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Page 1: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Time implicit-explicit high-order discontinuousGalerkin method with reduced evaluation costs

Florent Renac1, Frédéric Coquel2 & Claude Marmignon1

1ONERA/DSNA/NUMF2CMAP, Ecole Polytechnique

CEMRACSJuly 26th, 2011CIRM, Marseille

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Page 2: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Motivation and goal

DGM:

compact stencil: well-suited for unstructured meshes, algorithmparallelization, BC application.time explicit discretization: strongest CFL restriction associated toparabolic termtime implicit discretization: high computational cost induced by thelarge number of DOFs in practical app. (e.g. 3D Navier-Stokes eq.)

Aim:

ecient implicit-explicit procedure for the DGM:

partial uncoupling of DOFs in neighbouring elements

application to solution of nonlinear 2nd order PDEnumerical experiments with BR2 scheme

[Cockburn & Shu '01, Kroll et al. '10, Bassi et al. '97]

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Page 3: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Outline

1 Short introduction to DGM

2 Equations and Numerical ApproachNonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

3 Numerical experiments2D steady convection-diusion equation2D unsteady convection-diusion equation

4 Summary and outlook

[email protected] Implicit-explicit DG with reduced evaluation costs 3/43

Page 4: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Outline

1 Short introduction to DGM

2 Equations and Numerical ApproachNonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

3 Numerical experiments2D steady convection-diusion equation2D unsteady convection-diusion equation

4 Summary and outlook

[email protected] Implicit-explicit DG with reduced evaluation costs 4/43

Page 5: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Advection-reaction equation

Consider the linear scalar advection-reaction equation for u(x) ∈ R:

∇ · (cu) + µu = 0 in Ω ⊂ Rd (1a)

u = uD on Γ−D = ΓD ∪ Γ− (1b)

with µ ∈ L2(Ω) s.t. µ(x) +∇ · c(x) ≥ µ0 > 0, c ∈ L∞(Ω), uD ∈ L2(Γ−D ) andinow boundary:

Γ− = x ∈ ∂Ω : c(x) · n(x) < 0The exact solution u is sought in the function space

V = v ∈ L2(Ω) : ∇ · (cv) + µv ∈ L2(Ω)

Ω

ΓD

nc

Γ+

Γ-

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Page 6: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Space discretization

shape-regular subdivision of Ω into N cells (simplices): Ωh = κ s.t.Ω =

⋃κ∈Ωh

κ

sets of internal and boundary faces:

Ei = e ∈ Eh : e∩∂Ωh = ∅, Eb = e ∈ Eh : e ∈ ∂Ωh, E−D = ED∪E−

function space of discontinuous polynomials:

Vph = v ∈ L2(Ωh) : v |κ ∈ Pp(κ), ∀κ ∈ Ωh,

let (φ1, . . . , φNp ) be a basis of Pp(κ);

look for an approximate solution uh ∈ Vph of (1):

uh(x) =

Np∑l=1

φl (x)V lκ ∀x ∈ κ

Remark: function space dimension: dimVph = N × Np = N ×d∏i=1

p + i

i

Remark: non-conforming FE method: Vph 6⊂ [email protected] Implicit-explicit DG with reduced evaluation costs 6/43

Page 7: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Space discretization (cont.)

multiply (1a) by a test function vh ∈ Vph and integrate over a cell κ, thenintegrate by parts:

−∫κ

uhc · ∇vh dx +

∮∂κ

vhuhc · n ds +

∫κ

µuhvh dx = 0

sum over all cells and apply BCs:

−∫

Ωh

uh(c ·∇hvh−µvh) dx+∑κ∈Ωh

∮∂κ\E−D

vhuhc ·nds = −∫E−D

vhuDc ·nds

replace the physical ux (c · n)uh by a numerical ux h(u+h , u

−h , n) on Ei ;

κ+

κ-uh+uh

-

n+=n

e

n-

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Page 8: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Space discretization (cont.)

introducing [[vh]] = v+h − v−h , we get∑

κ∈Ωh

∮∂κ\Eb

v+h h(u+

h , u−h , n) ds =

∫Ei

[[vh]]h(u+h , u

−h , n) ds

one obtains the variational form: nd uh ∈ Vph s.t.

