Geometric Integration for Damped Hamiltonian PDEs€¦ · Geometric Integration for Damped...
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Geometric Integration for Damped Hamiltonian PDEs Brian E. Moore , Laura Nore ˜ na, and Constance Schober [email protected] University of Central Florida SIAM Conference on Nonlinear Waves and Coherent Structures, University of Washington, Seattle, June 13, 2011 Thanks to the National Science Foundation for partial funding Jul 2011 – p.1/16
Transcript of Geometric Integration for Damped Hamiltonian PDEs€¦ · Geometric Integration for Damped...
Geometric Integration forDamped Hamiltonian PDEs
Brian E. Moore, Laura Norena, and Constance Schober
University of Central Florida
SIAM Conference on Nonlinear Waves and Coherent Structures, University of
Washington, Seattle, June 13, 2011
Thanks to the National Science Foundation for partial funding
Jul 2011 – p.1/16
Multi-Symplectic PDE, Bridges (1997)
These equations can be written
Kzt + Lzx = ∇zS(z)
K and L are skew-symmetric matrices
z = z(x, t) is the vector of state variables
S is smooth
Examples include KdV, Boussinesq, Zakharov-Kuznetsov,nonlinear Schrödinger, nonlinear wave equations, etc.
Conservation laws are derived directly from the equation.An example is the multi-symplectic conservation law∂t 〈KU, V 〉+ ∂x 〈LU, V 〉 = 0, where U and V satisfy thevariational equation.
Jul 2011 – p.2/16
Conformal Multi-Symplectic PDE, M. (2009)
These equations can be written
Kzt + Lzx = ∇zS(z)−a
2Kz −
b
2Lz + F (x, t)
a and b are positive real constants
S is smooth and may depend on a and b
F is a forcing term
Equation satisfies a conformal multi-symplectic conservation law
∂t 〈KU, V 〉+ ∂x 〈LU, V 〉 = −a 〈KU, V 〉 − b 〈LU, V 〉.
Methods that preserve this property behave similarly tostandard multi-symplectic methods for small a and b.
Jul 2011 – p.3/16
General Conformal Conservation Laws
IfKzt + Lzx = ∇zS(z)
has a conservation law
∂tP + ∂xQ = 0,
then
Kzt + Lzx = ∇zS(z)−a
2Kz −
b
2Lz
has a conformal conservation law if it satisfies
∂tP + ∂xQ = −aP − bQ.
Jul 2011 – p.4/16
Examples of Conservation Laws for
Kzt + Lzx = ∇zS(z)
Energy: inner product of the equation with zt
∂t(
S(z) + 12(z
Tx Lz)
)
+ ∂x12(z
TLzt) = 0
Jul 2011 – p.5/16
Examples of Conservation Laws for
Kzt + Lzx = ∇zS(z)
Energy: inner product of the equation with zt
∂t(
S(z) + 12(z
Tx Lz)
)
+ ∂x12(z
TLzt) = 0
Momentum: inner product of the equation with zx
∂x(
S(z) + 12(z
Tt Kz)
)
+ ∂t12(z
TKzx) = 0
Jul 2011 – p.5/16
Examples of Conservation Laws for
Kzt + Lzx = ∇zS(z)
Energy: inner product of the equation with zt
∂t(
S(z) + 12(z
Tx Lz)
)
+ ∂x12(z
TLzt) = 0
Momentum: inner product of the equation with zx
∂x(
S(z) + 12(z
Tt Kz)
)
+ ∂t12(z
TKzx) = 0
Linear Symmetries: inner product with Bz
∂t(
zTKBz)
+ ∂x(
zTLBz)
= 0
Jul 2011 – p.5/16
Examples of Conformal Conservation Laws for
Kzt + Lzx = ∇zS(z)−a2Kz − b
2Lz
Energy: inner product of the equation with zt
∂t(
S(z) + 12(z
Tx Lz)
)
+ ∂x12(z
TLzt) =
−b2 (zTLzt)
Momentum: inner product of the equation with zx
∂x(
S(z) + 12(z
Tt Kz)
)
+ ∂t12(z
TKzx) =
−a2 (zTKzx)
Linear Symmetries: inner product with Bz
∂t(
zTKBz)
+ ∂x(
zTLBz)
= −a(
zTKBz)
− b(
zTLBz)
Jul 2011 – p.6/16
Numerical Preservation
Suppose we have the following PDE and Conformal CL
Kzt + Lzx = ∇S(z)−a
2Kz ∂tP + ∂xQ = −aP
Jul 2011 – p.7/16
Numerical Preservation
Suppose we have the following PDE and Conformal CL
Kzt + Lzx = ∇S(z)−a
2Kz ∂tP + ∂xQ = −aP
Integrating with appropriate boundary conditions, yields
∂tP = −aP ⇐⇒ P (t) = exp(−at)P (0) with P =
∫
Pdx.
Jul 2011 – p.7/16
Numerical Preservation
Suppose we have the following PDE and Conformal CL
Kzt + Lzx = ∇S(z)−a
2Kz ∂tP + ∂xQ = −aP
Integrating with appropriate boundary conditions, yields
∂tP = −aP ⇐⇒ P (t) = exp(−at)P (0) with P =
∫
Pdx.
A numerical method preserves this property if it satisfies
P i+1 = exp(−a∆t)P i with P i =∑
n
Pn,i∆x.
