A New Class of Uniformly Accurate Numerical Schemes for Highly Oscillatory Evolution Equations

Abstract

We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.

Appendix

This section is devoted to the proof of Proposition 3.2 and Theorem 3.3. We first recall the two following basic results from [4]:

Lemma A.1

(See Castella et al. [4]) Let $0< \delta < \rho \le 2R$. Assume that the function $(\theta ,u) \in {\mathbb {T}}\times \mathcal{K}_\rho \mapsto \varphi _\theta (u) \in X_{\mathbb {C}}$ is analytic, and that $\varphi _\theta$ is a near-identity mapping, in the sense that

$$\begin{aligned} \Vert \varphi - \mathrm{Id}\Vert _\rho \le \frac{\delta }{2}. \end{aligned}$$

Then $\Vert \partial _u \varphi \Vert _{\rho -\delta } \le \frac{3}{2}$, the mapping $\partial _u \langle \varphi \rangle ^{-1}$ is well defined and analytic on $\mathcal{K}_{\rho -\delta }$ with $\Vert \partial _u \langle \varphi \rangle ^{-1} \Vert _{\rho -\delta } \le 2$, and the mappings $(\theta ,u) \in {\mathbb {T}}\times \mathcal{K}_{\rho -\delta } \mapsto \Lambda (\varphi )_{\theta }(u)$ and $(\theta ,u) \in {\mathbb {T}}\times K_{\rho -\delta } \mapsto \Gamma ^\varepsilon (\varphi )_{\theta }(u)$ are well defined and analytic. In addition, the following bounds hold for all $\varepsilon \ge 0$,

$$\begin{aligned} \Vert \Lambda (\varphi )\Vert _{\rho -\delta } \le 4 \, M \quad \text{ and } \quad \Vert \Gamma ^\varepsilon (\varphi ) - \mathrm{Id}\Vert _{\rho -\delta } \le 4 \, M \, \varepsilon . \end{aligned}$$

Lemma A.1 shows that starting from a function $(\theta ,u) \in {\mathbb {T}}\times \mathcal{K}_{2R} \mapsto \varphi _\theta (u) \in X_{\mathbb {C}}$, we can consider iterates $\left( \Gamma ^\varepsilon \right) ^k (\varphi )_\theta$ at the cost of a gradual thinning of their domains of analyticity. The following contraction property holds.

Lemma A.2

(See Castella et al. [4]) Let $0< \delta < \rho \le 2R$ and consider two periodic, near-identity mappings $(\theta ,u) \in {\mathbb {T}}\times \mathcal{K}_\rho \mapsto \varphi _\theta (u)$ and $(\theta ,u) \in {\mathbb {T}}\times \mathcal{K}_\rho \mapsto {\tilde{\varphi }}_\theta (u)$, analytic on $\mathcal{K}_\rho$ and satisfying

$$\begin{aligned} \Vert \varphi -\mathrm{Id}\Vert _\rho \le \frac{\delta }{2} \quad \text{ and } \quad \Vert {\tilde{\varphi }}-\mathrm{Id}\Vert _\rho \le \frac{\delta }{2}. \end{aligned}$$

Then, the following estimates hold for all $\varepsilon \ge 0$,

$$\begin{aligned} \Vert \Lambda (\varphi ) - \Lambda ({\tilde{\varphi }})\Vert _{\rho -\delta }\le & {} \frac{16 \, M}{\delta } \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho } \quad \text{ and } \quad \Vert \Gamma ^\varepsilon (\varphi ) - \Gamma ^\varepsilon ({\tilde{\varphi }})\Vert _{\rho -\delta } \\\le & {} \frac{16 \, M \varepsilon }{\delta } \Vert \varphi - {\tilde{\varphi }}\Vert _\rho . \end{aligned}$$

Proof of Proposition 3.2

We assume that all $\Phi ^{[k]}$ are defined by (2.4) in the stroboscopic case (all functions $G^{[k]}$ are taken null). The following properties were already proved in [4]:

• (i) the maps $\Phi ^{[k]}$ and $F^{[k]} := \langle \partial _u \Phi ^{[k]} \rangle ^{-1} \langle f \circ \Phi ^{[k]} \rangle$ are well defined and analytic on $K_{R_k}$ for all $k=0,\ldots ,n+1$;

• (ii) the estimates,

  $$\begin{aligned} \left\| \Phi ^{[k]}-\mathrm{id}\right\| _{R_{k}}&\le \frac{r_n}{2}, \quad \left\| F^{[k]}\right\| _{R_{k+1}} \le 2M \; \text{ and } \; \left\| \Phi ^{[k+1]}-\Phi ^{[k]}\right\| _R \\&\le C \left( 2 (n+1) \frac{\varepsilon }{\varepsilon _0} \right) ^k, \end{aligned}$$

  hold for some positive constant C independent of n, k and $\varepsilon$.