Bh(uh, vh) = `h(vh), ∀vh ∈ Vph ” (2)

with bilinear and linear forms:

Bh(uh, vh) = −∫

Ωh

uh(c · ∇hvh − µvh) dx +

∫Ei

[[vh]]h(u+h , u

−h , n) ds

+

∫Eb\E

−D

v+h u

+h (c · n) ds

`h(vh) = −∫E−D

v+h uD(c · n) ds

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Page 9: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Consistency

Problem (2) is said to be consistent if the exact solution u ∈ V satises

Bh(u, vh) = `h(vh), ∀vh ∈ Vph

Lemma

Problem (2) is consistent i. the numerical ux is consistent:h(u, u, n) = u(c · n).

Proof: integrate by parts the volume integral in Bh(u, vh)− `h(vh) and replace uh by u, oneobtains∫

Ωh

vh((((((

(∇ · (cu) + µu) dx +

∫Ei

v+h

(h(u, u, n+)− u(c · n+)) ds

+

∫Ei

v−h

(h(u, u, n−)− u(c · n−)) ds

E−D

u(c · n+)v+hds +

E−D

uD(c · n+)v+hds = 0

Hence on internal faces we get h(u, u, n) = u(c · n).

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Page 10: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Conservation

Problem (2) is said to be conservative if the approximate solution uh satises∫Eb\E

−D

uh(c · n+) ds +

∫E−D

uD(c · n+) ds +

∫Ωh

µuhdx = 0 (3)

Lemma

Problem (2) is conservative i. the numerical ux is conservative:h(u+

h , u−h , n

+) = −h(u−h , u+h , n

−).

Proof: set vh ≡ 1 in (2), one obtains

−∫

Ωh

uhc ·∇h1− µuhdx +

∫Ei

1× h(u+h, u−h, n

+) + 1× h(u−h, u

+h, n−) ds

+

∫Eb\E

−D

1× u+h

(c · n) ds

+

∫E−D

1× uD(c · n) ds = 0

then, substract (3) which proves the lemma.

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

FE analysis for the upwind ux

DG-norm:

|||vh|||2 = µ0‖vh‖2L2(Ω) +1

2

∫Eb

|c · n|v2h ds +1

2

∫Ei|c · n|[[vh]]2 ds, ∀vh ∈ Vph

Galerkin orthogonality:

Bh(u − uh, vh) = 0, ∀vh ∈ Vph

coercivity:Bh(vh, vh) ≥ |||vh|||2, ∀vh ∈ Vph

continuity:

Bh(z , vh) ≤ |||z |||? × |||vh|||, ∀vh ∈ Vph , z ∈ V ⊕ Vp

h

stability:

|||vh||| ≤ supwh∈V

ph\0

Bh(vh,wh)

|||wh|||, ∀vh ∈ Vph

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Page 12: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Convergence analysis for the upwind ux

Theorem (a priori error estimate)

Let uh be solution to problem (2) and u ∈ Hp+1(Ωh), then ∃ C > 0 s.t.

|||u − uh||| ≤ Chp+ 12 |u|Hp+1(Ωh)

Theorem (exponential convergence)

Let u be elementwise analytic on Ωh, then ∃ C = C(u) > 0, dκ > 0 s.t.

|u|Hs (κ) ≤ C(u)d sκs!√|meas(κ)|, ∀s ≥ 0,

and suppose that uh is solution to problem (2), then ∃ C ′ = C ′(u, c, µ0) > 0and χκ > 0 s.t.

|||u − uh|||2 ≤ C ′∑κ∈Ωh

e−2p(χκ+

| ln hκ|√1+d2κ

)|meas(κ)|

[Houston et al. '02]

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Page 13: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Numerical experiments

rotational advection c = (x , 1− y)> and µ ≡ 0 with smooth inlet BC

Γ-

ΓD

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Numerical experiments (cont.)

rotational advection c = (x , 1− y)> and µ ≡ 0 with smooth inlet BC(cont.)