Jul 2011 – p.7/16
Numerical Preservation
Suppose we have the following PDE and Conformal CL
Kzt + Lzx = ∇S(z)−a
2Kz ∂tP + ∂xQ = −aP
Integrating with appropriate boundary conditions, yields
∂tP = −aP ⇐⇒ P (t) = exp(−at)P (0) with P =
∫
Pdx.
A numerical method preserves this property if it satisfies
P i+1 = exp(−a∆t)P i with P i =∑
n
Pn,i∆x.
Note: ∂tP < 0 is typically considered preserved if P i+1 < P i
Jul 2011 – p.7/16
Splitting Methods, McLachlan and Quispel (2002)
zt = FH(z) + FD(z)
Solve zt = fH(z) with conservative method zi+1 = Ψ∆t(zi).
Solve zt = Dz exactly with flow map Φt(z) = exp(Dt)z.
Original system is solved by composing these flow maps.
Jul 2011 – p.8/16
Splitting Methods, McLachlan and Quispel (2002)
zt = FH(z) + FD(z)
Solve zt = fH(z) with conservative method zi+1 = Ψ∆t(zi).
Solve zt = Dz exactly with flow map Φt(z) = exp(Dt)z.
Original system is solved by composing these flow maps.
Example: Define non-standard difference/average operators
Dat z =
zi+1 − e−a∆tzi
∆tand Aa
t z =zi+1 + e−a∆tzi
2
Implicit midpoint for zt = FH(z) is D0t z = FH
(
A0t z)
.
Splitting method for zt = FH(z) + FD(z) is Da/2t z = FH
(
Aa/2t z
)
Jul 2011 – p.8/16
Structure-Preservation
A non-standard 1-step FD method preserves a conformal CL, ifit has a discrete product rule and the corresponding CL ispreserved by the conservative method upon which it is based.
Example: The conformal box scheme
K
(
Da/2t Ab/2
x z)
+ L
(
Db/2x A
a/2t z
)
= ∇S(
Aa/2t Ab/2
x z)
satisfies Dat
⟨
Ab/2x z,KBAb/2
x z⟩
+Dbx
⟨
Aa/2t z,LBA
a/2t z
⟩
= 0.
Sum over spatial index with b = 0 and appropriate BCs gives
P i+1 = exp(−a∆t)P i with P i =∑
n
⟨
A0xz,KBA0
xz⟩
∆x.
Jul 2011 – p.9/16
Nonlinear Schrodinger with Dissipation
iψt + ψxx + V ′(|ψ|2)ψ + ia
2ψ = 0
Can be written Kzt + Lzx = ∇zS(z)−a2Kz with
J =
[
0 −1
1 0
]
K =
[
−J 0
0 0
]
, L =
[
0 −I
I 0
]
,
z = [v,w, p, q]T , S(z) = 12(p
2 + q2 + V (v2 + w2))
The conformal norm conservation law
∂t(w2 + v2) + 2∂x(vwx − wvx) = −a(w2 + v2)
Jul 2011 – p.10/16
Residual in Local Conformal Conservation Law
a = 0.01, h = 0.01,
05
1015
2025
−10
−5
0
5
100
0.2
0.4
0.6
0.8
1
1.2
x 10−13
MS−CC−CL−1: Γ = 0.010, |RaMAX
| = 1.19904e−13
Jul 2011 – p.11/16
Residual in Global Conformal Conservation
10 20 30 40 50 60 70 80 90 10010
−14
10−13
10−12
TIME
ln(D
)
PreissmanConformal
Jul 2011 – p.12/16
Dissipative Semi-Linear Wave Equation
utt = uxx − aut − f ′(u)
Can be written Kzt + Lzx = ∇zS(z)−a2Kz with
J =
[
0 −1
1 0
]
K =
[
J 0
0 J
]
, L =
[
0 −I
I 0
]
,
z = [u, v, w, p]T , S(z) = 12
(
a(uv + wp) + v2 − w2 + 2f(u))
momentum:∂t(vux+pwx)+∂x
(
f(u) + v2
2 − w2
2 − vut − pwt
)
= −a(vux+pwx)
angular momentum: ∂t(u× ut) + ∂x(ux × u) = −a(u× ut)
others: ∂t(vw + cup) + 1
2∂x
(
v2 + w2− c(u2 + p2)
)
= −a(vw + cup)
Jul 2011 – p.13/16
Error in dissipation rates for utt = uxx − aut − cu
c = 1, a = 34 , ∆t = 0.025, 80 grid points on [−π, π].
0 5 10 15 20 25 30 35 40 45 50−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
t
d
standard box schemeconformal box scheme
Jul 2011 – p.14/16
Solution Behavior for utt = uxx − aut − cu
c = 1, a = 34 , ∆t = 0.025, 80 grid points on [−π, π].
−4 −3 −2 −1 0 1 2 3 4−1
−0.5
0
0.5
1x 10
−8
x
uT = 50
−4 −3 −2 −1 0 1 2 3 4−1
−0.5
0
0.5
1x 10
−8
x
u
exactstandard
exactconformal
Jul 2011 – p.15/16
Solution Behavior for utt = uxx − aut − cu
c = 1, a = 34 , ∆t = 0.025, 80 grid points on [−π, π].
−4 −3 −2 −1 0 1 2 3 4−4
−2
0
2
4x 10
−14
x
uT = 100
exactstandard
−4 −3 −2 −1 0 1 2 3 4−1
−0.5
0
0.5
1x 10
−16
x
u
exactconformal
Jul 2011 – p.16/16
