modern discretization techniques for time domain maxwell

Modern Discretization Techniques for Time DomainMaxwell Equations

Christoph Lehrenfeld, Joachim Schoberl, Timo Euler

CCES, RWTH Aachen University & CST Computer Simulation Techniques, Darmstadt

Internship presentation, Aachen, December 4, 2009

content

1 DG formulations for Maxwell TD

2 Standard time stepping methods for Maxwell TD

3 Higher Order time stepping method for Maxwell TD

4 Local time stepping methods for Maxwell TD

Christoph Lehrenfeld, Joachim Schoberl, Timo Euler (RWTH)Modern Techniques for Maxwell TD Aachen, December 4, 2009 2 / 54

DG formulations for Maxwell TD


Maxwell Equations

in an isotropic, non-conducting medium

ε∂E

∂t= curlH

ν∂H

∂t= −curlE

+ boundary conditions

model problem: scalar wave equation

∂u

∂t= div(v)

∂v

∂t= ∇u



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 )


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 ?


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.


properties of the central flux

Writing

Gh(uh, η) = (∇uh, η)T − (1

2[[uh]], ηn)∂T

we get the semi-discrete formulation

∂

∂t

(uhvh

)=

(−GT

h

Gh

)(uhvh

)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


eigenvalue computation with the central flux

Nontrivial nonphysical kernel of Gh ⇒ spurious modes.


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


eigenvalue computation with the dissipative flux

Nonphysical damping of high frequencies

+ avoids spurios modes

− is not energy-conserving


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) (where α ∼ p(p + 1))


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+1(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


eigenvalue computation with the non-dissipative stabilizedflux

+ the zero eigenvalues vanished+ discretization is still energy-conserving


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 = − (curlEh, h)T + (

1

2[[Eh]]× n, h)∂T

h

α

∂

∂t(HF

h , hF )E = ([[Eh]]× n, hF )E


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)


the complete scheme


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 )


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


We use the covariant transformation (u(x) = F−T u(x)) (G. Cohen et al.2005) for the transformation from the reference element leading togeometry independent element matrices

On the reference element we use an orthogonal basis which is computed in atensor product form with Jacobi polynomials

This leads to several advantages

diagonal mass matrices for uncurved elements (orthogonal polynomials +covariant transformation)

...


Standard time stepping methods forMaxwell TD


Maxwells Equations

Maxwells Equations read as

ε∂E

∂t= −σEE +curlH + jE

ν∂H

∂t= −curlE −σMH + jM


If we use an energy-conserving discretization method in space (like our DGmethod or FIT) we end up with the semi-discrete form(

Mε

Mµ

)∂

∂t

(Eh

Hh

)=

(−MσE

CT

−C −MσM

)(Eh

Hh

)+

(jEjM

)


For ease of presentation we will - at the beginning - neglect loss terms andexcitations.Further we will transform the Fields Eh and Hh as well as the Operator CT and−C such that we can limit our investigations to the following problem

∂

∂t

(Eh

Hh

)=

(CT

−C

)(Eh

Hh

)On the time-continuous level the following energy is conserved:

‖Eh‖22 + ‖Hh‖2

2 = const


symplectic Euler method

tn

H E



tn

H E

1. ∆Hn+ 12 = −∆tCE n

tn+16

1



tn

H E

1. ∆Hn+ 12 = −∆tCE n

2. ∆E n+ 12 = ∆tCTHn+1

tn+16

1

6

2



tn

H E

1. ∆Hn+ 12 = −∆tCE n

2. ∆E n+ 12 = ∆tCTHn+1

tn+16

1

6

2

+ conserves a discrete energy(dependent of ∆t):

ε = ‖E n‖22 + (Hn+1,Hn) = ‖E n‖2

2 + ‖Hn‖22 −∆t(CE n,Hn)