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Page 15: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Numerical experiments (cont.)

rotational advection c = (x , 1− y)> and µ ≡ 0 with discontinuous inletBC

Γ-

ΓD

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Literature

[1] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unied analysis of discontinuous Galerkin methods forelliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 17491779.

[2] B. Cockburn, G. E. Karniadakis & C.-W. Shu (eds.), Discontinuous Galerkin methods. Theory, computationand applications, Lecture Notes in Computational Science and Engineering, 11. Springer-Verlag, Berlin, 2000.

[3] B. Cockburn and C. W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems,J. Sci. Computing, 16 (2001), pp. 173261.

[4] A. Ern & J.-L. Guermond, Theory and Practice of Finite Elements, vol. 159 of Applied Mathematical Series,Springer, New York, 2004.

[5] R. Hartmann, Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods. In H.Deconinck (ed.), VKI LS 2008-08: 35th CFD/ADIGMA course on very high order discretization methods, Oct.13-17, 2008. Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium (2008).

[6] J. S. Hesthaven & T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, andApplications. Springer, 2007.

[7] P. Huston, C. Schwab & E. Süli, Discontinuous hp-nite element methods for advection-diusion-reactionproblems, SIAM J. Numer. Anal., 39 (2002), pp. 21332163.

[8] G. Kanschat, Discontinuous Galerkin Methods for Viscous Incompressible Flow. Teubner Research, 2007.

[9] N. Kroll, H. Bieler, H. Deconinck, V. Couaillier, H. van der Ven & K. Sorensen (ed.), ADIGMA - A EuropeanInitiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications, vol. 113of Notes on numerical uid mechanics and multidisciplinary design, Springer, 2010.

[10] B. Riviere, Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory andImplementation. SIAM, 2008.

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Page 17: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Outline

1 Short introduction to DGM

2 Equations and Numerical ApproachNonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

3 Numerical experiments2D steady convection-diusion equation2D unsteady convection-diusion equation

4 Summary and outlook

[email protected] Implicit-explicit DG with reduced evaluation costs 17/43

Page 18: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Nonlinear advection-diusion equationmodel problem

Consider the scalar advection-diusion equation for u ∈ R:

∇ · (c(x)u)−∇ · (B(x, u)∇u) = s(x) in Ω ⊂ Rd (4)

with c(x) ∈ Rd and B(x, u) ∈ Rd×d a nonlinear function of u and BCs on∂Ω = ΓD ∪ ΓN :

u = uD on ΓD

∇u · n = gN on ΓN

We solve (4) with a fast time marching method:

ut +∇ · (c(x)u)−∇ · (B(x, u)∇u) = s(x) in Ω× (0,∞) (5)

and ICu(x, 0) = u0(x) in Ω

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Page 19: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

entropy solution

U(u) is an entropy function for (5) if

Uuu > 0U satises

B(x, u) = UuuC(x)

where C ∈ Rd×d is a positive denite matrix

Introduce the change of variable u(v) s.t.

Uuuuv = 1

one obtains the following problem for v :

u(v)t +∇ · (c(x)u(v))−∇ · (C(x)∇v) = s(x) in Ω× (0,∞)

[Degond et al. '97]

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Page 20: Time implicit-explicit high-order discontinuous Galerkin method …smai.emath.fr/cemracs/cemracs11/doc/talk-Renac.pdf · 2011-07-26 · Time implicit-explicit high-order discontinuous

THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Space discretizationvariational form

Partition of Ω into N simplicies:

Ωh = κ s.t. Ω =⋃κ∈Ωh

κ

Function space of discontinuous polynomials:

Vph

= ϕ ∈ L2(Ωh) : ϕ|κ ∈ Pp(κ), ∀κ ∈ Ωh,

with for d = 2:

Pp(κ) = ϕ ∈ L2(κ) : ϕ(x , y) =∑

0≤k+l≤pαlxky l

We look for a numerical approximation of the solution vh ∈ Vp

h:

vh(x, t) =

Np∑l=1

φl (x)V lκ(t) ∀x ∈ κ, t ∈ (0,∞)

Remark: number of DOFs per discretization element: Np = N ×d∏i=1

p + i

i

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Space discretizationupwind discretization ux

