energy preserving numerical integration methods · 2010. 9. 24. · reinout quispel energy...
TRANSCRIPT
Energy Preserving Numerical IntegrationMethods
Reinout Quispel
Department of Mathematics and Statistics, La Trobe UniversitySupported by Australian Research Council Professorial Fellowship
Collaborators: Celledoni (Trondheim)Grimm (Erlangen)McLachlan (Massey)McLaren (La Trobe)O’Neale (La Trobe)Owren (Trondheim)
PART I PART II
Outline of the talk
Energy-preserving integrators for ODEs
Energy-preserving integrators for PDEs
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Outline of the talk
Energy-preserving integrators for ODEs
Energy-preserving integrators for PDEs
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
PART I
Energy-preserving integrators for Ordinary DifferentialEquations
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Consider a Poisson ODE:
dx
dt= S(x)∇H(x).
Here S(x) is a skew matrix, and ∇H is the gradient of theHamiltonian energy function.
Definition (Skew matrix)S(x) is skew if
(a · S(x)b) = −(b · S(x)a) ∀a, b
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Consider a Poisson ODE:
dx
dt= S(x)∇H(x).
Here S(x) is a skew matrix, and ∇H is the gradient of theHamiltonian energy function.
Definition (Skew matrix)S(x) is skew if
(a · S(x)b) = −(b · S(x)a) ∀a, b
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
ENERGY-PRESERVING DISCRETE GRADIENTMETHOD
Definition (Discrete gradient)
A discrete gradient ∇H(xn, xn+1) is defined by
(i) (xn+1 − xn) · ∇H(xn, xn+1) ≡ H(xn+1)−H(xn)
and
(ii) limxn+1→xn ∇H(xn, xn+1) = ∇H(xn)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Theorem
Let ∇H be a discrete gradient. Then the discretization
xn+1 − xn∆t
= S(xn)∇H(xn, xn+1)
is energy-preserving.
Proof.
H(xn+1)−H(xn) = ∇H(xn, xn+1) · (xn+1 − xn)
= ∆t∇H(xn, xn+1) · S(xn)∇H(xn, xn+1)
= 0
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Theorem
Let ∇H be a discrete gradient. Then the discretization
xn+1 − xn∆t
= S(xn)∇H(xn, xn+1)
is energy-preserving.
Proof.
H(xn+1)−H(xn) = ∇H(xn, xn+1) · (xn+1 − xn)
= ∆t∇H(xn, xn+1) · S(xn)∇H(xn, xn+1)
= 0
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
TWO EXAMPLES OF DISCRETE GRADIENTS
Remember (xn+1 − xn) · ∇H(xn, xn+1) ≡ H(xn+1)−H(xn)
Example 1. (Itoh-Abe discrete gradient):
∇H :=
(H(xn+1,yn)−H(xn,yn)
xn+1−xnH(xn+1,yn+1)−H(xn+1,yn)
yn+1−yn
).
(This can be generalised to any dimension)
Example 2. (“Average" discrete gradient):
∇H :=∫ 10 ∇H(ξxn+1 + (1− ξ)xn) dξ
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
TWO EXAMPLES OF DISCRETE GRADIENTS
Remember (xn+1 − xn) · ∇H(xn, xn+1) ≡ H(xn+1)−H(xn)
Example 1. (Itoh-Abe discrete gradient):
∇H :=
(H(xn+1,yn)−H(xn,yn)
xn+1−xnH(xn+1,yn+1)−H(xn+1,yn)
yn+1−yn
).
(This can be generalised to any dimension)
Example 2. (“Average" discrete gradient):
∇H :=∫ 10 ∇H(ξxn+1 + (1− ξ)xn) dξ
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Proof.
(xn+1 − xn) · ∇H =
∫ 1
0(xn+1 − xn)∇H(ξxn+1 + (1− ξ)xn) dξ
=
∫ 1
0
d
dξH(ξxn+1 + (1− ξ)xn) dξ
= H(ξxn+1 + (1− ξ)xn)∣∣∣ξ=1
ξ=0
= H(xn+1)−H(xn)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Soxn+1 − xn
∆t= S(x)
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
is an energy preserving integrator for the ODE
dx
dt= S(x)∇H(x)
From now on we take S(x) = S (constant).Then
xn+1 − xn∆t
= S
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
→ xn+1 − xn∆t
=
∫ 1
0S∇H(ξxn+1 + (1− ξ)xn) dξ
But note that S∇H is the vector field! We get:
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Soxn+1 − xn
∆t= S(x)
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
is an energy preserving integrator for the ODE
dx
dt= S(x)∇H(x)
From now on we take S(x) = S (constant).
