In this part we introduce and analyze the sequential splitting and the Strang-Marchuk splitting which are the most traditional and widely used operator splitting methods.
Definition 3.2.1 The operator splitting corresponding to the choice
Φ(s1, s2, . . . , sd) = Yd
i=1
sd+1−i (3.2.1)
is called sequential splitting. The operator splitting obtained by the choice
Φ(s1, s2, . . . sd) =
If we apply the above approximations on the mesh ωτ, then the one step method (3.1.4) reads as
for the sequential splitting4, and rspl(τ A) = rSM(τ A) :=
3When the operator splitting is exact, i.e., the local splitting error is zero, then we setp=∞.
4It is also called Lie-Trotter or Yanenko splitting.
for the Strang-Marchuk splitting5.
In the realization of the operator splitting algorithm (3.2.3) the central query is the computation of the exponential, namely, the computation of w := exp(τ Ai)v with some given elementv. Since clearlyw=w(τ), wherew(t) is the solution of the Cauchy problem defined by the operator Ai and the initial value v on the time interval [0, τ], therefore the algorithm for the split solutions in the sequential splitting and Strang-Marchuk splitting are the following.
1. Sequential splitting [3]. For each n = 1,2, . . . , N we successively solve the Cauchy problems:
dwin
dt (t) =Aiwin(t), (n−1)τ < t≤nτ, win((n−1)τ) = wi−1n (nτ),
(3.2.6) where i= 1,2, . . . , d. The split solution is defined as
wseq(N)(nτ) =wnd(nτ). (3.2.7) Here wn0(nτ) = w(N)seq((n −1)τ), and w(Nseq)(0) = w(0) is known from the original continuous problem (3.1.1).
That is, the algorithm is the following:
A1 →A2 → · · ·Ad
| {z }
step 1
⇒A| 1 →A2{z→ · · ·A}d
step 2
⇒ · · · ⇒A|1 →A2{z→ · · ·A}d
step N
.
2. Strang-Marchuk splitting [93, 135]. For each fixed n = 1,2, . . . , N we solve the following Cauchy problems:
For the values i= 1,2, . . . , d−1 we solve the problems dwin
dt (t) =Aiwin(t), (n−1)τ < t≤(n−0.5)τ, win((n−1)τ) = wi−1n ((n−0.5)τ).
(3.2.8)
Then we define the solution of the problem dwdn
dt (t) =Adwdn(t), (n−1)τ < t≤nτ, wdn((n−1)τ) = wd−1n ((n−0.5)τ).
(3.2.9)
Finally, at a fixedn for the values i=d+ 1, d+ 2, . . .2d−1 we solve the problems dwni
dt (t) =A2d−iwni(t), (n−0.5)τ < t≤nτ, wni((n−0.5)τ) =wni−1(nτ).
(3.2.10)
5It is also called second order leapfrog, St¨ormer, Verlet splitting.
The split solution is
w(NSM)(nτ) = w2d−1n (nτ). (3.2.11) Here wn0((n−0.5)τ) = wNSM((n−1)τ), and wSM(N)(0) = w(0) is known from (3.1.1).
That is, the algorithm is the following:
1
In the following we analyze the order of the sequential splitting and the Strang-Marchuk splitting for d linear bounded sub-operators. (For d = 2 and d = 3 similar results can be found in [73].)
Theorem 3.2.2 For linear and bounded operatorsAi (i= 1,2, . . . , d) the sequential split-ting has first, while the Strang-Marchuk splitsplit-ting has second order of accuracy.
Proof. The exact solution of (3.1.1) at t=τ is where, as before, O denotes the Landau symbol (see p.35). Since A=Pd
i=1Ai, we get For the sequential splitting the split solution at t =τ is
w(Nseq)(τ) =
Using the expressions (3.2.13) and (3.2.15), for the local splitting error we obtain
Errseq(τ) = τ2
Hence, for arbitrary chosen operators Ai the right-hand side of (3.2.16) is O(τ2), which yields that the sequential splitting is of first order.
In order to prove the statement for the Strang-Marchuk splitting, we have to show that the operator rSM(τ A) in (3.2.5) approximates the operator exp(τ A) in third order.
Since After some simple but tedious computation we get
B1B2B3 =I+τ
In the following we give the conditions under which the order of the sequential splitting and the Strang-Marchuk splitting are higher.
The error formula (3.2.16) can be rewritten as Errseq(τ) = τ2 denotes the commutator of the operators Ai and Aj. Hence, we have
Theorem 3.2.3 The sequential splitting has higher than first order accuracy if and only
Theorem 3.2.3 shows that the pairwise commutativity of the operators is a sufficient condition for (3.2.25). (This is quite natural because in this case the sequential splitting is exact.) Moreover, for d= 2 the commutativity is a necessary and sufficient condition.
Therefore, when there are only two operators in the operator sum in (3.1.1), then there are only two cases: either the sequential splitting is exact or it has first order accuracy.
However, it is not yet clear whether
• the pairwise commutativity is a necessary condition of the higher-order for d >2;
• the local splitting error vanishes only under the pairwise commutativity condition.
We analyze these problems for d= 3 in a simple matrix case. We consider the matrix A=
In this example A1 and A2 do not commute, since [A1, A2] = and so the sequential splitting is exact. Hence, the above example implies the following fact:
Theorem 3.2.4 In the sequential splitting the commutation of each pair of sub-operators is not necessary for vanishing of the local splitting error or for achieving higher order when the number of sub-operators is greater than two
As before, for the Strang-Marchuk splitting the commutativity of the sub-operators is sufficient for zero splitting error. However the necessity of the commutativity condition is unclear even for d = 2. Let us observe that the Strang-Marchuk splitting with two operators A1 and A2 is the same as the sequential splitting for three operators 0.5A1, A2
and 0.5A1, respectively. Hence, by use of Theorem 3.2.4, we get
Theorem 3.2.5 The commutation of the sub-operators in the Strang-Marchuk splitting is not a necessary condition for vanishing local splitting error.
In the following, for d = 2 we give a condition under which the Strang-Marchuk splitting has third order of accuracy.
We have to analyze the O(τ3) term for the difference exp(τ(A1+A2))−exp For the first exponential we have
exp(τ(A1+A2)) = I+τ(A1+A2) + τ2
2 (A1+A2)2+τ3
6 (A1+A2)3 +O(τ4).
(3.2.32) For the second expression we obtain
exp Then, by use of the expressions (3.2.32) and (3.2.33), the local splitting error of the Strang-Marchuk splitting can be expressed by the commutators, namely we have
ErrSM(τ) = τ3 Hence, we have the following
Theorem 3.2.6 The Strang-Marchuk splitting is of third order if and only if the operator 0.5A1+A2 commutes with the commutator [A1, A2], i.e., when the condition
[0.5A1+A2,[A1, A2]] = 0 (3.2.35) holds.
Remark 3.2.7 With some cumbersome computation, one can show that under the con-dition (3.2.35) the coefficient of not only τ3, but also of τ4 vanishes, i.e., the condition (3.2.35) guarantees the fourth-order accuracy [41].