Convergence of Lie group integrators

Abstract

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie–Butcher theory of Lie group integrators leads to global error estimates.

Auxiliary constructions

In this appendix we collect several auxiliary results which enable us to construct smooth functions needed in the estimates. We begin with a technical Lemma concerning partitions of unity with some desirable properties:

Lemma 4

Let M be a paracompact finite dimensional manifold, $o\in M$ and B be an open o-neighborhood. There exists a locally finite open cover $\{U_i\}_{i\in I}$ of M, such that $I = J \cup \{i_o\}$ and the following holds:

• 1. $i_o$ is the unique index such that $o \in U_{i_o}$,

• 2. $U_{i_o} \subseteq B$,

• 3. Every $U_{i}$ is connected and relatively compact,

Proof

Since M is locally compact, we can choose a connected manifold chart $(U_{i_o},\varphi _{i_o})$ and compact o-neighborhoods $C_1,C_2$ of o such that the following inclusions hold:

$$\begin{aligned} o \in C_1 \subseteq U_{i_o} \subseteq \overline{U}_{i_o} \subseteq C_2^\circ \subseteq C_2 \subseteq B \end{aligned}$$

(where $\overline{U}_{i_o}$ is the closure and $C_2^\circ$ the interior). Then $U := M \setminus C_1$ is open and metrisable, whence paracompact. Following [16, II, §3 Theorem 3.3] there is a locally finite cover of U by charts $(U_j,\varphi _j)_{j\in J'}$ such that $U_j$ is connected and relatively compact. Let us now throw out all elements of the cover which are contained in $U_{i_o}$, i.e. define $J := \{ j \in J'\mid U_j \cap M \setminus U_{i_o} \ne \emptyset \}$ and set $I := J \cup \{i_o\}$. By construction $o \in U_i$ for $i\in I$ if and only if $i=i_o$, $U_{i_o} \subseteq B$ and every $U_i$ is connected and relatively compact. To prove that $\{U_i\}_{i\in I}$ is a locally finite cover of M, we observe that $\{U_i\}_{i\in I}$ covers M by construction. Now $K := C_2 \setminus U_{i_o} \subseteq U$ is compact, whence only finitely many elements of the locally finite cover $\{U_j\}_{j\in J}$ intersect it. This means that only finitely many of the sets $U_j, j\in J$ intersect $U_{i_o}$, whence $\{U_i\}_{i\in I}$ is locally finite.$\square$

We now prove Lemma 2 whose statement we repeat here for convenience.

Lemma 5

For $\varepsilon >0$ and (M, g) a connected complete Riemannian manifold, there exists $F_\varepsilon \in C^\infty (M)$ with the following properties.

• 1. $F_\varepsilon (o)=0$ and $F_\varepsilon (x) \ge 0,\ \forall x \in M$,

• 2. if $d(x,o)\ge \varepsilon$, then $F_\varepsilon (x) \ge d(x,o)$.

Proof

Let $\varepsilon >0$ and denote by $B := B_\varepsilon ^d (o)$ the metric ball of radius $\varepsilon$ around o. Apply now Lemma 4 with the above choice of B to obtain a locally finite open cover $\{U_i\}_{i\in I}$ of M with a unique element $U_{i_o}$ such that $o\in U_{i_o} \subseteq B$. Following [16, II, §3 Corollary 3.8] we pick a smooth partition of unity $\{\chi _i\}_{i\in I}$ subordinate to the cover $\{U_i\}_{i\in I}$. Note that by construction of the cover, we must have $\chi _{i_0} (o) = 1$. Define the constants $M_j := \max \{\varepsilon , \sup _{y \in \overline{U}_j} d(o,y)\}$ for $j\in J$. By compactness of $\overline{U}_j$ and continuity of the Riemannian distance (follows from [13, Theorem 1.9.5]), the $M_j$ are finite. Hence we can build a family of smooth function:

$$\begin{aligned} f_{i} (x) := {\left\{ \begin{array}{ll} \varepsilon (1- \chi _{i_0} (x)) &{} \text { for } i = i_o\\ M_j \chi _j(x) &{} \text { for } i=j\in J\end{array}\right. } \quad i \in I. \end{aligned}$$

Observe now that since the $\{\chi _i\}_{i\in I}$ is a partition of unity, their supports form a locally finite family $\{\text {supp} \chi _i\}_{i\in I}$. We deduce that the family of supports for the functions $f_i$ is also locally finite, whence we can define a smooth function

$$\begin{aligned} F_\varepsilon (x) := \sum _{i \in I} f_i (x) \quad x \in M \end{aligned}$$

which satisfies $F_\varepsilon (o)=0$ and $F_{\varepsilon }(x) \ge 0$ for all $x\in M$. If $x \in M \setminus B$, there is a finite non empty $L_x \subseteq J$ such that $x \in U_i$ if and only if $i \in L_x$. In particular $\sum _{i \in L_x} \chi _i (x) = 1$ and as $x \in U_i$ for every $i \in L_x$ by construction one has $d(x,o)\le M_i$ for all $i\in L_x$. Thus we deduce that

$$\begin{aligned} d(x,o) \le \min _{i\in L_x} \{M_i\} \sum _{i\in L_x} \chi _i (x) \le \sum _{i\in L_x} M_i \chi _i (x) \le \sum _{i \in L_x \cup \{i_o\}} f_i(x) = F_\varepsilon (x). \end{aligned}$$

$\square$

Finally, we construct a family of smooth functions which allows us to obtain estimates on the Riemannian distance for points close to o. This is Lemma 3 whose statement we repeat for the readers convenience.