Then
xn+1 − xn∆t
= S
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
→ xn+1 − xn∆t
=
∫ 1
0S∇H(ξxn+1 + (1− ξ)xn) dξ
But note that S∇H is the vector field! We get:
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Soxn+1 − xn
∆t= S(x)
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
is an energy preserving integrator for the ODE
dx
dt= S(x)∇H(x)
From now on we take S(x) = S (constant).Then
xn+1 − xn∆t
= S
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
→ xn+1 − xn∆t
=
∫ 1
0S∇H(ξxn+1 + (1− ξ)xn) dξ
But note that S∇H is the vector field! We get:
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Soxn+1 − xn
∆t= S(x)
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
is an energy preserving integrator for the ODE
dx
dt= S(x)∇H(x)
From now on we take S(x) = S (constant).Then
xn+1 − xn∆t
= S
∫ 1
0∇H(ξxn+1 + (1− ξ)xn) dξ
→ xn+1 − xn∆t
=
∫ 1
0S∇H(ξxn+1 + (1− ξ)xn) dξ
But note that S∇H is the vector field! We get:Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Theorem (The “average vector field method")The numerical integration method
xn+1 − xn∆t
=
∫ 1
0f(ξxn+1 + (1− ξ)xn) dξ
preserves the energy H(x) exactly for any Hamiltonian ODEwith constant symplectic structure, i.e. for
dx
dt= f(x)
withf(x) = S∇H(x)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (DOUBLE WELL POTENTIAL)
dx
dt= y
dy
dt= 2x(1− x2)
AVF integrator
1
∆t
(xn+1 − xnyn+1 − yn
)=
(12(yn+1 + yn)xn+1 + xn − 1
2(x3n + x2nxn+1 + xnx2n+1 + x3n+1)
)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (DOUBLE WELL POTENTIAL)
dx
dt= y
dy
dt= 2x(1− x2)
AVF integrator
1
∆t
(xn+1 − xnyn+1 − yn
)=
(12(yn+1 + yn)xn+1 + xn − 1
2(x3n + x2nxn+1 + xnx2n+1 + x3n+1)
)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
PART II
Energy-preserving integrators for Partial Differential Equations
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Hamiltonian PDEs
Definition
The PDE∂u
∂t= f
(u,∂u
∂x,∂2u
∂x2, . . .
)is Hamiltonian if it is of the form
f
(u,∂u
∂x,∂2u
∂x2, . . .
)= S δH
δu
where S is a constant skew operator, andδHδu
is the variationalderivative of H.
(Note: f, u, and x can also be taken to be vectors. Although uis usually real, it may also be complex).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Hamiltonian PDEs
Definition
The PDE∂u
∂t= f
(u,∂u
∂x,∂2u
∂x2, . . .
)is Hamiltonian if it is of the form
f
(u,∂u
∂x,∂2u
∂x2, . . .
)= S δH
δu
where S is a constant skew operator, andδHδu
is the variationalderivative of H.(Note: f, u, and x can also be taken to be vectors. Although uis usually real, it may also be complex).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Hamiltonian PDEsSo a Hamiltonian PDE is written
∂u
∂t= S
δHδu
Definition (Variational derivative)Let
H(u) :=
∫H
(u,∂u
∂x,∂2u
∂x2, . . .
)dx
thenδHδu
:=∂H
∂u− ∂
∂x
(∂H
∂ux
)+
∂2
∂x2
(∂H
∂uxx
)− · · ·
Definition (skew operator)S is skew if 〈a.Sb〉 = −〈b.Sa〉, ∀a, b where 〈a.c〉 :=
∫a(x)b(x)dx
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Hamiltonian PDEsSo a Hamiltonian PDE is written
∂u
∂t= S
δHδu
Definition (Variational derivative)Let
H(u) :=
∫H
(u,∂u
∂x,∂2u
∂x2, . . .
)dx
thenδHδu
:=∂H
∂u− ∂
∂x
(∂H
∂ux
)+
∂2
∂x2
(∂H
∂uxx
)− · · ·
Definition (skew operator)S is skew if 〈a.Sb〉 = −〈b.Sa〉, ∀a, b where 〈a.c〉 :=
∫a(x)b(x)dx
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Hamiltonian PDEsSo a Hamiltonian PDE is written
∂u
∂t= S
δHδu
Definition (Variational derivative)Let
H(u) :=
∫H
(u,∂u
∂x,∂2u
∂x2, . . .
