Discontinuous Collocation Methods for DAEs in Mechanics · Collocation methods (and discontinuous...
Transcript of Discontinuous Collocation Methods for DAEs in Mechanics · Collocation methods (and discontinuous...
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Discontinuous Collocation Methods for DAEs inMechanics
Scott Small Laurent O. Jay
The University of Iowa
Iowa City, Iowa, USA
July 11, 2011
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Differential-Algebraic Equations
We consider overdetermined mixed index 2 and 3 DAEs:
y = v(t, y , z)
z = f (t, y , z , ψ) + r(t, y , λ)
0 = g(t, y)
0 = gt(t, y) + gy (t, y)v(t, y , z)
0 = k(t, y , z)
The functions
v : R× Rny × R
nz → Rny
f : R× Rny × R
nz × Rnk → R
nz
r : R× Rny × R
ng → Rnz
g : R× Rny → R
ng
k : R× Rny × R
nz → Rnk
are assumed to be sufficiently differentiable.
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Differential-Algebraic Equations
The two matrices
gy (t, y)vz(t, y , z)rλ(t, y , λ)[gyvz(t, y , z)kz(t, y , z)
] [rλ(t, y , λ) fψ(t, y , z)
]
are assumed invertible. These assumptions allow for the system ofDAEs to be expressed as ODEs. These are the so called underlyingODEs.
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Examples from Mechanics
Well-known formulations of classical mechanics are HamiltonianMechanics and Lagrangian Mechanics. If the evolution of thesystem is constrained, the result is a DAE.
In the context of classical mechanics,
Holonomic constraints are restrictions on the coordinates of asystem. For example, the position of a pendulum in Cartesiancoordinates is constrained to a circle or sphere.
Nonholonomic constraints are (nonintegrable) restrictions onthe velocities of a system. For example, an ice skate isconstrained to move in the direction the blade is pointing.
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Examples from Mechanics
Lagrangian System
q = v
d
dt∇vL(t, q, v) = ∇qL(t, q, v)− gq(t, q)
Tλ− kv (t, q, v)Tψ
0 = g(t, q)
0 = gt(t, q) + gq(t, q)v
0 = k(t, q, v)
Hamiltonian System
q = ∇pH(t, q, p)
p = −∇qH(t, q, p)− gq(t, q)Tλ− kp(t, q, p)
Tψ
0 = g(t, q)
0 = gt(t, q) + gq(t, q)∇pH(t, q, p)
0 = k(t, q, p)
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Runge-Kutta Type Methods
We introduce two classes of Runge-Kutta type methods for solvingoverdetermined mixed index 2 and 3 DAEs.
Specialized partitioned additive Runge-Kutta (SPARK)methods were introduced by Jay in 1998, and can be appliedto systems with nonholonomic constraints. Methods weredeveloped in 2007 and can be applied to systems withholonomic constraints.
Murua’s partitioned Runge-Kutta methods were proposed byMurua in 1996 for index 2 DAEs, and can be applied tosystems with nonholonomic constraints.
We consider in this presentation an extension of the SPARKmethods for systems of mixed index 2 and 3 DAEs. We consideralso the extended Murua’s partitioned Runge-Kutta (EMPRK)methods to systems of mixed index 2 and 3 DAEs.
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An Example Method
(1, 1)-SPARK/EMPRK Midpoint-Trapezoidal Method
Y1 = y0 +h
2v(t0 +
1
2h,Y1,Z1)
Z1 = z0 +h
2f (t0 +
1
2h,Y1,Z1,Ψ1) +
h
2r(t0, y0,Λ0)
y1 = y0 + h v(t0 +1
2h,Y1,Z1)
z1 = z0 + h f (t0 +1
2h,Y1,Z1,Ψ1) +
h
2r(t0, y0,Λ0) +
h
2r(t1, y1,Λ1)
0 = g(t1, y1)
0 = gt(t1, y1) + gy (t1, y1)v(t1, y1, z1)
0 = k(t1, y1, z1)
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The (s, s)-SPARK/EMPRK Methods
The internal stages
Yi = y0 + hs∑
j=1
aijv(t0 + cjh,Yj ,Zj), i = 1, . . . , s
Zi = z0 + hs∑
j=1
aij f (t0 + cjh,Yj ,Zj ,Ψj) + hs∑
j=0
aij r(t0 + cjh, Yj ,Λj),
i = 1, . . . , s
Yi = y0 + h
s∑
j=1
aijv(t0 + cjh,Yj ,Zj), i = 0, . . . , s
Zi = z0 + hs∑
j=1
aij f (t0 + cjh,Yj ,Zj ,Ψj) + hs∑
j=0
aij r(t0 + cjh, Yj ,Λj),
i = 0, . . . , s
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The (s, s)-SPARK/EMPRK Methods, cont.
