Selection Strategies of Set-Valued Runge-Kutta Methods

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Selection Strategies

of Set-Valued Runge-Kutta Methods ^∗

Robert Baier

University of Bayreuth Applied Mathematics

D-95440 Bayreuth, Germany

e-mail: robert.baier@uni-bayreuth.de

∗Lecture at the NAA 2004 Conference in Rousse (30 June 2004)

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Contents

1 Introduction 3

1.1 Differential Inclusions and Set-Valued Integral . . . 3

1.2 Arithmetic Operations on Sets . . . 5

1.3 Modulus of Smoothness . . . 8

2 Quadrature and Combination Methods 9 2.1 Quadrature Methods . . . 9

2.2 Quadrature Method for the Approximation of Attainable Sets . . . 11

2.3 Combination Methods . . . 14

3 Set-Valued Runge-Kutta Methods 19 3.1 Euler’s Method . . . 23

3.2 Euler-Cauchy Method (or Heun’s Method) . . . 25

3.3 Modified Euler Method . . . 28

3.4 Runge-Kutta (4) . . . 40

4 Conclusions 50

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1. Introduction

1.1. Differential Inclusions and Set-Valued Integral Problem 1.1 Consider the nonlinear differential inclusion (DI)

x

⁰

(t)

∈

F (t, x(t))

(f. a. e.

t ∈ I := [t

₀

, T ]

)

,

(1)

x(t

₀

)

∈

X

₀ (2)

with the nonempty set

X

₀

∈ C (

Rⁿ

)

and the set-valued mapping

F : I ×

Rⁿ⇒Rⁿ _with images in

C (

Rⁿ

)

.

Hereby, C(Rⁿ) denotes the set of nonempty, convex, compact subsets of Rⁿ _and

x : I →

Rⁿ _fulfills

x( · ) ∈

AC(I), i.e.

x( · )

is absolutely continuous.

Definition 1.2 The attainable set R(t, t₀, X₀) at a given time t

∈ I

for Problem 1.1 is defined as

R (t, t

₀

, X

₀

) = { x(t) | x( · ) ∈ AC(I )

is solution of (1)–(2)

} .

Aim of the methods presented here:

approximation of the attainable set at time T by other sets

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simplification for main parts of the talk:

Problem 1.3 The linear differential inclusion (LDI) is stated as follows:

x

⁰

(t) ∈

A(t)x(t) + B(t)U (f. a. e.

t ∈ I = [t

₀

, T ]

)

,

(3)

x(t

₀

) ∈ X

₀ (4)

with matrix functions

A : I →

R^n×n_,

B : I →

R^n×m _{and sets}

X

₀

∈ C (

Rⁿ

)

,

U ∈ C (

R^m

)

. Definition 1.4 The fundamental solutionof the corresponding matrix differential equation

X

⁰

(t) = A(t)X (t)

(f. a. e.

t ∈ I

)

, X (τ ) = I .

to Problem 1.3 is denoted by Φ(·, τ) for

τ ∈ I

, where

I ∈

R^n×n is the unit matrix.

Definition 1.5 ([Aumann, 1965])

Consider a set-valued function

F : I ⇒

Rⁿ with images in

C (

Rⁿ

)

which is measurable and integrably bounded, i.e. there exists

k( · ) ∈ L

₁

(I )

with

F (t) ⊂ k(t)B

₁

(0)

f. a. e.

t ∈ I

. Then, Aumann’s integralis defined as

T

Z

t0

F(t)dt

:=

T

Z

t0

f (t)dt | f ( · )

is an integrable selection of

F ( · ) .

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1.2. Arithmetic Operations on Sets

Definition 1.6 Let

C, D ∈ C (

Rⁿ

)

. The Hausdorff distance between

C

and

D

is defined as d_H(C, D)

= max { d(C, D), d(D, C ) } ,

where

d(C, D) = sup

c∈C

dist(c, D) , dist(c, D) = inf

d∈D

k c − d k

2

(c ∈ C ) .

Notation 1.7 The arithmetic operationsof sets

λ

·

C := { λ

·

c | c ∈ C }

(scalar multiple)

, C

+D

:= { c+d | c ∈ C, d ∈ D }

(Minkowski sum)

,

A

·

C := { A

·

c | c ∈ C }

(image under a linear mapping) are defined as usual for

C, D ∈ C (

Rⁿ

)

,

A ∈

R^k×n _and

λ ∈

R_.