)dx
thenδHδu
:=∂H
∂u− ∂
∂x
(∂H
∂ux
)+
∂2
∂x2
(∂H
∂uxx
)− · · ·
Definition (skew operator)S is skew if 〈a.Sb〉 = −〈b.Sa〉, ∀a, b where 〈a.c〉 :=
∫a(x)b(x)dx
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (1. sine-Gordon equation (a))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn − ϕn−1
∆x
)
H = ∆x∑n
[1
2π2n +
1
2(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
),∇ denotes standard gradient
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (1. sine-Gordon equation (a))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn − ϕn−1
∆x
)
H = ∆x∑n
[1
2π2n +
1
2(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
),∇ denotes standard gradient
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (1. sine-Gordon equation (a))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn − ϕn−1
∆x
)
H = ∆x∑n
[1
2π2n +
1
2(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
),∇ denotes standard gradient
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 5 10 15 20 25 308
7.999
7.998
7.997
7.996
7.995
7.994
7.993
7.992
7.991
7.99
time
Ener
gy
SG equation (B.D.), Nspace= 101, ∆t = 0.01
avf2Midpointexact
0 5 10 15 20 250
0.005
0.01
0.015
time
Glo
bal
Err
or
SG equation (B.D.), Nspace= 101, ∆t = 0.01
avf2Midpoint
Figure: sine-Gordon equation, Backward Differences for spatialdiscretizations : Energy (left) and global error (right) vs time, forAVF and implicit midpoint integrators.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (2. sine-Gordon equation (b))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn+1 − ϕn−1
2∆x
)
H = ∆x∑n
[1
2π2n +
1
8(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (2. sine-Gordon equation (b))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn+1 − ϕn−1
2∆x
)
H = ∆x∑n
[1
2π2n +
1
8(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (2. sine-Gordon equation (b))
∂2ϕ
∂t2=∂2ϕ
∂x2+ α sinϕ
Continuous:
H =
∫ [1
2π2 +
1
2
(∂ϕ
∂x
)2
+ α(1− cos(ϕ))
]dx
S =
(0 1−1 0
).
Semi-discrete(∂ϕ
∂x→ ϕn+1 − ϕn−1
2∆x
)
H = ∆x∑n
[1
2π2n +
1
8(∆x)2(ϕn − ϕn−1)2 + α (1− cos(ϕn))
]S =
(0 id−id 0
).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 5 10 15 20 25 308
7.999
7.998
7.997
7.996
7.995
7.994
7.993
7.992
7.991
7.99
time
Ener
gy
SG equation (C.D.), Nspace= 101, ∆t = 0.01
avf2Midpointexact
0 5 10 15 20 250
0.005
0.01
0.015
time
Glo
bal
Err
or
SG equation (C.D.), Nspace= 101, ∆t = 0.01
avf2Midpoint
Figure: sine-Gordon equation, Central Differences for spatialdiscretizations : Energy (left) and global error (right) vs time, forAVF and implicit midpoint integrators.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (3. Korteweg-deVries equation (a))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → u3n)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − u3n
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (3. Korteweg-deVries equation (a))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → u3n)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − u3n
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (3. Korteweg-deVries equation (a))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → u3n)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − u3n
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 0.5 1 1.5 2 2.5 3 3.5 4211.27
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211.21
211.2
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time
Ener
gy
KdV equation, Nspace= 401, ∆t = 0.001
avf2Midpointexact
Figure: KdV equation: Exact energy, and energy vs time given byAVF and implicit midpoint methods, using discretization (a).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (4. Korteweg-deVries equation (b))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → unun+1un+2) (for illustrative purposes only)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − unun+1un+2
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Note: Large freedom in semi-discretisation H without destroyingenergy preservation. But S must be skew!
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (4. Korteweg-deVries equation (b))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → unun+1un+2) (for illustrative purposes only)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − unun+1un+2
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Note: Large freedom in semi-discretisation H without destroyingenergy preservation. But S must be skew!
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (4. Korteweg-deVries equation (b))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → unun+1un+2) (for illustrative purposes only)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − unun+1un+2
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Note: Large freedom in semi-discretisation H without destroyingenergy preservation. But S must be skew!
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (4. Korteweg-deVries equation (b))
∂u
∂t= −6u
∂u
∂x− ∂3u
∂x3
Continuous:
H =
∫ [1
2u2x − u3
]dx
S =∂
∂x
Semi-discrete: (u3 → unun+1un+2) (for illustrative purposes only)
H = ∆x∑n
[1
2(∆x)2(un − un−1)2 − unun+1un+2
]
S =1
2∆x
0 1 . . . −1−1 0 1 0
−1 0 1...
