Non-existence of generalized splitting methods with positive coefficients of order higher than four

TL;DR

This paper proves that generalized exponential splitting methods with positive coefficients cannot achieve an order higher than four, extending the known limitation from classical methods of order two.

Contribution

It establishes a fundamental limitation on the order of generalized exponential splitting methods with positive real coefficients, generalizing previous results.

Findings

Limit of order four for generalized exponential splitting methods with positive coefficients

Extension of classical order two restriction to higher-order methods

Theoretical proof of non-existence for higher-order methods with positive coefficients

Abstract

We prove that generalized exponential splitting methods making explicit use of commutators of the vector fields are limited to order four when only real coefficients are admitted. This generalizes the restriction to order two for classical splitting methods with only positive coefficients.

Equations

$${\partial }_{t}y(t)=A(y(t))+B(y(t)),t\ge t_0,y(t_0)=y_0.$$

$$y_{n+1}=\mathcal{S}(\tau )y_n={\mathrm{e}}^{b_s\tau B}{\mathrm{e}}^{a_s\tau A}\cdots {\mathrm{e}}^{b_1\tau B}{\mathrm{e}}^{a_1\tau A}{y}_n.$$

$$\mathcal{S}(\tau )y_n={\mathrm{e}}^{\tau (A+B)}{y}_n+O({\tau }^{p+1}).$$

$$\mathcal{S}(\tau )={\mathrm{e}}^{\frac{1}{2}\tau A}{\mathrm{e}}^{\tau B}{\mathrm{e}}^{\frac{1}{2}\tau A}.$$

$$y_{n+1}=\mathcal{S}(\tau )y_n={\mathrm{e}}^{c_s{\tau }^3[B,[B,A]]}{\mathrm{e}}^{b_s\tau B}{\mathrm{e}}^{a_s\tau A}\cdots {\mathrm{e}}^{c_1{\tau }^3[B,[B,A]]}{\mathrm{e}}^{b_1\tau B}{\mathrm{e}}^{a_1\tau A}{y}_n,$$

$$y_{n+1}=\mathcal{S}(\tau )y_n={\mathrm{e}}^{b_s\tau B+c_s{\tau }^3[B,[B,A]]}{\mathrm{e}}^{a_s\tau A}\cdots {\mathrm{e}}^{b_1\tau B+c_1{\tau }^3[B,[B,A]]}{\mathrm{e}}^{a_1\tau A}{y}_n,$$

$$\mathcal{S}(\tau )={\mathrm{e}}^{\frac{1}{6}\tau B}{\mathrm{e}}^{\frac{1}{2}\tau A}{\mathrm{e}}^{\frac{2}{3}\tau B-\frac{1}{72}{\tau }^3[B,[B,A]]}{\mathrm{e}}^{\frac{1}{2}\tau A}{\mathrm{e}}^{\frac{1}{6}\tau B}$$

$${\partial }_{t}u(t)=H(t)u(t)=(H_0+t{H}_1)u(t),u(0)=u_0$$

$$v(\tau )={\mathrm{e}}^{a_s\tau H_0+c_s{\tau }^2{H}_1}\cdots {\mathrm{e}}^{a_1\tau H_0+c_1{\tau }^2{H}_1}{u}_0$$

$$v(\tau )-u(\tau )=O({\tau }^6),$$

$$y(t)=\left(\genfrac{}{}{0ex}{}{s(t)}{u(t)}\right),{\partial }_{t}y(t)=\underset{A(y(t))}{\underbrace{\left(\genfrac{}{}{0ex}{}{0}{H(s(t))u(t)}\right)}}+\underset{B(y(t))}{\underbrace{\left(\genfrac{}{}{0ex}{}{1}{0}\right)}},y(0)=y_0=\left(\genfrac{}{}{0ex}{}{0}{u_0}\right)$$

