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Differential-Algebraic Equations(DAEs) and numerical methods
by Sirui Tan
Definition of DAEs
F (t, y , y ′) = 0,
where y:R→Rm is the solution, F :R×R
m ×Rm→R
m is a given function.∂ F
∂ y ′nonsingular ⇒ explicit ODE
y ′ = f(t, y).
semi-explicit DAE (or ODE with constraints)
x′ = f(t, x, z), (differential equation)
0 = g(t, x, z). (algebraic eqution)
2
Example 1. simple pendulum
Figure 1. simple pendulum
z(t) Lagrange multiplier
x1′′ = − z x1,
x2′′ = − z x2− g,
x12 + x2
2 = 1.
In standard form
x1′ = x3,
x2′ = x4,
x3′ = − z x1,
x4′ = − z x2− g,
0 = z12 + z2
2− 1.
3
Index
Minimum number of differentiations of the system which would be required tosolve for y ′ uniquely in terms of y and t (i.e., to obtain an explicit ODE).
It measures the distance from a DAE to its related ODE. It reveals the mathe-matical structure and potential complications in the analysis and the numericalsolution of the DAE.
The higher the index of a DAE, the more difficulties for its numerical solution.
Example:
• index-1 DAE
y = q(t).
since y ′= q ′(t) explicit ODE.
• index-2 DAE
y1 = q(t),
y2 = y1′.
since y2′ = y1
′′= q ′′(t).
4
Special DAE forms
• Hessenberg Index-1 system (or semi-explicit index-1 system)
x′ = f(t, x, z),
0 = g(t, x, z).
Jacobian matrix gz nonsingular.
x(t) differential variables, z(t) index-1 algebraic variables.
• Hessenberg Index-2 system
x′ = f(t, x, z),
0 = g(t, x).
gx fz is nonsingular.
z(t) index-2 variables. Pure index-2 DAE.
5
Example 2. Navier-Stokes equations
ut + u ux + v uy + px − ν(uxx + uyy) = 0,
vt + u vx + v vy + py − ν (vxx + vyy) = 0,
ux + vy = 0.
after spatial discretization (FD, FV, FE)
M u′+ (K + N(u)) u + C p = f ,
CT u = 0.
CT M−1 C is a nonsingular matrix with a bounded inverse⇒Hessenberg Index-2system.
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Consistent initial values
Index-2 system
x(t) = sin(t)
x′ (t)+ y(t) = 0.
consistent� satisfying the algebraic constraint
x(t) = sin(t)
and hidden constraints
y(t) = − x′(t)=− cos(t).
Consistent initial values
x(0) = sin(0),
y(0) = − cos(0).
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Applications
• constrained mechanical systems (e.g. simple pendulum, N-S equations)
• electrical circuits
large but sparse DAE of the form
M(y) y ′+ f(y) = q(t).
• chemical reaction kinetics
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Numerical methods
Semi-explicit index-1 or index-2 DAE
x′ = f(t, x, z),
0 = g(t, x, z).
limiting case of the singularly perturbedODE
x′ = f(t, x, z),
εz ′ = g(t, x, z),
where 0 6 ε≪ 1.
Observations: consider methods for stiff ODEs.
9
Crank-Nicolson method
Explicit ODE
y ′ = f(t, y).
C-N methodyn − yn−1
h=
1
2(f (tn, yn) + f(tn−1, yn−1)).
Index-2 DAE
x(t) = sin(t)
x′(t)+ y(t) = 0
exact solution
x(t) = sin(t),
y(t) = − cos(t).
C-N method1
2(xn + xn−1) =
1
2(sin(tn)+ sin(tn−1))
xn −xn−1
h+
1
2(yn + yn−1) = 0.
10
Exact initial values
x0 = sin(0),
y0 = − cos(0).
0 1 2 3 4 5 6 7 8 9−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x(t)
y(t)
Figure 2. C-N method, exact initial values, h = 0.04.
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Perturbed initial values
x0 = sin(0) + h3,
y0 = − cos(0).
0 1 2 3 4 5 6 7 8 9 10−3
−2
−1
0
1
2
3
x(t)
y(t)
Figure 3. C-N method, perturbed initial values, h = 0.04.