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Definition 1.8 Let

C ∈ C (

Rⁿ

)

,

l ∈

Rⁿ_{. The} support function resp. the supporting face for

C

in direction

l

is defined as

δ^∗(l, C)

:= max

c∈C

h l, c i

^resp. Y(l, C)

:= { c ∈ C | h l, c i = δ

^∗

(l, C ) } .

Remark that

C =

\

klk2=1

{ x ∈

Rⁿ

: h l, x i ≤ δ

^∗

(l, x) } .

Lemma 1.9 Let

C, D ∈ C (

Rⁿ

)

,

A ∈

R^k×n _and

λ ≥ 0

. Then,

C

⊂

D ⇐⇒ δ

^∗

(l, C)

≤

δ

^∗

(l, D)

for all

l ∈ S

_n−1

⊂

Rⁿ_{, i.e.}

k l k

2

= 1

and the following calculus rules are valid for

l ∈ S

_n−1:

δ

^∗

(l, C

+D) =

δ

^∗

(l, C )+δ

^∗

(l, D) , Y (l, C+D) = Y (l, C)+Y (l, D) , δ

^∗

(l,

λC

) =

λδ^∗

(l, C) , Y (l,

λC) = λY

(l, C) ,

δ

^∗

(l,

AC

) = δ

^∗

(A

^t

l, C) , Y (l,

AC) = AY

(A

^t

l, C) ,

d_H

(C, D) =

sup

klk2=1

|

δ

^∗

(l, C )

−

δ

^∗

(l, D)

| ⁽⁵⁾

and

d

_H

(AU,

B

U ) ≤

kA − Bk

· k U k

^with

k U k := sup

u∈U

k u k

²

,

(6)

d

_H

((A

+B)U

,

AU +BU)

≤

kA − Bk

· k U k .

(7)

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Theorem 1.10 ([Aumann, 1965])

Let

F : I ⇒

Rⁿ with nonempty, closed images be measurable and integrably bounded.

Then, the Aumann integral of

F ( · )

is compact, convex and nonempty with δ^∗(l,

Z

I

F (t)dt) =

Z

I

δ^∗(l,

F (t))dt .

Lemma 1.11 (e.g. [Sonneborn and van Vleck, 1965])

Given Problem 1.3, the attainable set at time

t ∈ I

can be rewritten as

R (T, t

₀

, X

₀

) = Φ(T, t

₀

)X

₀

+

T

Z

t₀

Φ(T, t)B (t)U dt .

Scalarization by support functions resp. supporting faces yields for

l ∈ S

_n−1:

δ

^∗

(l,

R(T, t₀, X₀)) =

δ

^∗

(Φ(T, t

₀)^t

l,

X₀

) +

Z

I

δ

^∗

(B (t)

^tΦ(T, t)^t

l,

U

)dt , Y (l,

R(T, t₀, X₀)) = Φ(T, t₀)Y

(Φ(T, t

₀)^t

l,

X₀

)

+

Z

I

Φ(T, t)B

(t)Y (B(t)

^tΦ(T, t)^t

l,

U

)dt

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1.3. Modulus of Smoothness

Definition 1.12 Let

f : I →

Rⁿ be bounded. The averaged modulus of smoothness of order k ∈ N is defined as

τ_k(f;h)

:= k ω

_k

(f ; · ; h) k

L1

,

ω

_k

(f ; x; h) := sup {| ∆

^k_δ

f (t) | : t, t + kδ ∈ [x − kh

2 , x + kh

2 ] ∩ I }

^for

x ∈ I ,

where

∆

^k_δ

f (t)

is the

k

-th forward difference of

f ( · )

in

t

with step-size

δ

.

Lemma 1.13 (cf. [Sendov and Popov, 1988]) Let

f : I →

Rⁿ be bounded and

p ∈

N_{. Then,}

τ

_p

(f ; h) =















o

(1),

if

f ( · )

is Riemann integrable

O (h),

if

f ( · )

has bounded variation o

(h

^p−1

),

ifp

≥ 2

and

f

^p−2

( · ) ∈ AC(I ) O (h

^p

),

ifp

≥ 2

,

f

^p−2

( · ) ∈ AC(I )

and

f

^p−1

( · )

has bounded variation

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2. Quadrature and Combination Methods

2.1. Quadrature Methods

Notation 2.1 Let

I := [t

₀

, T ]

and

f : I →

Rⁿ be given. We denote the point-wise quadrature formula by

Q(f; [t₀, T])

:=

s

X

µ=1

b

_µ

f (t

₀

+ c

_µ

(T − t

₀

)) ,

where

b

_µ

∈

Rare the weights and

c

_µ

∈ [0, 1]

determine the nodes (

µ = 1, . . . , s

).

