Geiser J. Decomposition Methods for Differential Equations: Theory and Applications

Подождите немного. Документ загружается.

Time-Decomposition Methods for Parabolic Equations for Parabolic Equations 55

Thus, the estimations (3.79) and (3.89) result in

E



∞

= Kτ

e

i−1

 + O(τ

(3.90)

Taking into account the deﬁnition of E

and the norm ·

∞

,weobtain

e

 = Kτ

e

i−1

 + O(τ

(3.91)

where K = ω

∈ R

. This proves our statement.

3.2.5 Stability Analysis for the Linear and Quasi-Linear

Operator-Splitting Method

We concentrate on a stability analysis for the linear ordinary diﬀerential

equations with commutative operators. Our goal is to derive the general case

with the eigenvalues of the operators and apply the recursive computation

of the underlying diﬀerential equations. In the next section, we discuss the

stability of the weighted iterative splitting method, which is an upper class

of iterative splitting methods. These upper-class methods can stabilize the

iterative operator-splitting methods for stiﬀ problems, see [89].

3.2.5.1 Stability for the Linear Weighted Iterative Splitting Method

The proposed unsymmetrical weighted iterative splitting method is a com-

bination between the sequential splitting method, see [68], and the iterative

operator-splitting method, see [70]. The weighting factor ω is used as an adap-

tive switch between lower- and higher-order splitting methods. The following

algorithm is based on an iteration with a ﬁxed equidistant step size τ for the

splitting discretization. On the time interval [t

n+1

], we solve the following

subproblems consecutively for i =1, 3,...2m +1:

∂c

(t)

∂t

= Ac

(t)+ωBc

i−1

(t), with c

)=c

, (3.92)

and c

)=c

∂c

i+1

(t)

∂t

= ωAc

(t)+Bc

i+1

(t), (3.93)

with c

i+1

)=ωc

+(1−ω) c

n+1

where c

(t)=0andc

is the known split approximation at the time level

t = t

. The split approximation at the time level t = t

n+1

is deﬁned as

n+1

= c

2m+2

n+1

). For parameter ω it holds that ω ∈ [0, 1]. For ω =0,

we use the sequential splitting, and for ω = 1 we use the iterative splitting

method, cf. [70].

Because of weighting between the sequential splitting and iterative splitting

methods, the initial conditions are also weighted. Thus, we have the ﬁnal

56Decomposition Methods for Diﬀerential Equations Theory and Applications

results of the ﬁrst equation (3.92) appearing in the initial condition for the

second equation (3.93).

Damped Iterative Splitting Method

The damped iterative splitting method is the next stable splitting method. In

this version, we concentrate on the examples with very stiﬀ operators such as

the B-operator. For initial solutions that are far away from the local solution,

strong oscillations occur, see [89] and [126]. For this reason, we damp the

B-operator in such a way that we relax the initial steps with factors ω ≈ 0.

The following algorithm is based on an iteration with a ﬁxed step size τ

for the splitting discretization. On the time interval [t

n+1

]wesolvethe

following subproblems consecutively for i =1, 3,...2m +1:

∂c

(t)

∂t

=2(1− ω)Ac

(t)+2ωBc

i−1

(t), with c

)=c

, (3.94)

and c

)=c

∂c

i+1

(t)

∂t

=2ωAc

(t)+2(1− ω)Bc

i+1

(t), (3.95)

with c

i+1

)=c

where c

(t)=0andc

is the known split approximation at the time level

t = t

. For our parameter ω it holds that ω ∈ (0, 1/2]. For 0 <ω<1/2, we

use the damped method and solve only the damped operators. For ω =1/2

we use the iterative splitting method, cf. [70].

As discussed earlier for the unsymmetric weighted iterative splitting case,

weighting between the sequential splitting and iterative splitting method re-

quires that the initial conditions be also weighted. Consequently, we have the

damped version results of the weighted operators for the ﬁrst equation (3.94)

and second equation (3.95), where the initial conditions are equal. This is

called the symmetric weighted iterative splitting case and symmetrization of

the operators, see [71], [90] and [91].

REMARK 3.18 The theoretical background of the proofs is based on

the reformulation with simpler scalar equations, performed with eigenvalue

problems. This reformulation allows our stability analysis to be accomplished

with the semigroup theory and the A-stability formulation of the ODEs.

Recursion for the Stability Results

First, we concentrate on the (unsymmetric) weighted iterative splitting method,

(3.92) and (3.93). For an overview, we treat the particular case of the initial

values with c

)=c

and c

i+1

)=c

for an overview. The general case

i+1

)=ωc

+(1− ω)c

n+1

) can be treated in the same manner.

