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Introduction to Simulation - Lecture 14
Multistep Methods II
Jacob White
Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Outline
Small Timestep issues for Multistep MethodsReminder about LTE minimizationA nonconverging exampleStability + Consistency implies convergence
Investigate Large Timestep IssuesAbsolute Stability for two time-scale examples.Oscillators.
Multistep MethodsGeneral Notation
Basic Equations
( ) ( ( ), ( ))d x t f x t u tdt
=Nonlinear Differential Equation:
k-Step Multistep Approach: ( )( )0 0
ˆ ˆ ,k k
l j l jj j l j
j jx t f x u tα β− −
−= =
= ∆∑ ∑
Solution at discrete points
Time discretization
Multistep coefficients
2ˆ lx −
lt1lt −2lt −3lt −l kt −
ˆ lx1ˆ lx −
ˆ l kx −
Multistep MethodsSimplified Problem for
Analysis
( ) ( ) 0( ), 0d v t v t v vdt
λ λ= = ∈
Must Consid ALLer
Scalar ODE:
0 0
ˆ ˆk k
l j l jj j
j jv t vα β λ− −
= =
= ∆∑ ∑Scalar Multistep formula:
λ ∈
Growing Solutions
DecayingSolutions
Oscillations
( )Im λ
( )Re λ
Multistep MethodsConvergence Analysis
Convergence Definition
Definition: A multistep method for solving initial value problems on [0,T] is said to be convergent if given any
initial condition
( )0,
ˆmax 0 as t 0lTlt
v v l t⎡ ⎤∈⎢ ⎥∆⎣ ⎦
− ∆ → ∆ →
exactv
tˆ computed with 2
lv ∆ˆ computed with tlv ∆
Multistep MethodsConvergence Analysis
Two Conditions for Convergence
1) Local Condition: “One step” errors are small (consistency)
Typically verified using Taylor Series
2) Global Condition: The single step errors do not grow too quickly (stability)
Multi-step (k > 1) methods require careful analysis.
Multistep MethodsConvergence Analysis
Global Error Equation
0 0
ˆ ˆ 0k k
l j l jj j
j jv t vα β λ− −
= =
− ∆ =∑ ∑Multistep formula:
Exact solution Almostsatisfies Multistep Formula: ( ) ( )
0 0
k kl
j l j j l jj j
dv t t v t edt
α β− −= =
− ∆ =∑ ∑Local Truncation Error
(LTE)( ) ˆl llE v t v≡ −Global Error:
Difference equation relates LTE to Global error( ) ( ) ( )1
0 0 1 1l l l k l
k kt E t E t E eα λ β α λ β α λ β− −− ∆ + − ∆ + + − ∆ =
Multistep MethodsMaking LTE Small
Exactness Constraints
Local Truncation Error:
LTE
( ) ( ) 1If p pdv t t v t ptdt
−= ⇒ =
( ) ( )Can't be from d v t v tdt
λ=
( ) ( )0 0
k kl
j l j j l jj j
dv t t v t edt
α β− −= =
− ∆ =∑ ∑
( )( )( )
( )( )
( )0 0
1
k jk j
k kk
j jj j
p pk j t t p k
dv t v t
t
d
e
t
jα β
−−
= =
−− ∆ −∆ − ∆ =∑ ∑
Multistep MethodsMaking LTE Small
Exactness Constraint k=2 Example
( ) ( )0 0
1Exactness Cons train 0ts:k k
j jj j
p pk j p k jα β= =
−⎛ ⎞− − − =⎜ ⎟
⎝ ⎠∑ ∑
0
1
2
0
1
2
1 1 1 0 0 0 02 1 0 1 1 1 04 1 0 4 2 0 08 1 0 12 3 0 0
16 1 0 32 4 0 0
αααβββ
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥=− − ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− − ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
For k=2, yields a 5x6 system of equations for Coefficients
p=0
p=1
p=2
p=3
p=4
i 0α =∑Note
Always
Multistep MethodsMaking LTE Small
Exactness Constraint k=2 example, generating methods
1
2
0
1
2
1 1 0 0 0 11 0 1 1 1 21 0 4 2 0 41 0 12 3 0 81 0 32 4 0 16
ααβββ
−⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥− − − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥=− − −⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦⎣ ⎦
0First introduce a normalization, for example 1α =
0 1 2 0 1 21, 0, 1, 1/ 3, 4 / 3, 1/ 3α α α β β β= = = − = = =
Solve for the 2-step explicit method with lowest LTE
Solve for the 2-step method with lowest LTE
( )5Satisfies all five exactness constraints LTE C t= ∆
( )4Can only satisfy four exactness constraints LTE C t= ∆0 1 2 0 1 21, 4, 5, 0, 4, 2α α α β β β= = = − = = =
10-4 10-3 10-2 10-1 10010-15
10-10
10-5
100
Multistep MethodsMaking LTE Small
( ) ( )d v t v td
=
Trap
Beste
LTE
FE
Timestep
LTE Plots for the FE, Trap, and “Best” Explicit (BESTE).