Set

h =

^T−_N^t0 as step-size for

N ∈

N and define the iterated quadrature formula as Q_N(f; [t₀, T])

:= h

N−1

X

j=0

Q(f ; [t

_j

, t

_j+1

]) = h

N−1

X

j=0 s

X

µ=1

b

_µf

(t

_j

+ c

_µ

h) .

Q(f ; I )

has precision p

∈

N₀, if all polynomials up to degree

p

are integrated exactly and there exists a polynomial

f

with degree

p + 1

and

Q(f ; I ) 6 =

R

I

f (t)dt

.

Definition 2.2 Consider a point-wise quadrature formula of Notation 2.1 and

F : I ⇒

Rⁿ with images in

C (

Rⁿ

)

. The iterated set-valued quadrature method is defined with the usual arithmetic operations

Q_N(F; [t₀, T])

:= h

N−1

X

j=0 s

X

µ=1

b

_µF

(t

_j

+ c

_µ

h) .

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Proposition 2.3

(cf. [Polovinkin, 1975], [Balaban, 1982], [Donchev and Farkhi, 1990], [Veliov, 1989a]), [Krastanov and Kirov, 1994], [B. and Lempio, 1994b], [B., 1995])

Consider

N ∈

N and a point-wise iterated quadrature formula of Notation 2.1 with non-negative weights

b

_µ

≥ 0

(

µ = 1, . . . , s

) and the remainder term

R

_N

(f ; I ) :=

Z

I

f (t)dt − Q

_N

(f ; I ) .

Then, the corresponding set-valued quadrature method fulfills for

F : I ⇒

Rⁿ with images in

C (

Rⁿ

)

:

d

_H

(

Z

I

F (t)dt, Q

_N

(F ; I )) = sup

klk2=1

| R

_N

(δ

^∗

(l, F ( · )); I ) | .

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2.2. Quadrature Method for the Approximation of Attainable Sets Proposition 2.4

(cf. [Donchev and Farkhi, 1990], [B. and Lempio, 1994b], [B., 1995])

Consider

N ∈

N and a point-wise iterated quadrature formula of Notation 2.1 with non-negative weights

b

_µ

≥ 0

(

µ = 1, . . . , s

). Assume that

•

^{the values}Φ(T, t_j + c_µh) are knownfor

j = 0, . . . , N − 1

and

µ = 1, . . . , s

•

the quadrature method has precisionp

− 1

,

p ∈

N

• τ

_p

(δ

^∗

(l, Φ(T, · )B( · )U ), h) ≤ Ch

^p uniformly in

l ∈ S

_n−1 Then,

d

_H

( R (T, t

₀

, X

₀

), Q

_N

(Φ(T, · )B ( · )U ); [t

₀

, T ]) = O (h

^p

).

Proof: In [Sendov and Popov, 1988, Theorem 3.4]:

| R

_N

(f ) | = |

Z

I

f (t)dt − Q

_N

(f ; [t

₀

, T ]) | ≤ 1 +

s

X

µ=1

b

_µ

T − t

₀

· W

_p

· sup

klk2=1

τ

_p

f, 2 p h

Since Lemma 1.11 and

δ^∗(l, Q_N

(F ; I )) =

Q_N

(δ

^∗(l,

F ( · )); I ) ,

one can apply the error estimation above to

f ( · ) = δ

^∗

(l, F ( · ))

with

F ( · ) = Φ(T, · )B( · )U

.

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Example 2.5 set-valued rectangular rule (special Riemannian sum) for

I = [t

₀

, T ]

:

Q(F ; I ) = (T − t

₀

)F (t

₀

), Q

_N

(F ; I ) = h

N−1

X

j=0

F (t

_j

),

Q

_N

(Φ(T, · )B( · )U ; I ) = h

N−1

X

j=0

Φ(T, t

_j

)B (t

_j

)U

in iterative form:

Q

^N_j+1

= Φ(t

_j+1

, t

_j

)Q

^N_j

+ hΦ(t

_j+1

, t

_j

)B (t

_j

)U, Q

^N₀

= X

₀

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Example 2.6 set-valued trapezoidal rule for

I = [t

₀

, T ]

:

Q(F ; I ) = T − t

₀

2 F (t

₀

) + F (T )