. . .
−1 0 11 −1 0
Note: Large freedom in semi-discretisation H without destroyingenergy preservation. But S must be skew!
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 0.5 1 1.5 2 2.5 3 3.5 4207.345
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time
Ener
gy
KdV equation, Nspace= 401, ∆t = 0.001
avf2Midpoint
Figure: KdV equation: Energy vs time given by AVF and implicitmidpoint methods, using discretization (b).
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (5. NLS equation)Continuous:
∂
∂t
(uu∗
)=
(0 i−i 0
)(δHδuδHδu∗
), (1)
where u∗ denotes the complex conjugate of u.
H =
∫ [−∣∣∣∣∂u∂x
∣∣∣∣2 +γ
2|u|4]dx, (2)
S =
(0 i−i 0
). (3)
Semi-discrete:
H =∑j
[− 1
(∆x)2|uj+1 − uj |2 +
γ
2|uj |4
], (4)
S = i
(0 id−id 0
). (5)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (5. NLS equation)Continuous:
∂
∂t
(uu∗
)=
(0 i−i 0
)(δHδuδHδu∗
), (1)
where u∗ denotes the complex conjugate of u.
H =
∫ [−∣∣∣∣∂u∂x
∣∣∣∣2 +γ
2|u|4]dx, (2)
S =
(0 i−i 0
). (3)
Semi-discrete:
H =∑j
[− 1
(∆x)2|uj+1 − uj |2 +
γ
2|uj |4
], (4)
S = i
(0 id−id 0
). (5)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 10 20 30 40 50−2
0
2
4
6
8
10
12
14
16x 10
−6
time
Energ
yE
rror
NLS equation, Nspace= 201, Δt = 0.05
avf2Midpoint
0 10 20 30 40 5010
−5
10−4
10−3
10−2
10−1
time
Glo
bal
Err
or
NLS equation, Nspace= 201, Δt = 0.05
avf2Midpoint
Figure: Nonlinear Schrödinger equation: Energy error (left) andglobal error (right) vs time, for AVF and implicit midpointintegrators.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 10 20 30 40 50−2
0
2
4
6
8
10
12
14x 10
−6
time
Pro
babil
ity
Err
or
NLS equation, Nspace= 201, Δt = 0.05
avf2Midpt
Figure: Nonlinear Schrödinger equation: Total probability error vstime, for AVF and implicit midpoint integrators.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (6. Nonlinear Wave Equation)
∂2ϕ
∂t2= (∂2x + ∂2y)ϕ− ϕ3
Continuous:
H =
∫ 1
−1
∫ 1
−1
[1
2(π2 + ϕ2
x + ϕ2y) +
1
4ϕ4
]dx dy
S =
(0 1−1 0
)
Semi-discrete: We discretize the Hamiltonian in space with a tensorproduct Lagrange quadrature formula based on p+ 1Gauss-Lobatto-Legendre (GLL) quadrature nodes in each spacedirection:
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Example (6. Nonlinear Wave Equation)
∂2ϕ
∂t2= (∂2x + ∂2y)ϕ− ϕ3
Continuous:
H =
∫ 1
−1
∫ 1
−1
[1
2(π2 + ϕ2
x + ϕ2y) +
1
4ϕ4
]dx dy
S =
(0 1−1 0
)Semi-discrete: We discretize the Hamiltonian in space with a tensorproduct Lagrange quadrature formula based on p+ 1Gauss-Lobatto-Legendre (GLL) quadrature nodes in each spacedirection:
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Nonlinear Wave Equation II
H =1
2
p∑j1=0
p∑j2=0
wj1wj2
(π2j1,j2 +
(p∑
k=0
dj1,kϕk,j2
)2
+
(p∑
m=0
dj2,mϕj1,m
)2
+1
2ϕ4j1,j2
)
where dj1,k = dlk(x)dx
∣∣∣x=xj1
, and lk(x) is the k-th Lagrange basis
function based on the GLL quadrature nodes x0, . . . , xp, and withw0, . . . , wp the corresponding quadrature weights.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
T = 0 T = 3.1250
−1−0.5
00.5
1
−1−0.5
00.5
1−5
0
5
10
15
20x 10−3
−1−0.5
00.5
1
−1−0.5
00.5
1−5
0
5
10
15x 10−3
T = 6.875 T = 10
−1−0.5
00.5
1
−1−0.5
00.5
1−2
0
2
4
6
8x 10−3
−1−0.5
00.5
1
−1−0.5
00.5
1−5
0
5
10
15
20x 10−3
Figure: Snapshots of the solution of the 2D wave equation at differenttimes. AVF method with step-size ∆t = 0.6250. Space discretizationwith 6 Gauss Lobatto nodes in each space direction. Numericalsolution interpolated on a equidistant grid of 21 nodes in each spacedirection.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
0 2 4 6 8 10
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5 x 10−11
time
Ener
gy e
rror
2D Wave equation avf and ode15s
Figure: The 2D wave equation. MATLAB routine ode15s withabsolute and relative tolerance 10−14 (dashed line), and AVF methodwith step size ∆t = 10/(25) (solid line). Energy error versus time.Time interval [0, 10]. Space discretization with 6 Gauss Lobatto nodesin each space direction.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Leapfrog
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Exponential Integrator
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• Instead of using finite differences for the (semi-)discretization ofthe spatial derivatives one may use e.g. a spectral discretization.
