E cient High Order DG Method for Time-domain Maxwells ... · spatial discretizations preserving the...
Transcript of E cient High Order DG Method for Time-domain Maxwells ... · spatial discretizations preserving the...
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Efficient High Order DG Method for Time-domainMaxwells Equations
Christoph Lehrenfeld, Joachim Schoberl
Center for Computational Engineering Sciences (CCES)RWTH Aachen University
MAFELAP 2009, June 11, 2009
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content
1 spatial discretizationDG formulationdissipative approachnon-dissipative stabilization
2 the complete schemeglobal time-steppingcomputational aspects
3 local time-steppingdefinition of a local time-stepping schemeimplementation aspectsnumerical examples
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 2 / 30
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spatial discretization
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 3 / 30
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Maxwell Equations
in an isotropic, non-conducting medium
ε∂E
∂t= curlH
ν∂H
∂t= −curlE
+ boundary conditions
model problem: wave equation
∂u
∂t= div(v)
∂v
∂t= ∇u
+ boundary conditions
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 4 / 30
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Maxwell Equations
in an isotropic, non-conducting medium
ε∂E
∂t= curlH
ν∂H
∂t= −curlE
+ boundary conditions
model problem: wave equation
∂u
∂t= div(v)
∂v
∂t= ∇u
+ boundary conditions
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 4 / 30
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Discontinuous Galerkin Discretization
We approximate the continuous problem
∂
∂t(u, ξ)Ω = (div(v), ξ)Ω
∂
∂t(v , η)Ω = (∇u, η)Ω + b.c .
with discontinuous uh and vh:
ξ, uh ∈ Uh := v ∈ L2 : v |T ∈ Pk+1(T )η, vh ∈ [Vh]3,Vh := v ∈ L2 : v |T ∈ Pk(T )
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 5 / 30
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Discontinuous Galerkin Discretization
discrete problem (∑
T ):
∂
∂t(uh, ξ)T = (div(vh), ξ)T = −(vh,∇ξ)T + (v∗n , ξ)∂T
∂
∂t(vh, η)T = (∇uh, η)T = (∇uh, η)T + (u∗−uh, ηn)∂T
How to choose the numerical fluxes u∗ and v∗n ?
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 6 / 30
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Discontinuous Galerkin Discretization
discrete problem (∑
T ):
∂
∂t(uh, ξ)T = (div(vh), ξ)T = −(vh,∇ξ)T + (v∗n , ξ)∂T
∂
∂t(vh, η)T = (∇uh, η)T = (∇uh, η)T + (u∗−uh, ηn)∂T
How to choose the numerical fluxes u∗ and v∗n ?
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 6 / 30
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central flux
If we choose
u∗ = uhv∗n = vhn
we get
∂
∂t(uh, ξ)T = −(vh,∇ξ)T + (vhn, ξ)∂T
∂
∂t(vh, η)T = (∇uh, η)T − (
1
2[[uh]], ηn)∂T︸ ︷︷ ︸
(uh,ηn)∂T
with [[uh]] = u+h − u−h
qAAA
q
q
qJJJJ q
q
@
@@ qj+ j−j−j−
⇒ skewsymmetric r.h.s.
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 7 / 30
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properties of the central flux
Writing
Gh(uh, η) = (∇uh, η)T − (1
2[[uh]], ηn)∂T
we get the semi-discrete formulation
∂
∂t
(uh
vh
)=
(−GT
h
Gh
)(uh
vh
)which represents the structure of the continuous problem(−∇ is the adjoint operator to div)
∂
∂t
(uv
)=
(div
∇
)(uv
)spatial discretizations preserving the antisymmetry of the PDE areenergy-conserving
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 8 / 30
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eigenvalue computation with the central flux
Nontrivial nonphysical kernel of Gh ⇒ spurious modes.