+ Hn and E n are stored at the same timelevel

+ needs only 2 steps

- only first order accurate

- non-reversible (A−1it (∆t) 6= Ait(−∆t))


symmetric symplectic Euler

tn

H E



tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

tn+ 12

6

1



tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2. ∆E n+ 14 = 1

2 ∆tCTHn+ 14 tn+ 1

26

16

2



tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2. ∆E n+ 14 = 1

2 ∆tCTHn+ 14

3. ∆E n+ 34 = 1

2 ∆tCTHn+ 14

tn+ 12

6

16

2

6

3



tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2. ∆E n+ 14 = 1

2 ∆tCTHn+ 14

3. ∆E n+ 34 = 1

2 ∆tCTHn+ 14

4. ∆Hn+ 34 = − 1

2 ∆tCE n+ 34

tn+ 12

6

16

2

6

3

tn+16

4


symmetric symplectic Euler → Verlet scheme

tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2 + 3. ∆E n+ 34 = ∆tCTHn+ 1

4

4. ∆Hn+ 34 = − 1

2 ∆tCE n+ 34

tn+ 12

6

1

6

2 + 3

tn+16

4


Verlet scheme

tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2. ∆E n+ 34 = ∆tCTHn+ 1

4

3. ∆Hn+ 34 = − 1

2 ∆tCE n+ 34

tn+ 12

6

1

6

2

tn+16

3


Verlet scheme

tn

H E

1. ∆Hn+ 14 = − 1

2 ∆tCE n

2. ∆E n+ 34 = ∆tCTHn+ 1

4

3. ∆Hn+ 34 = − 1

2 ∆tCE n+ 34

tn+ 12

6

1

6

2

tn+16

3

+ conserves a discrete energy( dependent of ∆t):

ε = ‖E‖22 + ‖H‖2

2 −∆t2

4‖CE‖2

2

+ Hn and E n are stored at the same timelevel

+ second order accurate

+ reversible (A−1it (∆t) = Ait(−∆t))

- needs 3 steps


multiple Verlet steps

tn

H E



tn

H E

1. ∆Hn+ 14 = − 1

2∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3. ∆Hn+ 34 = − 1

2∆tCE n+1

tn+ 12

61

tn+16

263



tn

H E

1. ∆Hn+ 14 = − 1

2∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3. ∆Hn+ 34 = − 1

2∆tCE n+1

4. ∆Hn+ 54 = − 1

2∆tCE n+1

5. ∆E n+ 32 = ∆tCTHn+ 3

2

6. ∆Hn+ 74 = − 1

2∆tCE n+2 tn+ 1

261

tn+16

263

tn+ 32

64

tn+26

566


multiple Verlet steps → LeapFrog

tn

H E

1. ∆Hn+ 14 = − 1

2∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3 + 4. ∆Hn+1 = −∆tCE n+1

5. ∆E n+ 32 = ∆tCTHn+ 3

2

6. ∆Hn+ 74 = − 1

2∆tCE n+2 tn+ 1

261

tn+16

2

tn+ 32

6

3 + 4

tn+26

566


multiple Verlet steps → LeapFrog

tn

H

E

0 + 1. ∆Hn = −∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3 + 4. ∆Hn+1 = −∆tCE n+1

5. ∆E n+ 32 = ∆tCTHn+ 3

2

tn+ 12

tn− 12

6

0 + 1

tn+16

2

tn+ 32

6

3 + 4

tn+26

5


LeapFrog

tn

H

E

1. ∆Hn = −∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3. ∆Hn+1 = −∆tCE n+1

4. ∆E n+ 32 = ∆tCTHn+ 3

2

tn+ 12

tn− 12

6

1

tn+16

2

tn+ 32

6

3

tn+26

4


LeapFrog

tn

H

E

1. ∆Hn = −∆tCE n

2. ∆E n+ 12 = ∆tCTHn+ 1

2

3. ∆Hn+1 = −∆tCE n+1

4. ∆E n+ 32 = ∆tCTHn+ 3

2

tn+ 12

tn− 12

6

1

tn+16

2

tn+ 32

6

3

tn+26

4

+ algorithmically equal to symplectic euler(except for start/end synchronization (Verlet))

+ conserves the same discrete energy as the symplectic euler method

+ needs only 2 steps

+ second order accurate

+ ≈ reversible (A−1it (∆t) = Ait(−∆t))

