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(s_{1}*, s*_{2}*, . . . , s** _{d}*) =
Y

*d*

*i=1*

*s** _{d+1−i}* (3.2.1)

*is called sequential splitting. The operator splitting obtained by the choice*

Φ(s_{1}*, s*_{2}*, . . . s** _{d}*) =

If we apply the above approximations on the mesh *ω**τ*, then the one step method (3.1.4)
reads as

for the sequential splitting^{4}, and
*r*_{spl}(τ A) = *r*_{SM}(τ A) :=

3When the operator splitting is exact, i.e., the local splitting error is zero, then we set*p*=*∞.*

4It is also called Lie-Trotter or Yanenko splitting.

for the Strang-Marchuk splitting^{5}.

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 A*i*)v with some
given element*v. Since clearlyw*=*w(τ*), where*w(t) is the solution of the Cauchy problem*
defined by the operator *A** _{i}* 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:

*dw*_{i}^{n}

*dt* (t) =*A*_{i}*w*_{i}* ^{n}*(t), (n

*−*1)τ < t

*≤nτ,*

*w*

_{i}*((n*

^{n}*−*1)τ) =

*w*

_{i−1}*(nτ),*

^{n}(3.2.6)
where *i*= 1,2, . . . , d. The split solution is defined as

*w*_{seq}^{(N)}(nτ) =*w*^{n}* _{d}*(nτ). (3.2.7)
Here

*w*

^{n}_{0}(nτ) =

*w*

^{(N)}seq((n

*−*1)τ), and

*w*

^{(N}seq

^{)}(0) =

*w(0) is known from the original*continuous problem (3.1.1).

That is, the algorithm is the following:

*A*_{1} *→A*_{2} *→ · · ·A*_{d}

| {z }

step 1

*⇒A*| _{1} *→A*_{2}{z*→ · · ·A*}_{d}

step 2

*⇒ · · · ⇒A*|_{1} *→A*_{2}{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
*dw*_{i}^{n}

*dt* (t) =*A*_{i}*w*_{i}* ^{n}*(t), (n

*−*1)τ < t

*≤*(n

*−*0.5)τ,

*w*

_{i}*((n*

^{n}*−*1)τ) =

*w*

_{i−1}*((n*

^{n}*−*0.5)τ).

(3.2.8)

Then we define the solution of the problem
*dw*_{d}^{n}

*dt* (t) =*A*_{d}*w*_{d}* ^{n}*(t), (n

*−*1)τ < t

*≤nτ,*

*w*

_{d}*((n*

^{n}*−*1)τ) =

*w*

_{d−1}*((n*

^{n}*−*0.5)τ).

(3.2.9)

Finally, at a fixed*n* for the values *i*=*d*+ 1, d+ 2, . . .2d*−*1 we solve the problems
*dw*^{n}_{i}

*dt* (t) =*A*_{2d−i}*w*^{n}* _{i}*(t), (n

*−*0.5)τ < t

*≤nτ,*

*w*

^{n}*((n*

_{i}*−*0.5)τ) =

*w*

^{n}*(nτ).*

_{i−1}(3.2.10)

5It is also called second order leapfrog, St¨ormer, Verlet splitting.

The split solution is

*w*^{(N}_{SM}^{)}(nτ) = *w*_{2d−1}* ^{n}* (nτ). (3.2.11)
Here

*w*

^{n}_{0}((n

*−*0.5)τ) =

*w*

^{N}_{SM}((n

*−*1)τ), and

*w*

_{SM}

^{(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 operatorsA**i* *(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*=P_{d}

*i=1**A** _{i}*, we get
For the sequential splitting the split solution at

*t*=

*τ*is

*w*^{(N}_{seq}^{)}(τ) =

Using the expressions (3.2.13) and (3.2.15), for the local splitting error we obtain

*Err*_{seq}(τ) = *τ*^{2}

Hence, for arbitrary chosen operators *A** _{i}* the right-hand side of (3.2.16) is

*O(τ*

^{2}), 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 *r*_{SM}(τ A) in (3.2.5) approximates the operator exp(τ A) in third order.

Since After some simple but tedious computation we get

*B*_{1}*B*_{2}*B*_{3} =*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
*Err*_{seq}(τ) = *τ*^{2}
denotes the commutator of the operators *A** _{i}* and

*A*

*. Hence, we have*

_{j}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 *A*_{1} and *A*_{2} do not commute, since
[A_{1}*, A*_{2}] =
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 *A*1 and *A*2 is the same as the sequential splitting for three operators 0.5A1, *A*2

and 0.5A_{1}, 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(τ*^{3}) term for the difference
exp(τ(A_{1}+*A*_{2}))*−*exp
For the first exponential we have

exp(τ(A_{1}+*A*_{2})) = *I*+*τ*(A_{1}+*A*_{2}) + *τ*^{2}

2 (A_{1}+*A*_{2})^{2}+*τ*^{3}

6 (A_{1}+*A*_{2})^{3} +*O(τ*^{4}).

(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

*Err*SM(τ) = *τ*^{3}
Hence, we have the following

Theorem 3.2.6 *The Strang-Marchuk splitting is of third order if and only if the operator*
0.5A_{1}+*A*_{2} *commutes with the commutator* [A_{1}*, A*_{2}], i.e., when the condition

[0.5A_{1}+*A*_{2}*,*[A_{1}*, A*_{2}]] = 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* *τ*^{3}*, but also of* *τ*^{4} *vanishes, i.e., the condition*
(3.2.35) *guarantees the fourth-order accuracy [41].*