The constraints
0 = g(t0 + cih, Yi ), i = 0, . . . , s
0 = g(t1, y1)
0 = gt(t1, y1) + gy(t1, y1)v(t1, y1, z1)
0 =
s∑
j=1
bjci−1j k(t0 + cjh,Yj ,Zj), i = 1, . . . , s − 1
0 = k(t0 + cih, Yi , Zi), i = 0, . . . , s
0 = k(t1, y1, z1)
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The (s, s)-SPARK/EMPRK Methods, cont.
The numerical solution
y1 = y0 + h
s∑
j=1
bjv(t0 + cjh,Yj ,Zj)
z1 = z0 + h
s∑
j=1
bj f (t0 + cjh,Yj ,Zj ,Ψj) + h
s∑
j=0
bj r(t0 + cjh, Yj ,Λj)
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Gauss-Lobatto Coefficients
An example of a class of methods is the (s, s)-Gauss-Lobattomethods.
Use s-stage Gauss coefficients for the aij , bi , ci
Use (s + 1)-stage Lobatto coefficients for the bi , ci
Use (s + 1)-stage Lobatto-IIIA for aij
Define aij and aij by
s∑
j=1
aijck−1j =
ckik, k = 1, . . . , s
a0j = 0, j = 1, . . . , ss∑
j=0
aij ck−1j =
ckik, k = 1, . . . , s
ai0 = b0, i = 1, . . . , s.
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Existence and Uniqueness
Theorem
Suppose that y0 = y0(h), z0 = z0(h), λ0 = λ0(h), ψ0 = ψ0(h) satisfy
o(h2) = g(t0, y0)
o(h) =d
dt(g(t, y))(t0, y0, z0)
o(1) =d2
dt2(g(t, y))(t0, y0, z0, λ0, ψ0)
o(h) = k(t0, y0, z0)
o(1) =d
dt(k(t, y , z))(t0, y0, z0, λ0, ψ0).
Then the SPARK and EMPRK methods possess a unique solution for hsufficiently small.
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Determining Local Order
Local order is defined through series expansions of thenumerical and exact solutions. However, using seriesexpansions to determine local order is tedious.
Collocation methods (and discontinuous collocation methods),by contrast, have a much cleaner derivation of their localorder.
We are thus interested in showing the equivalence of theSPARK and EMPRK methods to a class of discontinuouscollocation type methods.
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Collocation and Discontinuous Collocation
Continuous Collocation Discontinuous Collocation
Hairer, Nørsett, Wanner. Solving Ordinary Differential Equations I, Nonstiff
Problems. 2000
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Discontinuous Collocation Type Methods
Let c1, . . . , cs be distinct real numbers, and c0, . . . , cs also bedistinct real numbers, with c0 = 0 and cs = 1. Assume also thatb0, bs are positive real numbers. We then define the s-degreepolynomials Y (t), Λ(t), Ψ(t), Z f (t), the (s + 1)-degreepolynomials Z (t), Z r (t), and the (s + 2)-degree polynomials Z(t),Z r (t) as the polynomials satisfying the initial conditions
Y (t0) = y0,
Z f (t0) = z0, Z r (t0) = −hb0µ(t0),
Z r (t0) = 0,
Z (t0) = Z f (t0) + Z r (t0) = z0 − hb0µ(t0)
Z(t0) = Z f (t0) + Z r (t0) = z0,
where
µ(t) := Z r (t)− r(t,Y (t),Λ(t)),
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Discontinuous Collocation Type Methods
and the conditions
Y (t0 + cih) = v(t0 + cih,Y (t0 + cih),Z (t0 + cih)), i = 1, . . . , s
Z f (t0 + cih) = f (t0 + cih,Y (t0 + cih),Z (t0 + cih),Ψ(t0 + cih)),
i = 1, . . . , s
Z r (t0 + cih) = r(t0 + cih,Y (t0 + cih),Λ(t0 + cih)), i = 1, . . . , s − 1
Z (t) = Z f (t) + Z r (t)
˙Z
r
(t0 + cih) = r(t0 + cih,Y (t0 + cih),Λ(t0 + cih)), i = 0, . . . , s
Z(t) = Z f (t) + Z r (t)
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Discontinuous Collocation Type Methods
0 = g(t0 + cih,Y (t0 + cih)), i = 0, . . . , s
0 = gt(t1, y1) + gy (t1, y1)v(t1, y1, z1)
0 = k(t0 + cih,Y (t0 + cih), Z (t0 + cih)), i = 0, . . . , s
0 =
s∑
j=1
bjci−1j k(t0 + cjh,Y (t0 + cjh),Z (t0 + cjh)), i = 1, . . . , s − 1
0 = k(t1, y1, z1).
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Discontinuous Collocation Type Methods
The numerical solution at t1 = t0 + h is taken to be
y1 := Y (t1)
z1 := Z (t1)− hbsµ(t1).
These discontinuous collocation type methods can be shown to beequivalent to the Gauss-Lobatto SPARK and EMPRK methods.