- H and E are no longer stored at the same timelevel


What all three methods have in common:

They are explicit

They conserve a discrete energy (for ∆t = const)

They have the same stability limit (for one complete step)


Remark on simplified assumptions

6 6

6 6

If you go back to the “symmetric symplectic Euler” and interprete theexplicit/implicit updates (indicated by green/red arrows) as explicit/implicitupdates for the loss and excitation terms you end up with an algorithm, which

is computationally always explicit when handling curl terms

handles only mass terms implicit(they are cheap ((block-) diagonal) for the DG Solver or FIT)

The steps from “symmetric symplectic Euler” to “Verlet” to “LeapFrog” can begenerated as well by summing updates using the same curl-terms together


Remark on stability

If you can show that the conserved (discrete) energy is a continuous, symmetricpositiv definite bilinearform, you now

‖Hn‖22 + ‖E n‖2

2 ' εn = ε0 ' ‖H0‖22 + ‖E 0‖2

2

⇒ H,E are bounded (stability)

(Here a ' b means a ≤ c1b and a ≥ c2b for c1, c2 > 0 independent of n)

For ∆t = const the condition of this remark leads to an explicit time steprestriction (the CFL-condition from LinAlg point of view):

∆t ≤ 2√λmax(CTC )

⇔ |λ(AIt)| = 1


Remark on non-constant ∆t

If we choose ∆t 6= const, we can easily construct all three methods with theabove orders of consistency.

Here, looking at the LeapFrog scheme as an algorithmically enhanced Verletscheme is helpful to construct an extension of the LeapFrog scheme for arbitrarilychanging ∆t which is still of second order accurate.

But: For arbitrarily changing ∆t we can no longer assure stability or conservationof a discrete energy.


Higher Order time stepping methodfor Maxwell TD


Higher Order time stepping methods for Maxwell TD

Standard approaches for Higher Order time stepping methods for Maxwell’sEquations are:

explicit Runge-Kutta methods

The Modified Equation Approach

(only for the lossfree case)

explicit (partitioned) Runge-Kutta methods

Our method of choice is called Composition Method and is related to explicit(partitioned) Runge-Kutta methods

Main Idea:Combine several steps (with different step sizes) of a symmetric second ordermethod, such that truncation errors cancel out and you get a method of higherorder (even).


Higher Order time stepping methods for Maxwell TD

Standard approaches for Higher Order time stepping methods for Maxwell’sEquations are:

explicit Runge-Kutta methods (not energy-conserving)

The Modified Equation Approach (only for the lossfree case)

explicit (partitioned) Runge-Kutta methods

Our method of choice is called Composition Method and is related to explicit(partitioned) Runge-Kutta methods

Main Idea:Combine several steps (with different step sizes) of a symmetric second ordermethod, such that truncation errors cancel out and you get a method of higherorder (even).


Example: Suzuki Method

6

Each of those steps is a Verlet Step but can also be translated to an equivalentLeapFrog formulation.



6

6




6

6

?




6

6

?

6




6

6

?

6

6

Each of those steps is a Verlet Step

but can also be translated to an equivalentLeapFrog formulation.



6

6

?

6

6

Each of those steps is a Verlet Step . but can also be translated to anequivalent LeapFrog formulation.



6

6

?

6

6

Each of those steps is a Verlet Step , but can also be translated to an equivalentLeapFrog formulation.


properties of the Composition Method

The Composition Method

can be constructed of arbitrary high order

is explicit

conserves a discrete energy

For the Composition Method we need a second order accurate time steppingscheme which can handle changing step sizes. The Verlet and LeapFrog schemesfulfill this requirements.The nice thing about the Composition Method is, that we can use a LeapFrog orVerlet of arbitrary complexity, meaning:

+ all special features that are only implemented for second order,translate naturally to higher order!