, Q

_N

(F ; I ) = h 2

N−1

X

j=0

F (t

_j

) + F (t

_j+1

,

Q

_N

(Φ(T, · )B ( · )U ; I ) = h 2

N−1

X

j=0

Φ(T, t

_j

)B (t

_j

)U + Φ(T, t

_j+1

)B(t

_j+1

)U

in iterative form:

Q

^N_j+1

= Φ(t

_j+1

, t

_j

)Q

^N_j

+ h

2 Φ(t

_j+1

, t

_j

)B(t

_j

)U + Φ(t

_j+1

, t

_j+1

)B (t

_j+1

)U

, Q

^N₀

= X

₀ Remark 2.7

problems with quadrature methods:

•

no generalization for nonlinear differential inclusions possible

•

values of fundamental solutions

Φ(t

_j+1

, t

_j

)

resp.

Φ(T, t

_j

)

must be known in advance

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2.3. Combination Methods

Proposition 2.8 (cf. [B. and Lempio, 1994b], [B., 1995])

Consider

N ∈

N and a point-wise iterated quadrature formula of Notation 2.1 with non- negative weights

b

_µ

≥ 0

(

µ = 1, . . . , s

). Assume that

(i) the quadrature method has precisionp

− 1

,

p ∈

N

(ii)

τ

_p

(δ

^∗

(l, Φ(T, · )B( · )U ), h) ≤ Ch

^p uniformly in

l ∈ S

_n−1 (iii)

d

_H

(X

₀

,

X₀^N

) = O (h

^p

)

and uniformly in

j = 0, . . . , N − 1

and

µ = 1, . . . , s

(iv) Φ(te _j+1, t_j)

= Φ(t

_j+1

, t

_j

) + O (h

^p+1

)

(v)

d

_H

(

Ue_µ(t_j +c_µh),

Φ(t

_j+1

, t

_j

+ c

_µ

h)B(t

_j

+ c

_µ

h)U ) = O (h

^p

)

Then, the combination method defined as

X_j+1^N

=

Φ(te _j+1, t_j)X_j^N

+ h

s

X

µ=1

b

_µUe_µ(t_j + c_µh)

(j = 0, . . . , N − 1)

satisfies the global estimate

d

_H

( R (T, t

₀

, X

₀

), X

_N^N

) = O (h

^p

) .

Especially, (iv) is satisfied for

U

e_µ

(t

_j

+ c

_µ

h) :=

Φe_µ(t_j+1, t_j +c_µh)B

(t

_j

+ c

_µ

h)U ,

if Φe_µ(t_j+1, t_j +c_µh)

= Φ(t

_j+1

, t

_j

+ c

_µ

h) + O (h

^p

) .

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Proof: Define for

j = 0, . . . , N − 1

the iterations

R

_j+1^N

= Φ(t

_j+1

, t

_j

)R

^N_j

+

tj+1

Z

t_j

Φ(t

_j+1

, τ )B(τ )U dτ ,

Q

^N_j+1

= Φ(t

_j+1

, t

_j

)Q

^N_j

+ h

s

X

µ=1

b

_µ

Φ(t

_j+1

, t

_j

+ c

_µ

h)B (t

_j

+ β

_µ

h)U , R

^N₀

= Q

^N₀

= X

₀

.

Then,

R

_N^N

= R (T, t

₀

, X

₀

) , Q

^N_N

= Q

_N

(Φ(T, · )B( · )U ; [t

₀

, T ]) .

Show that

X

_j^N is bounded uniformly in

j = 0, . . . , N

and that

d

_H

(R

^N_j+1

, Q

^N_j+1

) ≤ k Φ(t

_j+1

, t

_j

) k · d

_H

(R

_j^N

, Q

^N_j

) + d

_H

(

t_j+1

Z

tj

Φ(t

_j+1

, t)B (t)U dt, h

s

X

µ=1

b

_µ

Φ(t

_j+1

, t

_j

+ c

_µ

h)B(t

_j

+ c

_µ

h)U )

≤ (1 + h C

e

) d

_H

(R

^N_j

, Q

^N_j

) + O (h

^p+1

)

⇒ d

_H

(Q

^N_j

, X

_j^N

) ≤ (1 + h C

e

)

^j

d

_H

(R

^N₀

, Q

^N₀

) + j O (h

^p+1

) ≤ N O (h

^p+1

) = O (h

^p

) .