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 9 / 30
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a dissipative flux
A choice which avoids spurious modes is
u∗ = uh − [[vh · n]]
v∗n = vhn −1
2[[uh]]
we get
∂
∂t(uh, ξ)T = −(vh,∇ξ)T + (vhn − 1
2 [[uh]], ξ)∂T
∂
∂t(vh, η)T = (∇uh, η)T − ( 1
2 [[uh]] + [[vh · n]], ηn)∂T
The difference to the central flux can be written as a sum over all facets:
−∑E
1
2([[uh]], [[ξ]])E + ([[vh · n]], [[η · n]])E
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 10 / 30
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eigenvalue computation with the dissipative flux
Nonphysical damping of high frequencies
+ avoids spurios modes
− is not energy-conserving
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 11 / 30
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problem in the frequency domain
The problem in frenquency domain reads
iωu = divv
iωv = ∇u
and the discretized version reads (with the discrete operator Gh):
iω(u, ξ) = −(Ghξ, v)
iω(v , η) = (Ghu, η)
or0 = (iω)2(u, ξ) + (Ghu,Ghξ) +
∑E
α
h([[u]], [[ξ]])∂T
(Ghu,Ghξ) ≈ (∇u,∇ξ) is an symmetric, positive definite bilinearform⇒ Add an Internior Penalty Term to avoid nontrivial kernel of Gh
(proposed by Warburton/Hesthaven)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 12 / 30
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problem in the frequency domain
The problem in frenquency domain reads
iωu = divv
iωv = ∇u
and the discretized version reads (with the discrete operator Gh):
iω(u, ξ) = −(Ghξ, v)
iω(v , η) = (Ghu, η)
or0 = (iω)2(u, ξ) + (Ghu,Ghξ) +
∑E
α
h([[u]], [[ξ]])∂T
(Ghu,Ghξ) ≈ (∇u,∇ξ) is an symmetric, positive definite bilinearform⇒ Add an Internior Penalty Term to avoid nontrivial kernel of Gh
(proposed by Warburton/Hesthaven)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 12 / 30
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non-dissipative stabilization approach
With αh [[u]] = iωvF we get a new unknown living only on the facets.
We introduce this variable analogously on the time domain level
vFh ∈ V F
h := vFh ∈ L2(F) : v |E ∈ Pk(E )
∂
∂t(vF
h , ξF ) = (
α
h[[uh]], ξF ) ∀ξF ∈ V F
h
and get the new non-dissipative spatial discretization(∑
T · · · ,∑
E · · · )
∂
∂t(uh, ξ)T = −(vh,∇ξ)T + (vhn, ξ)∂T − (vF
h , [[ξ]])E
∂
∂t(vh, η)T = (∇uh, η)T − (
1
2[[uh]], ηn)∂T
h
α
∂
∂t(vF
h , ξF )E = ([[uh]], ξF )E
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 13 / 30
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eigenvalue computation with the non-dissipative stabilizedflux
+ the zero eigenvalues vanished+ discretization is still energy-conserving
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 14 / 30
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discretization of Maxwell’s equation
Transfering the ideas of the wave equation to Maxwell’s equation, we get (∑
T · · ·,∑
E · · · )
∂
∂t(εEh, e)T = (Hh, curl e)T + (Hh × n, e)∂T + (HF
h × n, [[e]])E
∂
∂t(νHh, h)T = − (curl Eh, h)T + (
1
2[[Eh]]× n, h)∂T
h
α
∂
∂t(HF
h , hF )E = ([[Eh]]× n, hF )E
where α ∼ p(p + 1)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 15 / 30
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the complete scheme
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 16 / 30
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global time-stepping
For the investigations of the time-stepping scheme we consider the semi-discreteproblem
∂
∂t
(MεEMνH
)=
(−GT
h HGhE
)To maintain the energy-conserving property of our spatial discretization we use atime-stepping scheme that is also energy-conserving in a discrete sense. Here weuse the symplectic euler / leap frog:
Hn+1 = Hn+M−1ν Gh(E n
h ) Hn+ 12 = Hn− 1
2 +M−1ν Gh(E n
h )
E n+1 = E n−M−1ε GT
h (Hn+1h ) E n+1 = E n −M−1
ε GTh (H
n+ 12
h )
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 17 / 30
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numerical exampleAs an example of the opportunities of a higher order discretization, we will look atthe error of the smallest resonance frequencies (eigenvalues) for a cube vacuumresonator with side length 1mm and PEC boundary conditions.
6 elements
p-refinement
1e-08
1e-06
0.0001
0.01
1
500 1000 2000 4000 8000 16000
rela
tive
erro
r
#unknowns
relative error of resonant frequencies
Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6Mode 7Mode 8
exp(-n^1/3)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 18 / 30
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computational aspects
From point of view of implementation we have
full polynomials of degree k + 1 for Eh
full polynomials of degree k and additional facet unknowns for Hh
block-diagonal mass matrices
For further optimization we
don’t assemble global matrices Gh or −GTh
but implement the operations M−1ν Gh(E k
h ) and −M−1ε GT
h (Hkh )
The complexity of elementwise operations is dominated by
evaluating gradients
evaluating trace terms
For both we use non-standard techniques on tetrahedra elements leading to fastcomputations.
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 19 / 30
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local time-stepping
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 22 / 30
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For explicit schemes the stability limit of the global time step is limited by thematerial parameters, the mesh size and the polynomial order.
global time step depends (more or less) on the smallest element.