We consider the suitable vector norm ||·||being on R

, together with its

induced operator norm. The matrix exponential of Z ∈ R

M×M

is denoted by

exp(Z). We assume that there holds:

||exp(τA)|| ≤ 1and||exp(τB)|| ≤ 1 for all τ>0.

Time-Decomposition Methods for Parabolic Equations for Parabolic Equations 57

It can be shown, that the system (3.92)–(3.93) implies ||exp(τ (A + B))|| ≤ 1

and is itself stable.

For the linear problem (3.92)–(3.93), it follows by integration that there

holds

(t)=exp((t − t

)A)c



exp((t − s)A) ωBc

i−1

(s) ds, (3.96)

and

i+1

(t)=exp((t − t

)B)c



exp((t − s)B) ωAc

(s) ds. (3.97)

With elimination of c

,weget

i+1

(t)=exp((t − t

)B)c

+ ω



exp((t − s)B) A exp((s − t

)A) c

+ω



s=t





exp((t − s)B) A exp((s − s



)A) Bc

i−1



) ds



ds. (3.98)

For the following commuting case, we can evaluate the double integral



s=t











s=s



and can derive the weighted stability theory.

Commuting Operators

For more transparency of the formula (3.98), we consider a well-conditioned

system of eigenvectors and eigenvalues λ

of A and λ

of B instead of the

operators A and B themselves. Replacing the operators A and B by λ

and

, respectively, we obtain after some calculations

i+1

(t)=c

− λ

(ωλ

exp((t − t

)λ

) + ((1 − ω)λ

− λ

)exp((t − t

)λ

))

+ c

− λ



s=t

(exp((t − s)λ

) − exp((t − s)λ

)) ds. (3.99)

Note that this relation is commutative in λ

and λ

; hence, the scalar com-

mutator is given as [λ

,λ

]=[λ

,λ

Strong Stability

We deﬁne z

= τλ

, k =1, 2, τ = t −t

. We start with c

(t)=c

and obtain

n+1

)=S

) c

, (3.100)

where S

is the stability function of the scheme with m iterations. We use

Equation (3.99) and obtain after some calculations for the ﬁrst iteration step

) c

= ω

ωz

+ ω

− z

exp(z

) c

(3.101)

(1 − ω − ω

) z

− z

exp(z

) c

58Decomposition Methods for Diﬀerential Equations Theory and Applications

and for the second iteration step

) c

= ω

ωz

+ ω

− z

exp(z

) c

(3.102)

(1 − ω − ω

) z

− z

exp(z

) c

− z

)

((ωz

+ ω

)exp(z

)

+(−(1 − ω − ω

+ z

)exp(z

)) c

− z

)

((−ωz

− ω

)(exp(z

) − exp(z

))

+((1 − ω − ω

− z

)(exp(z

) − exp(z

))) c

Let us consider the stability given by the following eigenvalues in a wedge:

= {ζ ∈ C : |arg(ζ) ≤ α}. (3.103)

For the stability we have |S

)|≤1, whenever z

∈W

π/2

The stability of the two iterations is given in the following theorem.

THEOREM 3.5

We have the following stability for the weighted iterative splitting method,

(3.92)–(3.93).

For S

we have a strong stability with

max

≤0,z

∈W

)|≤1, ∀ α ∈ [0,π/2] with 0 ≤ ω ≤ 1.

For S

we have a strong stability with

max

≤0,z

∈W

)|≤1, ∀ α ∈ [0,π/2] with 0 ≤ ω ≤



8tan

(α)+1



1/8

PROOF

We consider a ﬁxed z

= z, Re(z) < 0, and z

→−∞. Then we obtain

(z,−∞)=ω

(1 − exp(z)), (3.104)

and

(z,−∞)=ω

(1 − (1 − z)exp(z)). (3.105)

If z = x + iy, x < 0, then it holds that

Time-Decomposition Methods for Parabolic Equations for Parabolic Equations 59

1) For S

we have

(z,−∞)|

= ω

|(1 − 2exp(x)cos(y)+exp(2x))|, (3.106)

and hence,

(z,−∞)|≤1 ⇔ ω

≤



1 − 2exp(x)cos(y)+exp(2x)



. (3.107)

Because of x<0andy ∈ R, we can estimate − 2 ≤ 2exp(x)cos(y), and

exp(2x) ≥ 0.

From (3.107) we obtain ω ≤

√

2) For S

we have

(z,−∞)|

= ω

{1 − 2exp(x)[(1 − x)cosy + y sin y] (3.108)

+exp(2x)[(1 − x)

+ y

]},

and after some calculations, we obtain

(z,−∞)|≤1 ⇔ exp(x) ≤ (

− 1)

exp(−x)

(1 − x)

+ y

|1 − x| + |y|

(1 − x)

+ y

. (3.109)

We can estimate for x<0, y ∈ R

|1−x|+|y|

(1−x)

≤ 3/2, and

2tan

(α)

exp(−x)

(1−x)

where tan(α)=y/x.