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Conclusion: A-stability is not enough.
x′ = f(t, x, z),
εz ′ = g(t, x, z),
where 0 6 ε≪ 1.
Set ε= 0 ⇒ the limit DAE.
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Methods with stiff decay
test equation
y ′ = λ(y − g(t)),
or
ε y ′ = λ̂ (y − g(t)),
where ε=1
|Re(λ)|, λ̂ = ε λ. |λ̂ |= 1.
Set ε= 0⇒ y(t) = g(t).
The method has stiff decay if tn > 0 fixed
yn→ g(tn) as ε→ 0 + .
C-N method does not have stiff decay.
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Backward Euler method
Index-2 DAE
x(t) = sin(t),
x′(t)+ y(t) = 0.
BE method
xn = sin(tn),xn − xn−1
h+ yn = 0.
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Exact initial values
x0 = sin(0),
y0 = − cos(0).
0 1 2 3 4 5 6 7 8 9 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x(t)
y(t)
Figure 4. BE method, exact initial values, h = 0.04.
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Perturbed initial values
x0 = sin(0) + h,
y0 = − cos(0).
0 1 2 3 4 5 6 7 8 9 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x(t)
y(t)
Figure 5. BE method, perturbed initial values, h = 0.04.
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Multistep methods
The only method with stiff decay is the BDF method.
Semi-explicit index-1 or index-2 DAE
x′ = f(t, x, z),
0 = g(t, x, z).
BDF method
1
β0 h
∑
j=0
k
αj xn−j = f (tn, xn, zn),
0 = g(tn, xn, zn).
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Convergence of BDF
Theorem: the k-step BDF method is convergent of order p= k for k 6 6, i.e.,
xn −x(tn) = O(hp), zn − z(tn) = O(hp),
whenever the initial values satisfy
• if k 6 2
xj −x(tj)= O(hp+1) for j = 0,� , k − 1.
• if k > 3
xj −x(tj)= O(hp) for j = 0,� , k − 1.
19
Multistage methods
If we consider collocation methods, the only method with stiff decay is theRadau method.
Semi-explicit index-1 or index-2 DAE
x′ = f(t, x, z),
0 = g(t, x, z).
Radau method
ti = tn−1 + ci h, i = 1,� , s,
Ki = f(ti, Xi, Zi),
Xi = xn−1 + h∑
j=1
s
aij Kj ,
0 = g(ti, Xi, Zi).
and (since asj = bj , j = 1,� , s.)
xn = Xs, zn = Zs.
20
Convergence of the Radau method
Theorem: the s-stage Radau method is convergent of order 2 s− 1 for xn and oforder s for zn. (No order reduction for xn.)
21
Practical difficulties
• obtaining a consistent set of initial conditions
Given x0, solve for z0 from 0 = g(0, x0, z0). What’s the initial guess?
• ill-conditioning of iteration matrix
e.g. backward Euler method, iteration matrix
(
hn−1 I − fx − fz
− gx − gz
)
condition number O(hn−1). If hn small, Newton iteration fails?
• Error estimation for index-2 DAEs
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Available codes
code method authors DAE index
DASSL/DASPK BDF Petzold 6 1
RADAU5 IRK Hairer, Wanner 6 3
MEBDF MEBDF Abdulla, Cash 6 3
Table 1. a list of available codes for DAEs
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Summary
• DAEs are generalizations of ODEs. The index indicates the distance fromits underlying ODE and thus the difficulty. We should be careful whenimposing initial conditions.
• Methods with stiff decay perform well for solving DAEs, e.g, BDF methodand the Radau collocation method.
24
References
• Uri M. Ascher and Linda R. Petzold, Computer Methods for OrdinaryDifferential Equations and Differential-Algebraic Equations.
• E. Hairer and G. Wanner, Solving Ordinary Differential Equations II:Stiff and Differential-Algebraic Equations.
• K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution ofInitial-Value Problems in Differential-Algebraic Equations.
• Series of lectures on DAEs, URL:http://www.win.tue.nl/casa/meetings/seminar/previous/
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