Idea of local time-stepping:
use different step sizes for different elements
So the global time step is only limited by the size of the largest elements.
For a local time-stepping scheme the mesh elements are divided in differentclasses representing different time steps:
Class 0 is the class with the largest time step
...
Class N is the class with the smallest time step
So we need an algorithm making the different classes compatible.We use only classes such that time steps of smaller classes are a certain multipleinteger of a larger class.
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 23 / 30
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recursive leap frog
One possible algorithm proposed by Montseny et al. (2008) is the recursiveleap-frog method which is based on the time integration method leap frog or asymplectic euler :
1 ComputeH(0,∆t)
2 ComputeE(0,∆t)
with the recursively defined functions
ComputeH(N,∆t): ComputeE(N,∆t):
UpdateH(N,∆t) UpdateE(N,∆t)ComputeH(N+1, 1
3 ∆t) ComputeE(N+1, 13 ∆t)
ComputeE(N+1, 13 ∆t) ComputeH(N+1, 1
3 ∆t)ComputeH(N+1, 1
3 ∆t) ComputeE(N+1, 13 ∆t)
where UpdateH and UpdateE are the actual computation and updates.
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 24 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
class 0: ComputeH(0,∆t)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
class 0: ComputeH(0,∆t) → UpdateH(0,∆t)
tn+ 12
6
1
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
class 0: ComputeH(0,∆t) → ComputeH(1, 13 ∆t)
class 1: ComputeH(1, 13 ∆t) → UpdateH(1, 1
3 ∆t)
tn+ 16
62
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
class 0: ComputeH(0,∆t) → ComputeE(1, 13 ∆t)
class 1: ComputeE(1, 13 ∆t) → UpdateE(1, 1
3 ∆t)
tn+ 13
63
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
tn+ 13
63
class 0: ComputeH(0,∆t) → ComputeH(1, 13 ∆t)
class 1: ComputeH(1, 13 ∆t) → UpdateH(1, 1
3 ∆t)
64
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
tn+ 13
63
64
class 0: ComputeE(0,∆t) → UpdateE(0, 13 ∆t)
6
5
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
tn+ 13
63
64
6
5
class 0: ComputeE(0,∆t) → ComputeE(1, 13 ∆t)
class 1: ComputeE(1, 13 ∆t) → UpdateE(1, 1
3 ∆t)
tn+ 23
6
6
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
tn+ 13
63
64
6
5
tn+ 23
6
6
class 0: ComputeE(0,∆t) → ComputeH(1, 13 ∆t)
class 1: ComputeH(1, 13 ∆t) → UpdateH(1, 1
3 ∆t)
tn+ 56
67
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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recursive leap frog example
tn− 12
H0
E0
tn
H1
tn− 16 E1
tn+ 12
6
1
tn+ 16
62
tn+ 13
63
64
6
5
tn+ 23
6
6
tn+ 56
67
class 0: ComputeE(0,∆t) → ComputeE(1, 13 ∆t)
class 1: ComputeE(1, 13 ∆t) → UpdateE(1, 1
3 ∆t)
tn+168
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 25 / 30
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implementation of a local time-stepping scheme
The implementation of a local time-stepping scheme for a DG method isstraight-forward:
The loop over all elements gets replaced by a loop over all elements of theactual class.
The loop over all facets gets replaced by a loop over all facets of the actualclass. (some facets may be in two classes)
So a code which features local time-stepping needs
1 local time step estimates for every element
2 a subroutine dividing elements and facets into several classes
3 an estimate for the global time step
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 26 / 30
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3 An estimate for the global time step can be obtained with a few steps of apower iteration for the matrix A of(
Eh
Hh
)k+1
= A ·(
Eh
Hh
)k
2 A subroutine dividing elements and facets into several classes is also easy toimplement
1 Good estimates for the maximum local time step are not so easy to get
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 27 / 30
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one example for the local time-stepping scheme
The Algorithm divides the elements and facets into 7 classes:total 0 1 2 3 4 5 6
elements 1467 131 341 305 260 227 155 48facets 3264 351 790 745 653 553 386 128
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 28 / 30
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Order 4max time step elements each (large) step facets each (large) step
w/o lts 4.10526 · 10−9 1467 3264w lts 2.18564 · 10−6 101963 258960factor 532.399 69.50 79.3382
⇒ Speedup ≈ 7with just rough estimates for the local time step(∆tlocal ∼ h + one smoothing step)
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 29 / 30
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Thank you for your attention!
Christoph Lehrenfeld, Joachim Schoberl (RWTH) Efficient High Ord. DGTD M. for Maxw. Equations MAFELAP 2009, June 11, 2009 30 / 30