$$({\mathrm{e}}^{t{D}_A}F)(y)=F({\mathcal{E}}_A(t,y)),({\mathrm{e}}^{t{D}_B}F)(y)=F({\mathcal{E}}_B(t,y)),$$

$${\mathcal{E}}_A(t,y_0)=({\mathrm{e}}^{t{D}_A}\mathrm{Id})(y_0),{\mathcal{E}}_B(t,y_0)=({\mathrm{e}}^{t{D}_B}\mathrm{Id})(y_0),$$

$$y_{n+1}=\mathcal{S}(\tau ,y_n)$$

$$\displaystyle=\big{(}\mathrm{e}^{a_{1}\tau D_{A}}\mathrm{e}^{b_{1}\tau D_{B}}\cdots\mathrm{e}^{a_{s}\tau D_{A}}\mathrm{e}^{b_{s}\tau D_{B}}\mathrm{Id}\big{)}(y_{n})$$

$$D_A=\sum _{i=1}^{d+1}{A}_i(y)\frac{\partial }{\partial y_i}=\sum _{i=1}^d{[H(s)u]}_i\frac{\partial }{\partial u_i}=\sum _{i=1}^d{[(H_0+s{H}_1)u]}_i\frac{\partial }{\partial u_i}$$

$$D_B=\sum _{i=1}^{d+1}{B}_i(y)\frac{\partial }{\partial y_i}=\frac{\partial }{\partial s}.$$

$$[D_B,D_A]=\sum _{i=1}^d{[H^{\prime }(s)u]}_i\frac{\partial }{\partial u_i}=\sum _{i=1}^d{[H_{1}u]}_i\frac{\partial }{\partial u_i}$$

$$[D_B,[D_B,D_A]]=\sum _{i=1}^d{[H^{\prime \prime }(s)u]}_i\frac{\partial }{\partial u_i}=0,$$

$${\mathrm{e}}^X{\mathrm{e}}^Y={\mathrm{e}}^{Y+[X,Y]]}{\mathrm{e}}^X\text{mbox}for[X,[X,Y]]=0$$

$$\mathcal{S}(\tau ,y_0)=$$

$$\displaystyle{}=\big{(}\mathrm{e}^{a_{1}\tau D_{A}}\mathrm{e}^{a_{2}\tau D_{A}+c_{2}\tau^{2}[D_{B},D_{A}]}\cdots\mathrm{e}^{a_{s}\tau D_{A}+c_{s}\tau^{2}[D_{B},D_{A}]}\mathrm{e}^{(b_{1}+\ldots+b_{s})\tau D_{B}}\mathrm{Id}\big{)}(y_{0})$$

$$\displaystyle\big{(}\mathrm{e}^{a_{j}\tau D_{A}+c_{j}\tau^{2}[D_{B},D_{A}]}F\big{)}\left({s\atop u}\right)$$

$$=F\left(\genfrac{}{}{0ex}{}{s}{{\mathrm{e}}^{a_j\tau H_0+a_j\tau s{H}_1+c_j{\tau }^2{H}_1}u}\right),$$

$$\big{(}\mathrm{e}^{a_{j}\tau D_{A}+c_{j}\tau^{2}[D_{B},D_{A}]}F\big{)}\left({0\atop u}\right)=F\left({0\atop\mathrm{e}^{a_{j}\tau H_{0}+c_{j}\tau^{2}H_{1}}u}\right).$$

$${\mathrm{e}}^{a_s\tau H_0+c_s{\tau }^2{H}_1}\cdots {\mathrm{e}}^{a_2\tau H_0+c_2{\tau }^2{H}_1}{\mathrm{e}}^{a_1\tau H_0}{u}_0,$$

Full text

††journal: Appl. Math. Lett.

url]www.asc.tuwien.ac.at/ winfried/

url]www.harald-hofstaetter.at

url]www.othmar-koch.org

1 Introduction

1.1 Splitting methods

We consider evolution equations on $\mathbb{R}^{d}$ or $\mathbb{C}^{d}$ where the right-hand side is split into two components,