Numerical approximation of the weak formulation: nd vh ∈ Vp

hs.t. ∀φ ∈ Vp

h∫Ωh

φu(vh)tdx −∫

Ωh

u(vh)c · ∇hφdx +∑e∈Ei

∫e

[[φ]]hc ds

+

∫Ωh

(Cθh) · ∇hφdx−∑e∈Ei

∫e

[[φ]]hv ds −∫

Ωh

φs(x)dx = 0

Upwind discretization ux

hc = u(vh)c · n +α

2[[vh]]

α = max|uv (v)c · n| : v = v±h

Jump and average operators

[[vh]] = v+h− v−

h

vh =1

2(v+h

+ v−h

)

κ+

κ-uh+uh

-

n+=n

e

n-

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Space discretizationBR2 scheme

Numerical approximation of the weak formulation: nd vh ∈ Vp

hs.t. ∀φ ∈ Vp

h∫Ωh

φu(vh)tdx −∫

Ωh

u(vh)c · ∇hφdx +∑e∈Ei

∫e

[[φ]]hc ds

+

∫Ωh

(Cθh) · ∇hφdx−∑e∈Ei

∫e

[[φ]]hv ds −∫

Ωh

φs(x)dx = 0

lifting operators

θh = ∇hvh + Rh

hv = C∇hvh + reh · n

with

Rh ,∑e∈∂κ

reh∫

κ+∪κ−φreh dx = −

∫e

φ[[vh]]n ds

[Bassi et al. '97][email protected] Implicit-explicit DG with reduced evaluation costs 22/43

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Implicit-explicit time discretizationbackward Euler scheme

Convective terms: explicit time discretization

Diusive terms: backward Euler scheme

A(V(n+1) − V(n)) + R(V(n)) = 0

with V(n) = V(n∆t) and

residual vector:

R(V(n)) = Lc(V(n)) + LvV

(n) + Ls

implicit matrix:

A =1

∆tM

(n) + Lv

mass matrix M(n) = diag(M1(n), . . . ,MN

(n)):

(Mj(n))kl =

∫κj

φkφluv (v(n)h )dx, 1 ≤ j ≤ N, 1 ≤ k, l ≤ Np

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Algorithm simplicationevaluation cost reduction (1/2)

Basics:

implicit matrix coecients represent coupling between DOFs

coupling appears through numerical uxes

Lowering coupling between modes by:

1 low order problem resolution: modes s.t. 0 ≤ q ≤ ps2 reconstruction of higher modes s.t. ps < q ≤ p

1D example for p = 2 and ps = 0:

x

vh

xj+½xj-½

Ω+

+

+

=

+

+

=

Ω-

(a) full coupling

x

vh

xj+½xj-½

Ω+ Ω-

(b) partial coupling

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Algorithm simplicationevaluation cost reduction (2/2)

Resolution algorithm:

1 low order problem resolution:

~A(~V(n+1) − ~V(n)) = −~R(V(n))

with ~V = (V0, . . . ,Vps )

2 local reconstruction of higher modes:

Vq(n+1)

= Vq(~V(n+1),Vq(n)

) , ps < q ≤ p

1D example for p = 2 and ps = 0:

x

vh

xj+½xj-½

Ω+ Ω-

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Algorithms comparisonoperation count

FULL method: A is a square matrix of size N × Np:

matrix inversion is a O(N2N2

p ) process

SIMP method: ~A is a square matrix of size N × Nps :

matrix inversion is a O(N2N2

ps ) processhigher DOFs reconstruction is a O(2NNp(Np − Nps )) process

FLOPs reduction for matrix inversion:

N2N2

p

NN2ps + 2NNp(Np − Nps )

∼( Np

Nps

)2

when N large

p 1 2 2 3 3 3 4 4 4 4ps 0 0 1 0 1 2 0 1 2 3(

NpNps

)2 1D 4 9 2.25 16 4 1.78 25 6.25 2.78 1.562D 9 36 4 100 11.1 2.78 225 25 6.25 2.25

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Nonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

Algorithms comparisonVon Neumann analysis for the 1D linear advection-diusion equation with periodic BCs

Amplication matrix eigenspectra (p = 1; σ = c∆th

= 10, 20 and 30; Reh = chν = 0.1):

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

(a) FULL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

(b) SIMP0

FULL: full implicit matrix

SIMP0: simplied implicit matrix with ps = 0

Necessary condition of stability for both methods:

σReh ≤ 2 and Reh ≤ 6

Renac, Marmignon & Coquel, Time implicit HO DG method with reduced

evaluation cost, submitted to SIAM J. Sci. Comput., Dec. 2010 (accepted).