Best Explicit Method has highest one-step
accurate
10-4 10-3 10-2 10-1 10010-8
10-6
10-4
10-2
100
Multistep MethodsMaking LTE Small
Global Error for the FE, Trap, and “Best” Explicit (BESTE).
[ ]( ) ( ) 0,1d v t v t td
= ∈
FE
Trap
Max Error
Where’s BESTE?
Timestep
10-4
10-3
10-2
10-1
10010
-100
100
10100
10200
Multistep MethodsMaking LTE Small
Global Error for the FE, Trap, and “Best” Explicit (BESTE).
( ) ( )d v t v td
=
FE Trap
Beste
Max Error
Best Explicit Method has lowest one-step error but global errror increases as
timestep decreases
worrysome
Timestep
Multistep MethodsStability of the method
Difference Equation
Why did the “best” 2-step explicit method fail to Converge?
( ) ( ) ( )10 0 1 1
l l l k lk kt E t E t E eα λ β α λ β α λ β− −− ∆ + − ∆ + + − ∆ =
Multistep Method Difference Equation
( ) ˆlv l t v∆ −Global Error
LTE
We made the LTE so small, how come the Global error is so large?
Multistep MethodsStability of the method
Stability Definition
Multistep Method Difference Equation( ) ( ) ( )1
0 0 1 1l l l k l
k kt E t E t E eα λ β α λ β α λ β− −− ∆ + − ∆ + + − ∆ =
Definition: A multistep method is stable if as
( )0, 0,
interval dependent
max max l lT Tl lt t
TE C T et⎡ ⎤ ⎡ ⎤∈ ∈⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦
≤∆
0t∆ →
Global Error is bounded by a constant times the sum of the LTE’sStability means:
Given a kth order difference eqn with zero initial conditions1
0 , 0, , 0l l k l kka x a x u x x− − −+ + = = =
convolution sum
0
ll l j j
jx h u−
=
= ∑
( ) ( )1
,1 0
qMQ
lq m q
q m
lmh l
x can be related to the input u by
γ ς−
= =
=∑∑1
0 1 0k kka z a z a−+ + + =
Roots of
Root multiplicity
Aside on difference Equations
Convolution Sum
Root Relation
Aside on difference Equations
Convolution Sum
Bounding Terms
( ) ( )1
,1 0 0
,
qMQ ll j
q m qq m j
l jm
q mR
x l j uγ ς−
−
= = =
⎛ ⎞= −⎜ ⎟
⎝ ⎠∑∑ ∑
,If then<1, max jq q m jR C uς ≤
Independent of l
( ) 0q ,If t< 1 hen+ , maxl
jq j
eR C uε
ς εε
≤
Bounds distinct Roots
Multistep MethodsStability of the method
Stability Theorem
Theorem: A multistep method is stable if and only if
10 1Roots of 0 either: k k
kz zα α α−+ + + =
1. Have magnitude less than one2. Have magnitude equal to one and are distinct
Multistep MethodsStability of the method