[TABLE]

In this introduction we only consider the linear case where $A,B$ are linear operators represented by real or complex matrices. For the numerical solution of (1) we consider $s$-stage splitting methods, where one step $(t_{n},y_{n})\mapsto(t_{n+1},y_{n+1})$ with step-size $\tau$ is given by

[TABLE]

A splitting method has convergence order $p$ if it holds

[TABLE]

1.2 Positive coefficients

For certain applications only splitting schemes with non-negative coefficients are suitable. In particular this is the case if $A$ is a discretized sectorial operator associated with a parabolic equation, because in this case the flow of $A$ is non-reversible and does not tolerate negative time increments in the numerical approximation, whence $a_{j}$ is required to be non-negative. A splitting method of order $p=2$ with all coefficients positive is given by Strang splitting

[TABLE]

It is known that $p=2$ is the maximum order of a splitting method with all coefficients positive, a fact which is established by the following theorem.

Theorem 1

If $\mathcal{S}$ is a splitting method (2) of order $p\geq 3$ with real coefficients, then at least one of the coefficients $a_{j}$ is strictly negative, and also at least one of the coefficients $b_{j}$ is strictly negative.

This theorem was first proved in [1], see also [2]. A weaker version stating that at least one of all coefficients $a_{j}$, $b_{j}$ combined is strictly negative, was proved earlier in [3].

1.3 Generalized splitting methods

In many applications the commutator $[B,[B,A]]$ and its exponential are readily computable, see [4]. This suggests to consider generalized splitting methods of the form

[TABLE]

or

[TABLE]

which possibly allow orders higher than 2 while involving only positive coefficients. Indeed, the scheme

[TABLE]

proposed in [5, 6] has order $p=4$ and positive coefficients $a_{j}$ and $b_{j}$. However, as established by the following theorem, $p=4$ is the maximum order of such a generalized splitting method with all coefficients $a_{j}$ positive. This holds even under the additional assumption $[B,[B,[B,A]]]=0$, which in many applications is satisfied, see [4].

Theorem 2

**

(i) If $\mathcal{S}$ is a generalized splitting method of the form (3) or (4) of order $p\geq 5$ with real coefficients, then at least one of the coefficients $a_{j}$ is strictly negative.

(ii) If $\mathcal{S}$ is a generalized splitting method (4) with real coefficients which is of order $p\geq 5$ if applied to an equation (1) where the operators $A,B$ satisfy $[B,[B,[B,A]]]=0$, then at least one of the coefficients $a_{j}$ is strictly negative.

It is clear111By logical transposition: If an object (here a generalized splitting method with all coefficients $a_{j}$ nonnegative) does not exist under some restrictive assumptions, then it cannot exist under more general assumptions.

that part (i) follows immediately from part (ii), which immediately follows from the following theorem, which may be interesting in itself.

Theorem 3

If $\mathcal{S}$ is a splitting method (2) with real coefficients which is of order $p\geq 5$ if applied to an equation (1) where the operators $A,B$ satisfy $[B,[B,A]]=0$, then at least one of the coefficients $a_{j}$ is strictly negative.

A proof of Theorem 2 was proposed in [7]. In Section 2 we will give a new independent proof by showing that Theorem 3 (and thus also Theorem 2) is an easy consequence of a recent result proved by the authors in [8].

2 Proof of Theorem 3

The essential step leading to the main result of [8] is comprised by the following proposition.222Note that we have changed some denotations: $H(t)$, $H_{0}$, $H_{1}$, $s$, $a_{j}$, $c_{j}$ correspond respectively to the denotations $A(t)$, $A_{0}$, $A_{1}$, $J$, $b_{j}$, $y_{j}$ of [8].