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

Outline

1 Short introduction to DGM

2 Equations and Numerical ApproachNonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

3 Numerical experiments2D steady convection-diusion equation2D unsteady convection-diusion equation

4 Summary and outlook

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationmodel problem

∇ · (cu)−∇ · (B(u)∇u) = s(x) in Ω = (0, 1)2

u = 0 on ∂Ω

with c = (1, 1)>, B(u) = ν(eu − 1)I with ν > 0

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.38

0.3

0.2

0.02

0.1

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationerror analysis

1 ≤ p ≤ 4; σ = 10; Reh = 0.1

h

||u-u

h|| 2

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110-8

10-7

10-6

10-5

10-4

10-3

10-2DG(p) FULLDG(p) SIMP

h2

h4

h5

h3

p=1

p=2

p=3

p=4

FULL: full implicit matrix

SIMP: simplied implicit matrix

[Arnold et al. '02.]

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationconvergence analysis (1/2)

1 ≤ p ≤ 3; σ = 10; Reh = 0.1

iteration

||u-u

h|| 2

0 50 100 150 20010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FULL (p=1, Re h=0.1)SIMP0 (p=1, Reh=0.1)FULL (p=2, Re h=0.1)SIMP0 (p=2, Reh=0.1)FULL (p=3, Re h=0.1)SIMP0 (p=3, Reh=0.1)

CPU time [s]

||u-u

h|| 2

0 50 100 150 200 250 30010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FULL (p=1, Re h=0.1)SIMP0 (p=1, Reh=0.1)FULL (p=2, Re h=0.1)SIMP0 (p=2, Reh=0.1)FULL (p=3, Re h=0.1)SIMP0 (p=3, Reh=0.1)

FULL: full implicit matrix

SIMP0: simplied implicit matrix with ps = 0

speedup =CPU time(FULL)

CPU time(SIMP)

p 1 2 3speedup 1.1 2.6 4.0

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationconvergence analysis (2/2)

p = 4; σ = 10; Reh = 0.1

iteration

||u-u

h|| 2

0 100 200 300 40010-10

10-8

10-6

10-4

10-2

100

FULL (p=4, Re h=0.1)SIMP0 (p=4, Reh=0.1)SIMP1 (p=4, Reh=0.1)SIMP2 (p=4, Reh=0.1)

CPU time [s]

||u-u

h|| 2

0 200 400 600 800 1000 1200 140010-10

10-8

10-6

10-4

10-2

100

FULL (p=4, Re h=0.1)SIMP0 (p=4, Reh=0.1)SIMP1 (p=4, Reh=0.1)SIMP2 (p=4, Reh=0.1)

FULL: full implicit matrix

SIMP0: simplied implicit matrix with ps = 0

SIMP1: simplied implicit matrix with ps = 1

SIMP2: simplied implicit matrix with ps = 2

p / ps 4 / 1 4 / 2speedup 2.3 4.8

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationmesh size eects

p = 3; ∆t = 6× 10−3

iteration

||u-u

h|| 2

0 100 200 300 400 50010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FULL (Re h=0.6, N=200)SIMP0 (Reh=0.6, N=200)FULL (Re h=0.3, N=800)SIMP0 (Reh=0.3, N=800)FULL (Re h=0.2, N=1800)SIMP0 (Reh=0.2, N=1800)FULL (Re h=0.15, N=3200)SIMP0 (Reh=0.15, N=3200)