From Eq. (2.4) with $G^{[k+1]} \equiv 0$ and Assumption 3.1, it is obvious that $\Phi ^{[k]}$ is $(p+1)$ times continuously differentiable w.r.t. $\theta$. We have furthermore, for $0 \le k \le n$

$$\begin{aligned} \partial _\theta \Phi ^{[k+1]}_\theta = \varepsilon \left( f_\theta \circ \Phi ^{[k]}_\theta - \partial _u \Phi ^{[k]}_\theta F^{[k]}\right) \end{aligned}$$

so that, by differentiating $\nu \le p$ times w.r.t. $\theta$ with the help of the Leibniz and the Faà di Bruno formulae, we obtain

$$\begin{aligned} \frac{1}{\varepsilon } \partial ^{\nu +1}_\theta \Phi ^{[k+1]}_\theta&= \partial _\theta ^\nu f_\theta \circ \Phi ^{[k]}_\theta + \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q \end{array} \right) \sum _\mathbf{i} C^{q}_\mathbf{i} (\partial ^{\nu -q}_\theta \partial ^{|\mathbf{i}|}_u f_\theta ) \circ \Phi ^{[k]}_\theta \\&\quad \left( \left( \partial _\theta \Phi ^{[k]}_\theta \right) ^{i_1}, \ldots ,\left( \partial ^q_\theta \Phi ^{[k]}_\theta \right) ^{i_q} \right) \\&\quad -\partial _\theta ^\nu \partial _u \Phi ^{[k]}_\theta F^{[k]} \end{aligned}$$

where the inner sum runs over multi-indices $\mathbf{i}=(i_1,\ldots ,i_q) \in {\mathbb {N}}^q$ such that $i_1+2i_2+\ldots +qi_q=q$ and where $|\mathbf{i}|=i_1+\cdots +i_q$ and

$$\begin{aligned} C^{q}_\mathbf{i}=\frac{q!}{i_1! i_2! 2!^{i_2}\ldots i_q!q!^{i_q}}. \end{aligned}$$

Owing to (ii), for $u \in \mathcal{K}_{R_{k+1}}$, $\Phi ^{[k]}_\theta (u)$ takes its values in $\mathcal{K}_{R_{k+1}+r_n/2}$, a set on which $\partial _\theta ^j \partial _u^i f_\theta$ is bounded (due to Assumption 3.1 and Cauchy estimates for analytic functions) as follows:

$$\begin{aligned} \left\| \partial _\theta ^j \partial _u^i f \right\| _{R_{k+1}+r_n/2} \le \frac{1}{(2R-(R_{k+1}+r_n/2))^i} M = \frac{M}{(k+\frac{1}{2})^i r_n^i}. \end{aligned}$$

Hence, if we denote $\alpha _{k,\nu }=\Vert \partial ^\nu _\theta \Phi ^{[k]}\Vert _{R_k}/(\varepsilon M)$, for $\nu \ge 1$, then we have

$$\begin{aligned} \forall k \ge 1, \quad \forall \nu \ge 1, \quad \alpha _{k+1,\nu +1} \le 1 + \frac{2 \varepsilon M \alpha _{k,\nu }}{r_n} + \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q \end{array} \right) \sum _\mathbf{i} C^{q}_\mathbf{i} \frac{(\varepsilon M)^{|\mathbf {i}|} \alpha _k^\mathbf{{i}}}{r_n^{|\mathbf i|} (k+\frac{1}{2})^{|\mathbf i|} } \end{aligned}$$

where we denote $\alpha _k^\mathbf{{i}} =\prod _{j=1}^q \alpha _{k,j}^{i_j}$ and use that (owing to (ii) and a Cauchy estimate)

$$\begin{aligned} \left\| \partial _u \partial _\theta ^\nu \Phi ^{[k]} \, F^{[k]} \right\| _{R_{k+1}} \le \frac{1}{r_n} \left\| \partial _\theta ^\nu \Phi ^{[k]} \right\| _{R_{k}} \left\| F^{[k]} \right\| _{R_{k+1}} \le \frac{2 M \left\| \partial _\theta ^\nu \Phi ^{[k]} \right\| _{R_{k}}}{r_n}. \end{aligned}$$

Finally, the assumption that $(n+1) \varepsilon \le \varepsilon _0 = \frac{R}{8M}$ leads to the inequality

$$\begin{aligned} \alpha _{k+1,\nu +1} \le 1 + \frac{\alpha _{k,\nu }}{4}+ \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q \end{array} \right) \sum _\mathbf{i} C^{q}_\mathbf{i} \frac{\alpha _k^\mathbf{{i}}}{(8 k+4)^{|\mathbf i|} } . \end{aligned}$$