Finally, we get the bound ω ≤



8tan

(α)+1



1/8

REMARK 3.19 The stability for the unsymmetric weighted iterative

splitting method is stable for the ﬁrst and second iteration steps. Recursively

we can show further A

-stability for more than two iteration steps with α ∈

[0,π/2], see [89].

The stability for the damped iterative operator-splitting method, see Equa-

tions (3.94) and (3.95), is given in the following theorem.

THEOREM 3.6

We have the following stability for the damped iterative splitting method,

(3.94)–(3.95):

For S

we have a strong stability with

max

≤0,z

∈W

)|≤1, ∀ α ∈ [0,π/2],

60Decomposition Methods for Diﬀerential Equations Theory and Applications

where the damping factor is given as ω ≤ 1/2.

PROOF

We consider a ﬁxed z

= z, Re(z) < 0andz

→−∞.Thenwe

obtain

(z,−∞)=

1 − ω

(1 − e

). (3.110)

If z = x + iy, x < 0, then it holds that

For S

we have

(z,−∞)|



1 − ω



(1 − 2exp(x)cos(y)+exp(2x)), (3.111)

and hence,

(z,−∞)|≤1 ⇔



1 − ω



≤



1 − 2exp(x)cosy +exp(2x)



. (3.112)

From (3.112) we obtain 0 ≤ ω ≤

Recursively we can also show the stability for the next iteration steps.

In the next section we derive an a posteriori error estimate for the iterative

splitting methods, starting with diﬀerent initial solutions.

3.2.5.2 A Posteriori Error Estimates for the Bounded Operators

and the Linear Operator-Splitting Method

We consider the a posteriori error estimates for the beginning time itera-

tions.

We can derive the following theorem for the a posteriori error estimates.

THEOREM 3.7

Let us consider the iterative operator-splitting method with bounded operators.

is the exact solution at t

, the time-step is τ = t − t

.Weassumethe

following diﬀerent starting solutions for c

(t) and obtain the a posteriori error

estimates for the ﬁrst iterations:

Case 1: c

(t)=0,

− c

= ||B|| τc

+ O(τ

), (3.113)

Case 2: c

(t)=c

− c

= ||BA + B|| τ

/2 c

+ O(τ

), (3.114)

Time-Decomposition Methods for Parabolic Equations for Parabolic Equations 61

Case 3: c

(t)=exp(B(t−t

)) exp(A(t−t

)) c

, (prestep with A-B splitting

method)

− c

= ||[A, B]|| τ

/2 c

+ O(τ

). (3.115)

PROOF

We apply the equations (3.96) and (3.97), and show the proof for the ﬁrst

case, c

(t)=0.

The ﬁrst iteration c

is given as

(t)=exp(A(t − t

)) c

. (3.116)

The second iteration is given as

(t)=exp(B(t −t

)) c





exp(−B(s − t

))A exp(A(s − t

)) c



(3.117)

The Taylor expansion for the two functions leads to

(t)=(I + Aτ +

+ O(τ

), (3.118)

and

(t)=(I + Bτ +

+ Aτ + A

+ BA

+ O(τ

). (3.119)

Subtracting the approximations we obtain

||c

− c

|| ≤ ||B|| τc

+ O(τ

). (3.120)

The same can be done recursively for the next iteration steps.

Based on this proof, we can also derive the a posteriori estimates for the

other initial cases.

REMARK 3.20 The a posteriori error estimates are related to the

iterative solutions of the method. Therefore, the time-step and the bounded-

ness of the operator B are correlated to the error of the next iteration step.

More iteration steps and the nonstiﬀ operator B and therefore a small oper-

ator norm, for example, spectral norm, ||B|| ≈  (for example  ≈ 10

−6

)are

essential for the reduction of the error.

3.2.5.3 Stability for the Quasi-Linear Iterative Splitting Method

The following stability theorem is given for the quasi-linear iterative split-

ting method, see (3.66)–(3.67).

62Decomposition Methods for Diﬀerential Equations Theory and Applications

The convergence is examined in a general Banach space setting and we can

prove the following stability theorem.

THEOREM 3.8

Let us consider the quasi-linear diﬀerential equation

∂

c(t)=A(c(t))c(t)+B(c(t))c(t), for t

≤ t ≤ t

n+1

c(t

)=c

(3.121)

where the operators A(c),B(c):X → X are linear and densely deﬁned in the

real Banach space X,see[201] and Subsection 3.2.4. We can deﬁne a norm

|| · || = || · ||

on the Banach space X.

We rewrite the equation (3.121) to obtain

∂

c(t)=

Ac(t)+

Bc(t), for t

≤ t ≤ t

n+1

c(t

)=c

(3.122)

where

A = A(c) and

B = B(c) with

B : X → X are given linear bounded

operators being generators of the C

semigroup and c

∈ X is a given element.