Stability Theorem “Proof”
Given the Multistep Method Difference Equation( ) ( ) ( )1
0 0 1 1l l l k l
k kt E t E t E eα λ β α λ β α λ β− −− ∆ + − ∆ + + − ∆ =
( ) ( )0 0If, as 0, roots o 0f lk kt z tt α λ β α λ β− ∆ +∆ + − ∆→ =
• less than one in magnitude or• are distinct and bounded by 1 , 0tκ κ+ ∆ >
Then from the aside on difference equations
( )0, 0, 0,
max max max l t
l l lT T Tl l lt t t
C T
TCE e e Tt
C et T
eκ κ
⎡ ⎤ ⎡ ⎤ ⎡ ⎤∈ ∈ ∈⎢ ⎥ ⎢ ⎥ ⎢ ⎥∆ ∆ ∆⎣ ⎦ ⎣ ⎦
∆
⎣ ⎦
≤ ≤∆ ∆
Multistep MethodsStability of the method
Stability Theorem Picture
1-1Re
Im
( )0
roots of 0 for a nonzero k
k jj j
jt z tα λ β −
=
− ∆ = ∆∑
As 0, roots t∆ →move inward tomatch polynomial α0
roots of 0k
k jj
jzα −
=
=∑
Multistep MethodsStability of the method
The BESTE Method
Best explicit 2-step method0 1 2 0 1 21, 4, 5, 0, 4, 2α α α β β β= = = − = = =
Re
Im2roots of 4 5 0z z+ − =
Method is Wildly unstable!
-1 1-5
Multistep MethodsStability of the method
Dahlquist’s First Stability Barrier
For a stable, explicit k-step multistep method, the maximum number of exactness constraints that can be satisfied is less than or equal to k (note there are 2k-1
coefficients). For implicit methods, the number of constraints that can be satisfied is either k+2 if k is even
or k+1 if k is odd.
Multistep MethodsConvergence Analysis
Conditions for convergence, stability and consistency
1) Local Condition: One step errors are small (consistency)
Exactness Constraints up to p0 (p0 must be > 0)
( ) 0 11 0
0,max for pl
Tlt
e C t t t+
⎡ ⎤∈⎢ ⎥∆⎣ ⎦
⇒ ≤ ∆ ∆ < ∆
2) Global Condition: One step errors grow slowly (stability)
0roots of 0
kk j
jj
zα −
=
=∑
20, 0,
max maxl lT Tl lt t
TE C et⎡ ⎤ ⎡ ⎤∈ ∈⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦
⇒ ≤∆
( ) 0
0,max pl
Tlt
E CT t⎡ ⎤∈⎢ ⎥∆⎣ ⎦
≤ ∆Convergence Result:
Inside the unit circle or on the unit circle and distinct
Backward-Euler Computed Solution
With Backward-Euler it is easy to use small timesteps for the fast dynamics and then switch to large timesteps for the
slow decay
small t∆
large t∆
( ) ( )d x t Ax tdt
=
( ) 2.1, 0.1eig A = − −
Circuit Example
Multistep MethodsTwo time-constant circuit
Large timestep stability
Forward-Euler Computed Solution
Multistep MethodsFE on two time-constant
circuit?