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Furthermore,

d

_H

(Q

^N_j+1

, X

_j+1^N

) ≤k Φ(t

_j+1

, t

_j

) k · d

_H

(Q

^N_j

, X

_j^N

)

+ k Φ(t

_j+1

, t

_j

) − Φ(t

e _j+1

, t

_j

) k · k X

_j^N

k + h

s

X

µ=1

b

_µ

d

_H

(Φ(t

_j+1

, t

_j

+ c

_µ

h)B(t

_j

+ c

_µ

h)U, U

e_µ

(t

_j

+ c

_µ

h))

≤ (1 + h C

e

) d

_H

(Q

^N_j

, X

_j^N

) + O (h

^p+1

)

⇒ d

_H

(Q

^N_j

, X

_j^N

) ≤ (1 + h C

e

)

^j

d

_H

(Q

^N₀

, X

₀^N

) + j O (h

^p+1

)

≤ (1 + h C

e

)

^N

d

_H

(X

₀

, X

₀^N

) + N O (h

^p+1

)

≤ e

^(T−t0)Ce

O (h

^p

) + O (h

^p

) = O (h

^p

) .

⇒ d

_H

(R

^N_j

, X

_j^N

) ≤ d

_H

(R

^N_j

, Q

^N_j

) + d

_H

(Q

^N_j

, X

_j^N

) = O (h

^p

)

uniformly in

j = 0, . . . , N

.

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Example 2.9 combination method: iter. Riemannian sum/Euler for matrix differ. equation

X

⁰

(t) = A(t)X (t) (t ∈ [t

_j

, t

_j+1

]) ,

X (t

_j

) = I

X

_j+1^N

= Φ(t

e _j+1

, t

_j

)X

_j^N

+ h Φ

e₁

(t

_j+1

, t

_j

)B(t

_j

)U , (j = 0, . . . , N − 1) Φ(t

e _j+1

, t

_j

) = Φ(t

e _j

, t

_j

) + hA(t

_j

) Φ(t

e _j

, t

_j

) ,

Φ

e₁

(t

_j+1

, t

_j

) = Φ(t

e _j+1

, t

_j

) .

Hence,

X

_j+1^N

= (I + hA(t

_j

))X

_j^N

+ h(I + hA(t

_j

))B(t

_j

)U (j = 0, . . . , N − 1).

Other possibility for calculation: Euler for adjoint equation

Y

⁰

(t) = − Y (t)A(t) (t ∈ [t

₀

, T ]) , Y (T ) = I

gives

X

_N^N

= Φ(T, t

e _j

)X

₀^N

+ h

N−1

X

j=0

Φ

e₁

(T, t

_j

)B (t

_j

)U ,

Φ(T, t

e _j

) = N − j

(backward) steps of Euler for adjoint equation,

Φ

e₁

(T, t

_j

) = Φ(T, t

e _j

) .

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Example 2.10 usual combination of set-valued quadrature method and pointwise DE solver which provides approximations to the values of the fundamental solution at the quadrature nodes:

set-valued solver for step-size overall

quadrature method differential equations of DE solver order

iter. Riemannian sum Euler

h O (h)

iter. trapezoidal rule Euler-Cauchy/Heun

h O (h

²

)

iter. midpoint rule modified Euler ^h₂

O (h

²

)

iter. Simpson’s rule classical RK(4) ^h₂

O (h

⁴

)

Romberg’s method extrapolation of midpoint rule (with Euler as starting procedure)

h

_i

=

^T−_2i^t0

O (

j

Q

ν=0

h

²_i−ν

)

(under suitable smoothness assumptions)

Remark 2.11

problems with these combination methods:

•

no generalization for nonlinear differential inclusions possible

•

values of fundamental solutions

Φ(t

_j+1

, t

_j

), Φ

_µ

(t

_j

+ c

_µ

h, t

_j

)

resp.

Φ(T, t

_j

), Φ

_µ

(T, t

_j

+ c

_µ

h)

must be calculated additionally

•

approximation for

Φ

_µ

(t

_j

+ c

_µ

h, t

_j

)

resp.

