time implicit-explicit high-order discontinuous galerkin method...
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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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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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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
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
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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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
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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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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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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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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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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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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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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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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
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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
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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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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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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