CPU time [s]

||u-u

h|| 2

0 500 1000 150010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FULL (Re h=0.6, N=200)SIMP0 (Reh=0.6, N=200)FULL (Re h=0.3, N=800)SIMP0 (Reh=0.3, N=800)FULL (Re h=0.2, N=1800)SIMP0 (Reh=0.2, N=1800)FULL (Re h=0.15, N=3200)SIMP0 (Reh=0.15, N=3200)

FULL: full implicit matrix

SIMP0: simplied implicit matrix with ps = 0

p 3 3 3 3N 200 800 1800 3200speedup 2.0 3.0 6.9 9.4

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationspeedup of the residuals

iteration0 50 100 150 200 250

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

FULL (p=2, Re h=0.1) - ||u-u h||2

FULL (p=2, Re h=0.1) - ||R(uh(n))||2

SIMP0 (p=2, Reh=0.1) - ||u-u h||2

SIMP0 (p=2, Reh=0.1) - ||R(uh(n))||2

CPU time [s]0 50 100

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

FULL (p=2, Re h=0.1) - ||u-u h||2

FULL (p=2, Re h=0.1) - ||R(uh(n))||2

SIMP0 (p=2, Reh=0.1) - ||u-u h||2

SIMP0 (p=2, Reh=0.1) - ||R(uh(n))||2

FULL: full implicit matrixSIMP0: simplied implicit matrix with ps = 0

p 1 2 3 4ps 0 0 0 2speedup 1.1 2.6 4.0 4.8speedup(res.) 1.2 2.5 4.5 5.2

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D steady convection-diusion equationeects of mesh aspect ratio (AR)

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) AR = 16, N = 800

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1uh

1.151.050.950.850.750.650.550.450.350.250.150.05

(b) p = 4

AR 8 16 32 64 128 8 16 32 64 128p 2 2 2 2 2 4 4 4 4 4ps 1 1 0 0 0 3 3 3 3 2speedup 1.6 1.6 2.2 2.1 2.0 2.8 2.9 2.9 3.2 4.1

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D unsteady convection-diusion equationmodel problem

ut +∇ · (cu)−∇ · (B(u, x)∇u) = s(x, t) in Ω× (0,T ]

u(x , y , 0) = u0(x) in Ω

u(0, y , t) = u(1, y , t) ∀y ∈ [0, 1], t ∈ (0,T ]

u(x , 0, t) = u(x , 1, t) ∀x ∈ [0, 1], t ∈ (0,T ]

with c = (c, 0)>; B(u, x) = ν(eu − 1)C(x), ν > 0 and C(x) = 1

4(3 + cosπy)I

Exact solution:

u(x , y , t) = exp(e

(x− 12 +cT )2+(y− 1

2 )2

R2−4πνt

)− 1

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D unsteady convection-diusion equationpseudo-time stepping method

Add a pseudo-time derivative:

vτ + u(v)t +∇ · (cu(v))−∇ · (C(x)∇v) = s(x, t)

Space-time discretization:

A∆V(n+1,m+1) = −R(V(n+1,m))− 1

∆t

(U(v

(n+1,m)h )− U(v

(n)h ))

where V(n+1) = limm→∞ V(n+1,m)

Implicit matrix:

A =1

∆τN +

1

∆tM

(n+1,m) + Lv

Diagonal mass matrix N = diag(N1, . . . ,NN):

(Nj)kl =

∫κ

φkφldx

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D unsteady convection-diusion equationerror analysis

1 ≤ p ≤ 3; Reh = 0.1; T = 2.5

parameters FULL method SIMP methodp σ ‖u − uh‖2 order ps ‖u − uh‖2 order1 10.0 0.34570E − 02 − 0 0.34568E − 02 −1 5.0 0.15807E − 02 1.13 0 0.15806E − 02 1.131 2.5 0.69140E − 03 1.19 0 0.69137E − 03 1.191 1.25 0.27143E − 03 1.35 0 0.27144E − 03 1.352 10.0 0.34570E − 02 − 0 0.34567E − 02 −2 5.0 0.15807E − 02 1.13 0 0.15806E − 02 1.132 2.5 0.69142E − 03 1.19 0 0.69147E − 03 1.192 1.25 0.27147E − 03 1.35 0 0.27174E − 03 1.353 10.0 0.34570E − 02 − 0 0.34578E − 02 −3 5.0 0.15807E − 02 1.13 0 0.15810E − 02 1.133 2.5 0.69142E − 03 1.19 0 0.69173E − 03 1.193 1.25 0.27147E − 03 1.35 0 0.27171E − 03 1.35

