rieben ima poster, 05/11/2004 1 uc davis /llnl/ iscr high order symplectic integration methods for...

Rieben IMA Poster, 05/11/2004 1UC Davis /LLNL/ ISCRUC Davis /LLNL/ ISCR

High Order Symplectic Integration Methods for Finite High Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell EquationsElement Solutions to Time Dependent Maxwell Equations

Robert N. Rieben, UC Davis Applied Science / ISCR

Daniel A. White, LLNL / DSED

Garry H. Rodrigue, UC Davis Applied Science / ISCR

Special Thanks to: Joe Koning, Paul Castillo and Mark Stowell

www.llnl.gov/casc/emsolve

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract no. W-7405-Eng-48, UCRL-JC-152872


Maxwell’s Equations: Continuum and DiscreteMaxwell’s Equations: Continuum and Discrete

1 ( )E B J tt

B Et

T

t

t

M e K M b M j

b K e

We begin with the coupled first order Maxwell equations, then discretize in space via a Galerkin Finite Element Method to yield a linear system of ODEs:

1, ( ) ( ), ( )i j i hi j f f f F H Div

M

, ( ) ( ), ( ) i j i hi j w w w W H Curl

M

, ( )i ji j A w K i.e. the projection of the curl of a 1-form onto the discrete 2-form space

We use discrete differential form basis functions of arbitrary polynomial degree:


Second Order Accurate Leap Frog MethodSecond Order Accurate Leap Frog Method

Consider the very popular second order accurate “Leap-Frog” method applied to the system of ODEs (ignoring the current source):

11

1 1

Tn n n

n n n

t

t

e e M K M b

b b Ke

1

1

n n

n n

t t

t t

e ee

b bb

This explicit method is well known to be

energy conserving energy conserving and conditionally stable conditionally stable

Given the high order accuracy of our compatible spatial discretization method, can we apply high order accurate time integration methods to the discrete Maxwell equations that are still energy conserving and conditionally stable?


Symplectic Time IntegrationSymplectic Time Integration

Consider the case of a simple undamped harmonic oscillator:

Symplectic Method Non-Symplectic RK Method

Harmonic OscillatorHarmonic Oscillator

2 2 1

p q

q p

p q

Spatially-Discrete MaxwellSpatially-Discrete Maxwell

constantT T

e Cb

b Ce

e M e b M b

Traditional integration methods (such as Runge-Kutta) introduce numerical dissipation. Higher order accurate and non-dissipative symplectic methods have been developed for Hamiltonian systems with applications in astrophysics and molecular dynamics.


General Symplectic Integration AlgorithmGeneral Symplectic Integration Algorithm

Coefficients for symplectic integration methods of order 1 through 4 have been derived*. Note that the leap-frog method corresponds to the first order case.

*Forest & Ruth ‘90, Candy & Rozmus ‘91

1

1

1

1

1

1

1

j

j nn

Tout in j in

o

in i

in i

ut in j

i out

i ou

t

t

ou

t i t t

t

i nstep

j orde

t

r

e e

e e

b b

e e

b b

M K M b

b b Ke

for to do

en

for to do :

d

:

end


High Order Update Scheme and Numerical StabilityHigh Order Update Scheme and Numerical Stability

The generalized kth order symplectic update method applied to the discrete Maxwell equations can be written as a product of amplification matrices:

1

2 1

Ti

i Ti i i

t

t t

I M K MQ

K I KM K M

1

11

kn n

iin n

e eQ

b b

A necessary condition for stability is then:

( ) 1i Q 1

2

( )Ti i

t

KM K M


Now consider the similar amplification matrix:

Ti i i

i Ti

a b a

b

I AA AQ

A I

1) ( )

2) The eigenvalues of satisfy 0 2Ti i

Rank m

a b

A

AA

1 2

1 2

In order for ( ) 1 to be a condition for stability,

we need to exhibit linearly independent eigenvectors of ,

where ( ) and ( )

i

in n

n Dim n Dim

suffiQ

Q

e

cient

b

Numerical Stability (cont.)Numerical Stability (cont.)

Suppose:


Then:

1

1 221

2( (

By analyzing the eigenvalues and eigenvectors of:

and since and

we can generate 2

linearly independent eigenvectors of t

( (

he

))

(

)

a

(

mplifica

)

))T

Ti i

Dim NulDim Null n m

a b

n mm n

n

n

ml

nm

A

AA x

A

x

tion matrix iQ

Proof:

Numerical Stability (cont.)Numerical Stability (cont.)

1 2

1) The eigenvalues of lie on the unit circle

2) The eigenvectors of form an eigenbasis in

i

n ni R

Q

Q

Conservation of Numerical Energy:Conservation of Numerical Energy:

Numerical Stability:Numerical Stability:


Higher Order Time Integration:Higher Order Time Integration:

Global Phase and Energy ErrorGlobal Phase and Energy Error

• 1st Order Method:• Time Step = 0.005 sec• # Steps = 60,000• CPU time / step = 0.0941 sec• Total Run Time = 94.1min

• 3rd Order Method:• Time Step = 0.015 sec• # Steps = 20,000• CPU time / step = 0.2976 sec• Total Run Time = 99.2 min

20x More 20x More Effective!Effective!

.015 .00075

1.01 1.0001


Resonant Cavity AnalysisResonant Cavity Analysis

1st Order 3rd Order

Physical Time 300 sec 300 sec

Time Step 0.005 sec 0.015 sec

No. Steps 60,000 20,000

Avg CPU time/step 0.0941 sec 0.2976 sec

Total Run Time 94.1 min 99.2 min

Error in 1st Mode 1.3809e-3 1.0935e-4

Error in 2nd Mode 8.9125e-4 3.8032e-4

Error in 3rd Mode 5.3780e-4 5.3780e-4

Error in 4th Mode 1.5442e-3 2.7264e-4

Error in 5th Mode 3.2044e-3 6.1035e-4

Here we compute the resonant modes of a cubic cavity using two different integration methods in conjunction with a high order (p = 4) compatible spatial discretization. Use of high order in both time and space is required to achieve maximal accuracy.


Conductivity Terms and Implicit Time SteppingConductivity Terms and Implicit Time Stepping

In order to introduce conductivity terms while still maintaining numerical stability, we can treat the problem implicitly:

1 1 11

* 1 * 1

(1 ) (1 )2 2

(1 ) (1 )2 2

Tn n n

n n n

t tt

t tt

M M e M M e M K M b

M b M b Ke

Explicit 4Explicit 4thth Order Order Symplectic:Symplectic:

Implicit 4Implicit 4thth Order Order Symplectic:Symplectic:

rieben ima poster, 05/11/2004 1 uc davis /llnl/ iscr high order symplectic integration methods for...

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