(A.1)

We now introduce the generating functions for $\xi \in {\mathbb {C}}$,

$$\begin{aligned} g_k(\xi ) = \sum _{\nu \ge 1} \alpha _{k,\nu } \frac{\xi ^\nu }{\nu !}, \quad k=1,\ldots \end{aligned}$$

A direct estimation gives $\alpha _{1,1} \le 2$ and $\alpha _{1,\nu } \le 1$ for $\nu \ge 2$, so that $0 \le g_1(\xi ) \le \xi +e^\xi -1$ for all $\xi \in {\mathbb {R}}_+$ and from (A.1), we obtain (owing to the Leibniz and the Faà di Bruno formulae)

$$\begin{aligned} \alpha _{k+1,\nu +1} \le \frac{\alpha _{k,\nu }}{4}+ \sum _{q=0}^\nu \left( \begin{array}{c} \nu \\ q \end{array} \right) \left. \frac{d^q}{d\xi ^q} \exp \left( \frac{g_k(\xi )}{8 k+4}\right) \right| _{\xi =0} = \frac{\alpha _{k,\nu }}{4} + G^{(\nu )}_k(0) \end{aligned}$$

where

$$\begin{aligned} G_k(\xi ) = \exp \left( \xi +\frac{g_k(\xi )}{8 k+4}\right) \end{aligned}$$

so that, using $\alpha _{k+1,1} \le 4$, we have for $\xi \in {\mathbb {R}}_+$

$$\begin{aligned} g_{k+1}(\xi ) \le 3 \xi + \frac{1}{4} \int _0^\xi g_k(\mathrm{s}) {\text {d}}s + \int _0^\xi G_k(s) {\text {d}}s. \end{aligned}$$

Noticing that $\sup _{0 \le \xi \le 1} g_k(\xi ) = g_k(1)$, the previous inequality leads to

$$\begin{aligned} g_{k+1}(1) \le 3 + \frac{1}{4} g_k(1) + (e-1) \exp \left( \frac{g_k(1)}{8 k +4}\right) \end{aligned}$$

and an easy induction shows that the sequence $(g_k(1))_{k \ge 1}$ is upper bounded by 8. We conclude the proof by applying Cauchy estimates as follows:

$$\begin{aligned} \forall 1\le & {} k \le n+1, \, \forall 1 \le \nu \le p+1, \quad \alpha _{k,\nu } = \left| \frac{d^\nu g_k}{d\xi ^\nu } (0) \right| \le \frac{\nu ! \, \sup _{|\xi | \le 1} |g_k(\xi )|}{1^\nu } \\\le & {} \nu ! \, g_k(1) \le 8 \, \nu ! \end{aligned}$$

$\square$

Proof of Lemma 3.4

Under assumptions (3.9), it has been shown in [4] that:

• (a) $\Vert \partial _u \varphi \Vert _{\rho -\delta } \le \frac{3}{2}$, the mapping $\langle \partial _u \varphi \rangle ^{-1}$ is analytic on $\mathcal{K}_{\rho -\delta }$ and obeys $\Vert \langle \partial _u \varphi \rangle ^{-1} \Vert _{\rho -\delta } \le 2$;

• (b) the mappings $\Lambda (\varphi )_{\theta }(u)$ and $\Gamma ^\varepsilon (\varphi )_{\theta }(u)$ are analytic on $\mathcal{K}_{\rho -\delta }$ for all $\theta \in {\mathbb {T}}$, and

  $$\begin{aligned} \forall 0< \varepsilon < \varepsilon _0, \quad \Vert \Lambda (\varphi )\Vert _{\rho -\delta } \le 4 \, M \quad \text{ and } \quad \Vert \Gamma ^\varepsilon (\varphi ) - \mathrm{Id}\Vert _{\rho -\delta } \le 4 \, M \, \varepsilon . \end{aligned}$$

Now, on the one hand, for a multi-index $\mathbf{i}=(i_1,\ldots ,i_q) \in {\mathbb {N}}^q$, the algebraic identity

$$\begin{aligned} \prod _{j=1}^q X_j^{i_j} - \prod _{j=1}^q Y_j^{i_j} = \sum _{j=1}^q (X_j-Y_j) \prod _{l=1}^{j-1} Y_l^{i_l} \prod _{l=j+1}^{q} X_l^{i_l} \sum _{r=1}^{i_j} X_j^{i_j-r} Y_j^{r-1} \end{aligned}$$

leads, for any $(\Delta _1, \ldots , \Delta _q) \in X_{{\mathbb {C}}}^q$ and any $({\tilde{\Delta }}_1, \ldots , {\tilde{\Delta }}_q) \in X_{{\mathbb {C}}}^q$, to