Φ

_µ

(T, t

_j

+ c

_µ

h)

is calculated too accurately (

O (h

^p+1

)

instead of

O (h

^p

)

)

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3. Set-Valued Runge-Kutta Methods

Runge-Kutta methods could be expressed by theButcher array (cf. [Butcher, 1987]):

c

₁

a

₁₁

a

₁₂

. . . a

_1,s−2

a

_1,s−1

a

_1,s

c

₂

a

₂₁

a

₂₂

. . . a

_2,s−2

a

_2,s−1

a

_2,s

... ... ... ... ... ... ...

c

_s−1

a

_s−1,1

a

_s−1,2

. . . a

_s−1,s−2

a

_s−1,s−1

a

_1,s

c

_s

a

_s,1

a

_s,2

. . . a

_s,s−2

a

_s,s−1

a

_s,s with

c

₁

:= 0 . b

₁

b

₂

. . . b

_s−2

b

_s−1

b

_s

Explicit Runge-Kutta methods satisfy

a

_µ,ν

= 0

, if

µ ≤ ν

and

c

₁

= 0

. The set-valued Runge-Kutta methodfor LDI is defined as follows:

Choose a starting set

X

₀^N

∈ C (

Rⁿ

)

and define for

j = 0, . . . , N − 1

and

µ = 1, . . . , s

:

η

_j+1^N

= η

_j^N

+ h

s

X

µ=1

b

_µξ_j^(µ)

,

(8)

ξ

_j^(µ)

= A(t

_j

+ c

_µ

h) η

_j^N

+ h

µ−1

X

ν=1

a

_µ,ν

ξ

_j^(ν)

+ B(t

_j

+ c

_µ

h)u

^(µ)_j

,

(9)

u^(µ)_j ∈ U

,

(10)

η₀^N∈ X₀^N

,

(11)

X_j+1^N

= { η

^N_j+1

| η

_j+1^N is defined by (8)–(11)

} .

(12)

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Remark 3.1 If nonlinear DIs are considered with F(t, x) = S

u∈U

{f(t, x, u)}, equation (9) must be replaced by

ξ

_j^(µ)

= f (t

_j

+ c

_µ

h, η

_j^N

+ h

µ−1

X

ν=1

a

_µ,ν

ξ

_j^(ν)

, u

^(µ)_j

) .

For some selection strategies, some of the selections

u

^(µ)_j depend on others (e.g., they could be all equal).

If

f (t, x, u) = f (t, u)

, i.e. F(t, x) = F(t), and

X

₀^N

= { 0

Rⁿ

}

, we arrive at the underlying quadrature method

η

^N_j+1

= η

_j^N

+ h

s

X

µ=1

b

_µ

f (t

_j

+ c

_µ

h, u

^(µ)_j

), u

^(µ)_j

∈ U , X

_j+1^N

= X

_j^N

+ h

s

X

µ=1

b

_µ

F (t

_j

+ c

_µ

h) ,

X

_N^N

= h

N−1

X

j=0 s

X

µ=1

b

_µ

F (t

_j

+ c

_µ

h) = Q

_N

(F ; [t

₀

, T ])

of the Runge-Kutta method.

If

f (t, x, u) = f (t, x)

, i.e. F(t, x) = {f(t, x)}^{, then}

X

_j^N

= { η

_j^N

}

coincides with the pointwise Runge-Kutta method.

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Remark 3.2 Grouping in equation (8) by matrices multiplied by

η

_j^N and

u

^(µ)_j ,

µ = 1, . . . , s

we arrive at the form

X

_j+1^N

= Φ(t

e _j+1

, t

_j

)X

_j^N

+ h

[

u^(µ)_j ∈U

{

s

X

µ=1

b

_µ

Ψ

e_µ

(t

_j+1

, t

_j

+ c

_µ

h)u

^(µ)_j

}

with suitable matrices

Φ(t

e _j+1

, t

_j

)

(involving matrix values of

A( · )

) and

Ψ

e_µ

(t

_j+1

, t

_j

+ c

_µ

h)

(involving matrix values of

A( · )

and

B ( · )

).

Φ(t

e _j+1

, t

_j

)

is the same matrix as in the pointwise case for

f (t, x, u) = A(t)x

, hence it approximates

Φ(t

_j+1

, t

_j

)

from the same order as in the pointwise case.

Questions:

•

What is the order of the set-valued Runge-Kutta method, i.e.

d

_H

( R (T, t

₀

, X

₀

), X

₀^N

) = O (h

^p

)

?

Does the order coincide with the single-valued case?

•

What selection strategy is preferrable?

•

Should the chosen selection strategy depend on the Runge-Kutta method?

•

What smoothness assumptions do we need?