Large Timestep Stability
The Forward-Euler is accurate for small timesteps, but goes unstable when the timestep is enlarged
Multistep MethodsLarge Timestep Stability
( ) ( ) 0( ), 0d v t v t v vdt
λ λ= = ∈
FE, BE and Trap on the scalar ode problem
Scalar ODE:
( )1ˆ ˆ ˆ ˆ1l l l lv v t v t vλ λ+ = + ∆ = + ∆Forward-Euler:
Backward-Euler: ( )1 1 1 1ˆ ˆ ˆ ˆ ˆ
1l l l l lv v t v v v
tλ
λ+ + += + ∆ ⇒ =
−∆
If 1 1 the solution grows even if <0tλ λ+ ∆ >
1If 1 the solution decays even if 01 t
λλ
< >−∆
( ) ( )( )
1 1 1 1 0.5ˆ ˆ ˆ ˆ ˆ ˆ0.5
1 0.5ll l l l lt
v v t v v v vtλ
λλ
+ + + + ∆= + ∆ + ⇒ =
− ∆Trap Rule:
Multistep MethodsLarge Timestep Stability
FE large timestep region of absolute stability
Difference EqnStability region 1-1
( )1z tλ= + ∆
Im(z)
Re(z)
( )Im λ
( )Re λ
Forward Euler
ODE stability region
2t
−∆
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
FE large timestep stability, circuit example
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λ
( )Re λ
ODE stability region
2t
−∆
Circuit example with t = 0.1, 2.1, 0.1λ∆ = − −
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
FE large timestep stability, circuit example
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λ
( )Re λ
ODE stability region
2t
−∆
Circuit example with t=1.0, 2.1, 0.1λ∆ = − −
Unstable Difference Equation
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
BE large timestep region of absolute stability
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λBackward Euler ( ) 11z tλ −= − ∆
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
BE large timestep stability, circuit example
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λCircuit example with t = 0.1, 2.1, 0.1λ∆ = − −
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
BE large timestep stability, circuit example
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λCircuit example with t =1.0, 2.1, 0.1λ∆ = − −
Stable Difference Equation
Region ofAbsolute Stability
Multistep MethodsLarge Timestep Stability
Stability Definitions
Region of Absolute Stability for a Multistep method:
A method is A-stable if its region of absolute stability
( )0
Values of where roots of 0k
k jj j
jt t zλ α λ β −
=
∆ − ∆ =∑are inside the unit circle.
includes the entire left-half of the complex plane
A-stable:
Dahlquist’s second Stability barrier:There are no A-stable multistep methods of convergenceorder greater than 2, and the trap rule is the most accurate.
0 5 10 15 20 25 30-6
-4
-2
0
2
4
Multistep methodsNumerical Experiments
Oscillating Strut and Mass
Why does FE result grow, BE result decay and the Trap rule preserve oscillations
Trap rule
Forward-Euler
Backward-Euler
0.1t∆ =
Multistep MethodsLarge Timestep Stability
FE large timestep oscillator example
Difference EqnStability region 1-1
( )1z tλ= + ∆
Im(z)
Re(z)
( )Im λ
( )Re λ
Forward Euler
ODE stability region
2t
−∆
Region ofAbsolute Stability
unstable oscillating
Multistep MethodsLarge Timestep Stability
BE large timestep oscillator example
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λBackward Euler ( ) 11z tλ −= − ∆
Region ofAbsolute Stability
decaying oscillating
Difference EqnStability region 1-1
Im(z)
Re(z)
( )Im λTrap Rule
Region ofAbsolute Stability
oscillatingoscillating
( )( )1 0.51 0.5
tz
tλλ
+ ∆=
− ∆
Multistep MethodsLarge Timestep Stability
Trap large timestep oscillator example
Multistep Methods Large Timestep Issues
Two Time-Constant Stable problem (Circuit)FE: stability, not accuracy, limited timestep size.BE was A-stable, any timestep could be used.Trap Rule most accurate A-stable m-step method
Oscillator ProblemForward-Euler generated an unstable difference
equation regardless of timestep size.Backward-Euler generated a stable (decaying)
difference equation regardless of timestep size.Trapezoidal rule mapped the imaginary axis
Summary
Small Timestep issues for Multistep MethodsLocal truncation error and Exactness.Difference equation stability.Stability + Consistency implies convergence.
Investigate Large Timestep IssuesAbsolute Stability for two time-scale examples.Oscillators.
Didn’t talk aboutRunge-Kutta schemes, higher order A-stable methods.