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

2D steady convection-diusion equation2D unsteady convection-diusion equation

2D unsteady convection-diusion equationconvergence acceleration (σ = 1; T = 2.5)

Reh = 2× 10−3 Reh = 10−2 Reh = 10−1 Reh = 1p N ps speedup ps speedup ps speedup ps speedup

1 200 0 2.1 0 1.5 0 1.6 0 1.61 800 0 1.8 0 1.4 0 1.3 0 1.21 1800 0 1.7 0 1.2 0 1.3 0 1.11 3200 0 1.5 0 1.2 0 1.2 0 1.12 200 0 1.6 0 1.2 0 1.2 1 1.32 800 0 1.4 0 1.2 0 1.3 1 1.42 1800 0 1.6 0 1.3 0 1.2 1 1.32 3200 0 2.5 0 1.9 0 1.7 1 1.43 200 0 2.2 1 1.7 1 1.6 1 1.53 800 0 2.7 1 1.8 1 1.8 1 1.43 1800 0 5.2 1 3.7 1 2.8 1 1.93 3200 0 11.1 1 4.0 1 2.8 1 1.84 200 2 1.4 2 1.3 2 1.6 2 1.74 800 2 4.5 2 4.4 2 5.6 2 3.94 1800 2 7.1 2 7.4 2 7.3 2 9.44 3200 2 5.6 2 13.3 2 5.7 2 3.9

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Outline

1 Short introduction to DGM

2 Equations and Numerical ApproachNonlinear advection-diusion equationEntropy solutionSpace discretizationImplicit-explicit procedure

3 Numerical experiments2D steady convection-diusion equation2D unsteady convection-diusion equation

4 Summary and outlook

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Summary and outlook

Derived a fast implicit-explicit time discretization for high-order DG method:

partial mode uncoupling in neighbouring elementsimplicit treatment for low order modeslocal reconstruction of high order modes

Application to nonlinear 2D scalar equations:

convergence accelerationspeedup increases with p and Nsimilar results for steady and unsteady ow problems

Outlook:

independent of the DG method (successfully applied to BR1 scheme)extension to systems of conservation laws

[Bassi & Rebay, '97], [Dairay, MSc Thesis, '10]

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Thank you for your attention

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THE FRENCH AEROSPACE LAB

Short introduction to DGMEquations and Numerical Approach

Numerical experimentsSummary and outlook

Bibliography

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unied analysis of discontinuous Galerkin methods forelliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 17491779.

F. Bassi and S. Rebay, A high-order accurate discontinuous nite element method for the numerical solution of thecompressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267279.

F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti and M. Savini, A High-order accurate discontinuous nite elementmethod for inviscid and viscous turbomachinery ows, In proceedings of the 2nd European Conference onTurbomachinery Fluid Dynamics and Thermodynamics, R. Decuypere, G. Dibelius (eds.), Antwerpen, Belgium,1997.

B. Cockburn and C. W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J.Sci. Computing, 16 (2001), pp. 173261.

T. Dairay, Développement et évaluation d'une méthode implicite à coût réduit appliquée aux schémas de typeGalerkin discontinu, MSc Thesis, ONERA, 2010.

P. Degond, S. Génieys and A. Jüngel, Symmetrization and entropy inequality for general diusion equations, C. R.Acad. Sci., 325 (1997), pp. 963968.

N. Kroll, H. Bieler, H. Deconinck, V. Couaillier, H. van der Ven & K. Sorensen (ed.), ADIGMA - A EuropeanInitiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications, vol. 113of Notes on numerical uid mechanics and multidisciplinary design, Springer, 2010.

F. Renac, C. Marmignon & F. Coquel, Time implicit high-order discontinuous Galerkin method with reducedevaluation cost, submitted to SIAM J. Sci. Comput., Dec. 2010 (accepted).

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