As mentioned before, a changing step size Verlet scheme in general is notunconditionally stable.


stability of the Composition Method

A sufficient stability criteria for the Composition Method of a Verlet scheme canbe found by looking at the corresponding scalar problem

∂

∂t

(eh

)=

(0 −

√a√

a 0

)(eh

)If the discretization of the scalar problem is stable for all a ∈ ρ(CTC ), thediscretization of the whole system is stable.


Stabilty of the Verlet scheme (2nd order) [scalar]


Stabilty of the Suzuki method (4th order) [scalar]


Convergence in the L2-Error for a Vacuum Cube Resonator


Local time stepping methods forMaxwell TD


Local time stepping methods for Maxwell TD

For every explicit time stepping method for Maxwell TD you have to fulfill acertain CFL-criteria which depends linearly on the mesh size.

If you have a mesh with different mesh size scales, large elements have to step atsmall time scales because of other (small) elements.


Idea:Use different step sizes for different elements to circumvents this problem.

Aim:A Method which is

explicit

second order accurate

has few additional cost (for example for interpolation in time)

conserves a discrete energy

stable


recursive Verlet method

The Std. Verlet scheme consists of three steps.After each step on the smallest level (level with the largest elements) request thenext larger level to step the same field to the same time level.


recursive Verlet example

H0 E0

tn

H1 E1



H0 E0

tn

H1 E1

tn+ 12

6

1



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

3



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7

tn+ 56

68



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7

tn+ 56

68

69



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7

tn+ 56

68

69

6

10



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7

tn+ 56

68

69

6

10

tn+ 16

611



H0 E0

tn

H1 E1

tn+ 12

6

1tn+ 16

62

tn+ 13

6

364

65

tn+16

6

tn+ 23

6

7

tn+ 56

68

69

6

10

tn+ 16

6116

12


examplary part of the algorithm

First H Update(class i, ∆t)

1 Update H-Field on my level

2 If (i<=#classes)

3 Call First H Update(i+1, 13 ∆t)

4 Call E Update(i+1, 13 ∆t)

5 Call Second H Update(i+1, 13 ∆t)

6 Call First H Update(i+1, 13 ∆t)


This method is

explicit

reversible (symmetric)

cheap (no additional synchronization/interpolation overhead)

second order accurate

conservative (conserves a discrete energy (which is unknown to us))

again Verlet → LeapFrog:Sum up step 4,5 and step 8,9 at the inner part and step 0,1 and -1,2This works only if the factor between two classes is odd (here three).


recursive LeapFrog example

tn− 12

H0

E0

tn

H1

tn− 16 E1



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3

6

4



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3

6

4

6

5



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3

6

4

6

5

tn+ 23

6

6



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3

6

4

6

5

tn+ 23

6

6

tn+ 56

6

7



tn− 12

H0

E0

tn

H1

tn− 16 E1

tn+ 12

6

1

tn+ 16

6

2

tn+ 13

6

3

6

4

6

5

tn+ 23

6

6

tn+ 56

6

7

tn+16

8


To realize a local time stepping scheme we need:

local operators that are only acting on arbitrary parts of the domain

a strategy subdividing the domain into several classes

an algorithm giving a stable time step


Subdivision strategy

Our approach to divide the elements into the classes is as follows:

1 Estimate elementwise a local time step

2 First Division: [min ∆tlocal , 3 min ∆tlocal ] is the largest class, ...

3 Overlap: To “stabilize” the interaction between small to larger elements addone neighbouring layer of larger elements to the small class

4 Avoid large jumps in step size, meaning: the classes of neighbouring elementsmay only differ by one

5 Estimate the necessary time step for the classes


Example Glider (continued): local time steps


Example Glider (continued): (before Overlap procedure)


Example Glider (continued): (Overlapped class 4)


Example Glider (continued): (finished Overlapping)


Example Glider (continued): (avoided jumps)


Ongoing work and open question

Although accuracy of the method and the existence of a discrete energy canbe proven for the recursive local time stepping scheme, there is no explicitcriteria available that describes the stability limit! (⇒ “Fummelfaktor”)

heuristic methods (as long as needed) still have to be improved and developed


Thank you for your attention!


modern discretization techniques for time domain maxwell

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