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Discontinuous Collocation Type Methods
Theorem
A SPARK/EMPRK method with distinct values for cj and for cj isa discontinuous collocation method iff the coefficients satisfy
s∑
j=1
aijck−1j =
ckik,
s∑
j=1
bjck−1j =
1
k, k = 1, . . . , s
s∑
j=0
aij ck−1j =
ckik,
s∑
j=0
bj ck−1j =
1
k, k = 1, . . . , s − 1
ai0 = b0, ais = 0s∑
j=1
aijck−1j =
ckik, k = 1, . . . , s − 1
s∑
j=0
aij ck−1j =
ckik, k = 1, . . . , s + 1.
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Local Error for Gauss-Lobatto SPARK/EMPRK Methods
Theorem
The (s, s)-Gauss-Lobatto SPARK and EMPRK methods withconsistent initial values have local order 2s, i.e., for |h| ≤ h0,
y1 − y(t1) = O(h2s+1), z1 − z(t1) = O(h2s+1).
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Convergence
Theorem
Consider the (s, s)-Gauss-Lobatto SPARK / EMPRK methods withconsistent initial conditions (y0, z0) at time t0. Then the(s, s)-Gauss-Lobatto SPARK / EMPRK methods are convergent oforder 2s, i.e.
yn − y(tn) = O(h2s), zn − z(tn) = O(h2s),
where yn and zn are the numerical solution at time tn := t0 + nh,for nh ≤ Const.
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Numerical Example - Simple Pendulum
Consider a simple pendulum of mass m and length ℓ.
L(t, q, v) := T − U, T :=1
2m(v21 + v22 ), U := −mγq2,
0 = g(t, q) =1
2(q21 + q22 − ℓ2).
m is the mass (m = 1)
ℓ is the length of the bob (ℓ = 1)
γ is the acceleration due to gravity (γ = 1)
The initial conditions are
q0 = (1 0)T , v0 = (0 0)T .
The system is integrated from 0 to 100.
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Numerical Example - Simple Pendulum
10−2
10−1
100
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Global Error for Gauss−Lobatto Methods − Pendulum
Step Size
Err
or (
in 2
−no
rm)
1 Stage2 Stage
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Numerical Example - Simple Pendulum
(2, 2)–Gauss-Lobatto, h = .05
0 10 20 30 40 50 60 70 80 90 100−7
−6
−5
−4
−3
−2
−1
0
1x 10
−7 Energy Error
Time
Ene
rgy
− E
0
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Numerical Example - Simple Pendulum
(2, 2)–Gauss-Lobatto, h = .01
0 10 20 30 40 50 60 70 80 90 100−7
−6
−5
−4
−3
−2
−1
0
1
2x 10
−11 Energy Error
Time
Ene
rgy
− E
0
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Numerical Example - Skate on an Inclined Plane
Consider a skate sliding on an inclined plane without friction.
L(t, q, v) := T − U
T :=1
4m(v21 + v2
2 + v23 + v2
4
)U := −
1
2mγ sin(β)(q1 + q3)
0 = g(t, q) =1
2
((q3 − q1)
2 + (q4 − q2)2 − ℓ2
)
0 = k(t, q, v) = −(q4 − q2)(v1 + v3) + (q3 − q1)(v2 + v4)
m is the mass of the skate (m = 1)
ℓ is the length of the skate (ℓ = 2)
γ is the acceleration due to gravity, β is the incline of the plane(γ sin(β) = 1)
The initial conditions are
q0 = (−1/2 0 1/2 0)T
v0 = (0 − 1/2 0 1/2)T.
The system is integrated from 0 to 10.
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Numerical Example - Skate on an Inclined Plane
10−2
10−1
100
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Step Size
Err
or (
in 2
−no
rm)
Global Error for Gauss−Lobatto SPARK Methods − Skate on Inclined Plane
1 Stage2 Stage
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Numerical Example - Skate on an Inclined Plane
10−2
10−1
100
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Global Error for Gauss−Lobatto Murua Methods − Skate on Inclined Plane
Step Size
Err
or (
in 2
−no
rm)
1 Stage2 Stage
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Numerical Example - Skate on an Inclined Plane
(2, 2)–Gauss-Lobatto SPARK Method, h = .1
0 1 2 3 4 5 6 7 8 9 10−7
−6
−5
−4
−3
−2
−1
0x 10
−7 Energy Error
Time
En−
E0
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Numerical Example - Skate on an Inclined Plane
(2, 2)–Gauss-Lobatto EMPRK Method, h = .1
0 1 2 3 4 5 6 7 8 9 10−3
−2
−1
0x 10
−10 Energy Error
Time
En−
E0
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Outline
1 Introduction of the DAEs
2 SPARK and EMPRK Methods
3 Discontinuous Collocation Methods
4 Local Order and Convergence
5 Numerical Experiments
6 Conclusion
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Conclusion
SPARK and EMPRK methods are Runge-Kutta style methodsfor solving overdetermined mixed index 2 and 3 DAEs.
These methods have a unique solution.
These methods can be expressed as discontinuous collocationmethods.
For the Gauss-Lobatto coefficients, the SPARK and EMPRKmethods are of order 2s.