Lemma 6

Let $\varepsilon >0$ be sufficiently small that the closure of the metric ball $B_\varepsilon ^d (o)$ is contained in a manifold chart $(U,\varphi )$. Then there is $N \in \mathbb {N}$ and a family $\{f_n\}_{1 \le n\le N} \subseteq C^\infty (M)$ with the following properties

• 1. $f_n (o)=0$,

• 2. If $d(x,o)< \varepsilon$ then there is $1\le n_x\le N$ such that $f_n (x) \ge d(x,o)$.

Proof

As a homogeneous Riemannian manifold, (M, g) is complete, see 1.3. Thus the closed and bounded set $K := \overline{B_\varepsilon ^d (o)}$ is compact by the Hopf–Rinow theorem [2, Theorem 1.65]. We may assume without loss of generality that $\varphi (o)=0$. Now by standard arguments⁸ for every smooth Riemannian manifold the charts are locally bi-Lipschitz to Euclidean space. Since $K \subseteq U$ is compact, we may (after shrinking U if necessary) assume that $\varphi$ is bi-Lipschitz with respect to the euclidean distance $d_{2}$ on $\mathbb {R}^n$ and the geodesic distance on U, i.e.

$$\begin{aligned} d_{2} (\varphi (x),\varphi (y)) \lesssim d(x,y) \lesssim d_{2}(\varphi (x),\varphi (y)),\quad \forall x,y \in U, \end{aligned}$$

(9)

where “$\lesssim$” is used to denote an inequality up to a (multiplicative) constant. Using the equivalence of norms on $\mathbb {R}^n$, we now replace the euclidean distance $d_{2}$ in (9) by the distance $d_1$, induced by the $\ell ^1$-norm $||x||_1 := \sum _{i=1}^n |x_i|$. We claim now, that there is $N\in \mathbb {N}$ and a family of smooth functions $\{P_n\}_{1\le n\le N} \subseteq C^\infty (\varphi (U))$ which satisfy the following properties for all $1 \le n\le N$:

• 1. $P_n (\varphi (o)) = H_n (0) = 0$

• 2. If $x \in \varphi (K)$ then there exists $1\le n_x \le N$ such that $||x||= d_1 (x,\varphi (o)) \le P_{n_x} (x)$.

If this were true, then the proof can be finished as follows: Let L be the (smallest) Lipschitz constant such that $d(x,y) \le Ld_1(\varphi (x),\varphi (y)), \quad \forall x,y\in U$. Since U is an open neighborhood of K, we can choose a smooth cut-off function $\xi :M \rightarrow [0,1]$ such that $\xi |_K \equiv 1$ and $\xi |_{M\setminus U} \equiv 0$. Then we set

$$\begin{aligned} f_n :M \rightarrow \mathbb {R},\ x \mapsto {\left\{ \begin{array}{ll} L\xi (x) \cdot P_n \circ \varphi (x) &{}\quad \text { if } x \in U \\ 0 &{}\quad \text {otherwise}.\end{array}\right. }, \end{aligned}$$

Clearly we have $f_n \in C^\infty (M)$ and $f_n(o)= 0$ for all $1\le n \le N$. If $d(x,o) <\varepsilon$, then $x \in K$, whence there is $n_x := n_{\varphi (x)}$ as in property 2. of the family $\{P_n\}_n$ such that

$$\begin{aligned} f_{n_x} (x) = L\underbrace{\xi (x)}_{=1} \cdot P_{n_{\varphi (x)}} \circ \underbrace{\varphi (x)}_{\in \varphi (K)} \ge L d_1 (\varphi (x),\varphi (o)) \ge d(x,o). \end{aligned}$$

Proof of the claim We have to construct smooth functions which satisfy properties 1. and 2. To this end, consider for $1\le k \le n$ the smooth (linear) functions

$$\begin{aligned} p_{k,0} :\mathbb {R}^n \rightarrow \mathbb {R}, \quad (x_1, \ldots , x_n) \mapsto x_k, \qquad p_{k,1} :\mathbb {R}^n \rightarrow \mathbb {R},\quad p_{k,1}(x) := -p_{k,0} (x) \end{aligned}$$

Construct for every multiindex $\alpha = (\alpha _1, \ldots , \alpha _n) \in \{0,1\}^n$ a function

$$\begin{aligned} P_{\alpha } (x) := \sum _{i=1}^n p_{i,\alpha _i} (x), \quad x\in \mathbb {R}^n \end{aligned}$$

Set $N := 2^n$ and choose an arbitrary order of the multiindices $\alpha$ (naming the ith $\alpha ^i$, to define the desired family $P_n := P_{\alpha ^n}|_{U}$ for $1\le n\le N$. Obviously $P_n (0)=0$ and from the construction it is clear that the $P_n$ satisfy property 2.$\square$