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Answers in the literature:

set-valued iter. quadrature global disturbance local order overall RK-method method order term for . . . of disturbance global order

Euler Riemannian sum

O (h) η

_j^N

O (h

²

) O (h)

u

⁽¹⁾_j

O (h)

Euler/Cauchy midpoint rule

O (h

²

) η

_j^N

O (h

³

) O (h

²

)

(constant sel.)

u

⁽¹⁾_j

O (h

²

)

Euler/Cauchy trapezoidal rule

O (h

²

) η

_j^N

O (h

³

) O (h

²

)

(2 free sel.)

u

⁽¹⁾_j

O (h

²

)

u

⁽²⁾_j

O (h

²

)

Euler’s method (see Subsection 3.1):

cf. [Nikol’skiˇı, 1988], [Dontchev and Farkhi, 1989], [Wolenski, 1990] for nonlinear DIs, for extensions see [Artstein, 1994], [Grammel, 2003])

Euler-Cauchy method (see Subsection 3.2):

cf. [Veliov, 1992] as well as [Veliov, 1989b]

for strongly convex nonlinear DIs

modified Euler method (see Subsection 3.3) Runge-Kutta(4) method (see Subsection 3.4)

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3.1. Euler’s Method

Remark 3.3 Consider Euler’s method, i.e. the Butcher array

0 0 .

1

underlying quadrature method = special Riemannian sum:

Q

_N

(F ; [t

₀

, T ]) = h

N−1

X

j=0

F (t

_j

)

Grouping by

η

_j^N and the single selection

u

⁽¹⁾_j yields

X

_j+1^N

= I + hA(t

_j

)

X

_j^N

+ hB(t

_j

)U (j = 0, . . . , N − 1) .

Proposition 3.4 Euler’s method is a combination method with the following settings:

Q

_N

(F ; [t

₀

, T ]) = h

N−1

X

j=0

F (t

_j

) ,

Φ(t

e _j+1

, t

_j

) = I + hA(t

_j

) ,

Φ

e₁

(t

_j+1

, t

_j

) = I .

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Proposition 3.5 (cf. [Nikol’skiˇı, 1988], [Dontchev and Farkhi, 1989], [Wolenski, 1990], see also [Artstein, 1994], [Grammel, 2003])

If

• A( · )

is Lipschitz,

• B( · )

is bounded,

• τ

₁

(δ

^∗

(l, Φ(T, · )B ( · )U ), h) ≤ C

huniformly in

l ∈ S

_n−1, e.g., if

B ( · )

is Lipschitz,

• d

_H

(X

₀

, X

₀^N

) = O (h)

,

then Euler’s method converges at least with order O(h). Proof: The quadrature method has precision 0.

If

B( · )

is Lipschitz, then

Φ(T, · )B( · )

and hence also

δ

^∗

(l, Φ(T, · )B( · )U )

(uniformly in

l ∈ S

_n−1) are Lipschitz.

The following estimations are valid:

k Φ(t

e _j+1

, t

_j

) − Φ(t

_j+1

, t

_j

) k = k I + hA(t

_j

)

− Φ(t

_j+1

, t

_j

) k = O (h

²

) , k Φ

e₁

(t

_j+1

, t

_j

) − Φ(t

_j+1

, t

_j

) k = k I − Φ(t

_j+1

, t

_j

) k = O (h) .

Hence, Proposition 2.8 can be applied yielding

O (h)

.

For order of convergence 1, it is sufficient that

A( · )

and

B ( · )

(resp.

δ

^∗

(l, Φ(T, · )B( · )U )

, uniformly in

l ∈ S

_n−1) have bounded variation.

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3.2. Euler-Cauchy Method (or Heun’s Method)

Remark 3.6 Consider method of Euler-Cauchy (orHeun’s method), i.e. the Butcher array

0 0 0

1 1 0 .

1 2

1 2

underlying quadrature method = iterated trapezoidal rule:

Q

_N

(F ; [t

₀

, T ]) = h 2

N−1

X

j=0

F (t

_j

) + F (t

_j+1

)

Grouping by

η

_j^N and the two selections

u

⁽¹⁾_j and

u

⁽²⁾_j yields

X

_j+1^N

=

I + h

2 A(t

_j

) + A(t

_j+1

)

+ h

²

2 A(t

_j+1

)A(t

_j

)

X

_j^N

+ h

2

[

u⁽¹⁾_j ,u⁽²⁾_j ∈U

I + hA(t

_j+1

)

B(t

_j

)u

⁽¹⁾_j

+ B (t

_j+1

)u

⁽²⁾_j

for

j = 0, . . . , N − 1

.