We assume also

||A(c)y||

≤ λ

||y||

, ||B(c)y||

≤ λ

||y||

, (3.123)

where λ

,λ

are constants in R

and the embedding Y ⊂ X is assumed.

Then the quasi-linear iterative operator-splitting method (3.66)–(3.67) is

stable with the following result:

c

i+1

n+1

)≤

K c





i+1

j=0

max

(3.124)

where

K>0 is a constant and c

= c(t

) is the initial condition, τ =

n+1

− t

) is the time-step and λ

max

is the maximal eigenvalue of the linear

and bounded operators

A and

We discuss this theorem in the following proof.

PROOF

Let us consider the iteration (3.66)–(3.67) on the subinterval [t

n+1

We obtain the eigenvalues of the following linear and bounded operators.

Because of the well-posed problem we have, λ

eigenvalue of

A, λ

eigenvalue

B, see [125] and [201].

Then our iteration methods are given with the eigenvalues as follows:

(t)

= λ

(t)+λ

i−1

(t),t∈ (t

n+1

)=c

(3.125)

Time-Decomposition Methods for Parabolic Equations for Parabolic Equations 63

and

i+1

(t)

= λ

(t)+λ

i+1

(t),t∈ (t

n+1

i+1

)=c

(3.126)

for i =1, 3, 5,...,withc

= c(t

) as the approximated solution of the time

level t = t

The equations can be estimated as follows:

n+1

)=exp(λ

τ)c



n+1

exp(λ

(t − s))λ

i−1

(s)ds, (3.127)

which leads to

||c

n+1

)|| ≤ K

||c

||+ τK

||c

i−1

n+1

)||. (3.128)

Further, the second equation can be estimated as

i+1

n+1

)=exp(λ

τ)c



n+1

exp(λ

(t − s))λ

(s)ds, (3.129)

which can further be estimated as

||c

i+1

n+1

)|| ≤ K

||c

||+ τK

||c

n+1

)||. (3.130)

With the recursive argument and the maximum of the eigenvalues we can

estimate the equations:

||c

i+1

n+1

)|| ≤ ||c

i+1



j=0

max

, (3.131)

||c

i+1

n+1

)|| ≤ ||c

i+1



j=0

max

, (3.132)

where

K is the maximum of all constants and λ

max

=max{λ

,λ

REMARK 3.21 We have stability for suﬃcient small time-steps τ.

In addition, accurate estimates can be derived by using the techniques of the

mild or weak solutions, see [201].

3.2.6 Detection of the Correct Splitting Operators

The goal of this section is to detect the correct combination for the operators

in the splitting methods.

We propose the idea of using the underlying eigenvalues of the used

operators.

64Decomposition Methods for Diﬀerential Equations Theory and Applications

For our given linear operator equation,

dc(t)

= Ac(t)+Bc(t), for t ∈ [0,T], (3.133)

c(0) = c

where rank(A)=rank(B)=r, we solve the following eigenvalue problems

by assuming that A and B are supposed to be diagonalizable with a well-

conditioned eigensystem:

A = X

−1

X, D

= diag

k=1

1,k

max

k=1

(Re(d

1,k

)) ≤ C

, ||X|| ≤ C

, ||X

−1

|| ≤ C

, (3.134)

B = Y

−1

Y, D

= diag

k=1

2,k

max

k=1

(Re(d

2,k

)) ≤ C

, ||Y || ≤ C

, ||Y

−1

|| ≤ C

, (3.135)

and C

are appropriate constants of moderate size. Re(d

1,k

)isthe

real value of d

1,k

and diag

k=1

1,k

) is a diagonal matrix with entries d

1,k

,k∈

{1,...,r}.

For the characterization of the diﬀerent scales of the operators, we apply

the maximum of the eigenvalues at each time-step and decide the values for

the next time-steps.

We split the operators, if we have

max

k=1

(Re(d

1,k

)) <<

max

k=1

(Re(d

2,k

)), (3.136)

max

k=1

(Re(d

2,k

)) <<

max

k=1

(Re(d

1,k

)), (3.137)

and our local time-steps for operator A are proposed as τ

max

k=1

(Re(d

1,k

))

and for operator B, the time-steps are proposed as τ

max

k=1

(Re(d

2,k

))

We do not split if we have

max

k=1

(Re(d

1,k

)) ≈

max

k=1

(Re(d

2,k

)), (3.138)

and our time-steps are given for the operators as τ

≈ τ

max

k=1

(Re(d

1,k

))

3.3 Iterative Operator-Splitting Methods for Unbounded

Operators

In this section, our goal is to generalize the results of the iterative splitting

methods for unbounded operators.