• The method presented also applies to linear PDEs, e.g. the(Linear) Time-dependent Schrödinger equation, 3D Maxwell’sequation.
• The method presented can be generalised to
∂u
∂t= N δH
δu
where N is a constant negative (semi) definite operator, and whereH is a Lyapunov function, i.e. ∂H
∂t ≤ 0.(e.g. Allen-Cahn eq, Cahn-Hilliard eq, Ginzburg-Landau eq.)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• Instead of using finite differences for the (semi-)discretization ofthe spatial derivatives one may use e.g. a spectral discretization.
• The method presented also applies to linear PDEs, e.g. the(Linear) Time-dependent Schrödinger equation, 3D Maxwell’sequation.
• The method presented can be generalised to
∂u
∂t= N δH
δu
where N is a constant negative (semi) definite operator, and whereH is a Lyapunov function, i.e. ∂H
∂t ≤ 0.(e.g. Allen-Cahn eq, Cahn-Hilliard eq, Ginzburg-Landau eq.)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• Instead of using finite differences for the (semi-)discretization ofthe spatial derivatives one may use e.g. a spectral discretization.
• The method presented also applies to linear PDEs, e.g. the(Linear) Time-dependent Schrödinger equation, 3D Maxwell’sequation.
• The method presented can be generalised to
∂u
∂t= N δH
δu
where N is a constant negative (semi) definite operator, and whereH is a Lyapunov function, i.e. ∂H
∂t ≤ 0.(e.g. Allen-Cahn eq, Cahn-Hilliard eq, Ginzburg-Landau eq.)
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• The method can also be generalised to preserving any number ofintegrals (not necessarily energy).
• If one replaces the integral in the Average Vector Field byquadrature of a certain order, the resulting method will preserveenergy to a certain order.
• The Average Vector Field Method is a B-series method.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• The method can also be generalised to preserving any number ofintegrals (not necessarily energy).
• If one replaces the integral in the Average Vector Field byquadrature of a certain order, the resulting method will preserveenergy to a certain order.
• The Average Vector Field Method is a B-series method.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
GENERALIZATIONS AND FURTHER REMARKS• The method can also be generalised to preserving any number ofintegrals (not necessarily energy).
• If one replaces the integral in the Average Vector Field byquadrature of a certain order, the resulting method will preserveenergy to a certain order.
• The Average Vector Field Method is a B-series method.
Reinout QuispelEnergy Preserving Numerical Integration Methods
PART I PART II
Some references to our workODEs1. McLachlan, Quispel & Robidoux, A unified approach to Hamiltoniansystems, Poisson systems, gradient systems and systems with Lyapunovfunctions and/or first integrals, Phys. Rev. Lett. 81(1998)2399–2103.2. McLachlan, Quispel & Robidoux, Geometric integration usingdiscrete gradients, Phil. Trans. Roy. Soc. A357(1999)1021–1045.3. Quispel & McLaren, A new class of energy-preserving numericalintegration methods, J. Phys A41(2008)045206 (7pp).
PDEs4. Celledoni, Grimm, McLachlan, McLaren, O’Neale & Quispel,Preserving energy resp. dissipation in numerical PDEs, using the‘Average Vector Field’ method, Submitted for publication.
Of course there are also many important publications byFurihata, Matsuo, Yaguchi, and collaborators.
Reinout QuispelEnergy Preserving Numerical Integration Methods