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Proposition 3.7 The method of Euler-Cauchy with two free selections ”

u

⁽¹⁾_j

, u

⁽²⁾_j ∈

U

” is a combination method with the following settings:

Q

_N

(F ; [t

₀

, T ]) = h 2

N−1

X

j=0

F (t

_j

) + F (t

_j+1

) ,

Φ(t

e _j+1

, t

_j

) = I + h

2 A(t

_j

) + A(t

_j+1

)

+ h

²

2 A(t

_j+1

)A(t

_j

) , Φ

e₁

(t

_j+1

, t

_j

) := I + hA(t

_j+1

) ,

Φ

e₂

(t

_j+1

, t

_j+1

) := I .

Proposition 3.8 The method of Euler-Cauchy with constant selection strategy ”

u

⁽¹⁾_j =u⁽²⁾_j ” is a combination method with the following settings:

Q

_N

(F ; [t

₀

, T ]) = h

N−1

X

j=0

F (t

_j

+ h 2 ) , Φ(t

e _j+1

, t

_j

) = I + h

2 A(t

_j

) + A(t

_j+1

)

+ h

²

2 A(t

_j+1

)A(t

_j

) , U

e₁

(t

_j

+ h

2 ) := 1

2 B (t

_j

) + B(t

_j+1

) + hA(t

_j+1

)B (t

_j

)

U .

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Proposition 3.9

(cf. [Veliov, 1992] as well as [Veliov, 1989b] for strongly convex nonlinear DIs) If

• A

⁰

( · )

and

B( · )

are Lipschitz,

• τ

₂

(δ

^∗

(l, Φ(T, · )B ( · )U ), h) ≤ C

h² uniformly in

l ∈ S

_n−1, e.g., if

B

⁰

( · )

is Lipschitz,

• d

_H

(X

₀

, X

₀^N

) = O (h

²

)

,

then the method of Euler-Cauchy with constant or with two free selections converges at least with order O(h²).

For order of convergence 2, it is sufficient that

A

⁰

( · )

and

B

⁰

( · )

(resp. _dt^d

δ

^∗

(l, Φ(T, · )B( · )U )

, uniformly in

l ∈ S

_n−1) have bounded variation.

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3.3. Modified Euler Method

Remark 3.10 Consider modified Euler method, i.e. the Butcher array

0 0 0

1 2

1

2

0 .

0 1

underlying quadrature method = iterated midpoint rule:

Q

_N

(F ; [t

₀

, T ]) = h

N−1

X

j=0

F (t

_j

+ h 2 )

Grouping by

η

_j^N and the two selections

u

⁽¹⁾_j and

u

⁽²⁾_j yields

X

_j+1^N

=

I + hA(t

_j

+ h

2 ) + h

²

2 A(t

_j

+ h

2 )A(t

_j

)

X

_j^N

+ h

[

u⁽¹⁾_j ,u⁽²⁾_j ∈U

h

2 A(t

_j

+ h

2 )B (t

_j

)u

⁽¹⁾_j

+ B(t

_j

+ h 2 )u

⁽²⁾_j

for

j = 0, . . . , N − 1

.

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Proposition 3.11 Modified Euler method with constant selection strategy ”

u

⁽¹⁾_j =u⁽²⁾_j ” is a combination method with the following settings:

Q

_N

(F ; [t

₀

, T ]) = h

N−1

X

j=0

F (t

_j

+ h 2 ) , Φ(t

e _j+1

, t

_j

) = I + hA(t

_j

+ h

2 ) + h

²

2 A(t

_j

+ h

2 )A(t

_j

) , U

e₁

(t

_j

+ h

2 ) :=

B (t

_j

+ h

2 ) + h

2 A(t

_j

+ h

2 )B(t

_j

)

U .

constant approximation by the quadrature method (midpoint rule) on

[t

_j

, t

_j+1

]

⇒

^constant selection in modified Euler is appropriate Proposition 3.12 If

• A

⁰

( · )

and

B( · )

are Lipschitz,

• τ

₂

(δ

^∗

(l, Φ(T, · )B ( · )U ), h) ≤ C

h² uniformly in

l ∈ S

_n−1, e.g., if

B

⁰

( · )

is Lipschitz,

• d

_H

(X

₀

, X

₀^N

) = O (h

²

)

,

then modified Euler method with constant selection strategy converges at least with order O(h²).

For order of convergence 2, it is sufficient that

A

⁰

( · )

and

B

⁰

( · )

(resp. _dt^d

δ

^∗

(l, Φ(T, · )B( · )U )

, uniformly in

l ∈ S

_n−1) have bounded variation.