns

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P5 in Space and 4th-order in time : Convergence study

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P5 in Space and 4th-order in time : Number of operations

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P5 in Space and 4th-order in time : Number of unknowns

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Number of operations LF vs ADER vs ∇

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Number of unknowns: LF vs ADER vs ∇

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Using ADER or ∇ scheme for RTM?

Do not forget the memory ties...

If B matrices are computed once at the beginning, LF, ADERand ∇ induce the same costs. But ∇ is the more accurate.

If B matrices are computed at each iteration, for a givenaccuracy level, ∇ requires more than twice less unknowns

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Numerical results: 2D

The domain is a square 1000 m × 1000 m with simuationtime equal to 50 s.

The coarsest space step is 100 m and the finest is 6.75 m

We consider periodic boundary conditions

Velocity c = 1500m/s

Initial data is such that

U(x , y , t) = sin (2π (2x + 2y − 4ct))

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Using P3 elements

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Using P4 elements

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Using P5 elements

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Using P6 elements

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Using P7 elements

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Summary

1.10−2 < Error Implement. Memory Comput. costsLF 1 1 1

ADER 2 2 2

Nabla 3 3 3

1.10−4 <Error< 1.10−3 Implement. Memory Comput. costsLF 1 2 2

ADER 2 2 1Nabla 3 1 2

Error < 1.10−4 Implement. Memory Comput. costsLF 1 3 3

ADER 2 2 2

Nabla 3 1 1

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Work in progress

Validation on industrial benchmarks

Dispersion analysis

and...

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Numerical Noise and Penalization

Seismogramms have been obtained with P1

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Seismogramms obtained with P2

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Seismogramms obtained with P3

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Solution : adding a penalization parameter

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

∑K

∫K

∂v

∂t·w =

∑K

∫K

p∇ ·w −∑Γint

∫Γp [[w]]

+∑Γint

∫Γα[[p]] [[w]]

∑K

∫K

∂p

∂tq =

∑K

∫K


∫Γv [[q]]

−∑Γint

∫Γα[[p]] [[w]]

Contrary to the second order formulation, the penalizationparameter does not depend neither on the geometry of the cells,nor on the polynomial degree of the approximation.

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Numerical noise and Penalization


α = 0 α = 0.5

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α = 0 α = 0.5

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α = 0 α = 0.5

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The CFL is halved

Dispersion analysis is needed to explain the spurious modes

Penalization seems to restore the order of convergence

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Thanks for your attention!

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