Proposition 1

Let $u(t)$ be the exact solution of

[TABLE]

with $H_{0},H_{1}\in\mathbb{C}^{d\times d}$ and

[TABLE]

with given coefficients $a_{j},c_{j}\in\mathbb{R}$. If

[TABLE]

then at least one of the coefficients $a_{j}$ is strictly negative.333See Remark 2 below.

Here (6) can be interpreted as one step with step-size $\tau$ of a commutator-free exponential integrator applied to the special non-autonomous equation (5). To show that Theorem 3 follows from Proposition 1 we first use the standard reformulation

[TABLE]

of the non-autonomous problem (5) as an autonomous problem by adding the component $s(t)=t$. Here the operators $A,B\colon\mathbb{R}^{d+1}\to\mathbb{R}^{d+1}$ are nonlinear, therefore a direct application of the splitting method (2) is not possible. However, by associating the flows $\mathcal{E}_{A}(t,y_{0})$, $\mathcal{E}_{B}(t,y_{0})$ of the subproblems $\partial_{t}y(t)=A(y(t))$, $\partial_{t}y(t)=B(y(t))$ with exponentials of Lie derivatives444We adopt the notation from [9, Chapter III]. $\mathrm{e}^{tD_{A}}$, $\mathrm{e}^{tD_{B}}$, which act on a smooth map $F:\mathbb{R}^{d+1}\to\mathbb{R}^{d+1}$ as

[TABLE]

and thus

[TABLE]

each splitting method (2) of order $p$ for linear problems (1) can be promoted to a splitting method

[TABLE]

of the same order for nonlinear problems, see [9, Section III.5.1].

Remark 1

The convergence order of a (generalized) splitting method is determined by order conditions, which are polynomial equations in the coefficients of the method. Usually these conditions are derived in a purely formal way in the abstract algebra of formal power series in the non-commuting variables $\mathtt{A}$, $\mathtt{B}$ and its embedded Lie algebra with Lie bracket defined by $[X,Y]=XY-YX$, see [10, 11]. By associating $\mathtt{A}$, $\mathtt{B}$ with the matrices $A$, $B$ in the linear case, and with the Lie derivatives $D_{A}$, $D_{B}$ in the nonlinear case, it follows that (2) and (8) indeed have the same order [9].

For the special problem (7) the Lie derivatives are given by

[TABLE]

and

[TABLE]

A straightforward calculation leads to

[TABLE]

and

[TABLE]

which shows that the condition $[B,[B,A]]=0$ of Theorem 3 promoted to the nonlinear case is satisfied. Repeated application of the formal identity

[TABLE]

to (8) yields

[TABLE]

with well-defined coefficients $c_{2},\dots,c_{s}\in\mathbb{R}$. Here a single exponential acts as

[TABLE]

and thus, substituting $s=0$,

[TABLE]

It follows that for $y_{0}=\left({0\atop u_{0}}\right)$ the lower components of (9) can be written as

[TABLE]

which is of the form (6). From Proposition 1 it follows that if the splitting method (8) has order $p\geq 5$ if applied to the special problem (7), or, a fortiori555See footnote 1., if applied to nonlinear problems with $[B,[B,A]]=0$ in general, then at least one of the coefficients $a_{j}$ is strictly negative. We have thus proved the nonlinear version of Theorem 3. For similar formal reasons as in Remark 1, the linear version of Theorem 3 follows as well.

Remark 2

Strictly speaking, only a version of Proposition 1 with the weaker conclusion that at least one of the coefficients $a_{j}$ is non-positive has been proved in [8]. Since we may assume form the outset that $a_{j}\neq 0$ for $j=2,\dots,s$ in (2), it is clear that this weaker version already suffices for the proof of Theorem 3. Conversely, Proposition 1 follows from Theorem 3, as can be shown by a similar reasoning as before. Thus, the version of Proposition 1 given here follows from the weaker version proved in [8].

Acknowledgements

This work was supported in part by the Vienna Science and Technology Fund (WWTF) [grant number MA14-002].