$$\begin{aligned}&\left\| \partial _u^{|\mathbf {i}|} f \circ {\tilde{\varphi }} \; (\Delta _1^{i_1}, \ldots ,\Delta _q^{i_q}) - \partial _u^{|\mathbf {i}|} f \circ {\tilde{\varphi }} \; ({\tilde{\Delta }}_1^{i_1}, \ldots ,{\tilde{\Delta }}_q^{i_q}) \right\| _{\rho -\delta } \nonumber \\&\le \Vert \partial _u^{|\mathbf {i}|} f \Vert _{\rho -\delta /2} \, \sum _{j=1}^{q} \Vert \Delta _j -{\tilde{\Delta }}_j\Vert \, \prod _{l=1}^{j-1} \Vert \Delta _l\Vert ^{i_l} \, \prod _{l=j+1}^{q} \Vert {\tilde{\Delta }}_l\Vert ^{i_l} \, \sum _{r=1}^{i_j} \Vert \Delta _j\Vert ^{i_j-r} \Vert {\tilde{\Delta }}_j\Vert ^{r-1}. \end{aligned}$$

(A.2)

On the other hand, the Faà di Bruno’s formula with $1 \le \nu < p$ gives

$$\begin{aligned}&\partial _\theta ^\nu \left( f_\theta \circ \varphi _\theta - f_\theta \circ {\tilde{\varphi }}_\theta \right) = \partial _\theta ^\nu f_\theta \circ \varphi _\theta -\partial _\theta ^\nu f_\theta \circ {\tilde{\varphi }}_\theta \\&\quad + \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} C_\mathbf{i}^{q} \partial ^{\nu -q}_\theta \partial ^{|\mathbf{i}|}_u f_\theta \circ \varphi _\theta \left( (\partial _\theta \varphi _\theta )^{i_1}, \ldots ,(\partial ^q_\theta \varphi _\theta )^{i_q} \right) \\&\quad - \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} C_\mathbf{i}^{q} \partial ^{\nu -q}_\theta \partial ^{|\mathbf{i}|}_u f_\theta \circ {\tilde{\varphi }}_\theta \left( (\partial _\theta {\tilde{\varphi }}_\theta )^{i_1}, \ldots ,(\partial ^q_\theta {\tilde{\varphi }}_\theta )^{i_q} \right) \end{aligned}$$

with the notations of Theorem 3.2. Hence, using the decomposition in (A.2) we have

$$\begin{aligned}&\Vert \partial _\theta ^\nu \left( f_\theta \circ \varphi _\theta - f_\theta \circ {\tilde{\varphi }}_\theta \right) \Vert _{\rho -\delta } \le \Vert \partial _u \partial _\theta ^\nu f_\theta \Vert _{\rho -\delta /2} \; \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho -\delta }\\&\quad + \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} C_\mathbf{i}^{q} \; \beta ^\mathbf{i} \; \Vert \partial ^{\nu -q}_\theta \partial ^{|\mathbf{i}|+1}_u f \Vert _{\rho -\delta /2} \, \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho -\delta } \\&\quad + \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} C_\mathbf{i}^{q} \; \beta ^\mathbf{i} \; \Vert \partial ^{\nu -q}_\theta \partial ^{|\mathbf{i}|}_u f \Vert _{\rho -\delta /2} \sum _{j=1}^q \frac{i_j}{\beta _j} \Vert \partial _\theta ^j (\varphi - {\tilde{\varphi }})\Vert _{\rho -\delta } \end{aligned}$$

Upon using Cauchy estimates, we obtain

$$\begin{aligned}&\Vert \partial _\theta ^\nu \left( f \circ \varphi - f \circ {\tilde{\varphi }} \right) \Vert _{\rho -\delta } \le \frac{2M}{\delta } \; \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho -\delta } + \frac{2M}{\delta } \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho -\delta } \;\\&\qquad \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} {\tilde{C}}_\mathbf{i}^{q} \, \left( \frac{16 M \varepsilon }{\delta }\right) ^{|\mathbf{i}|} \\&\qquad + \frac{1}{8 \varepsilon } \sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} {\tilde{C}}_\mathbf{i}^{q} \, \left( \frac{16 M \varepsilon }{\delta }\right) ^{|\mathbf{i}|} \sum _{j=1}^q \frac{i_j}{j!} \, \Vert \partial _\theta ^j (\varphi - {\tilde{\varphi }})\Vert _{\rho -\delta } \\&\quad \le \frac{2M}{\delta } P_\nu \left( \frac{16 M \varepsilon }{\delta }\right) \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho ,\nu } \end{aligned}$$

where we denote ${\tilde{C}}_\mathbf{i}^q = C_\mathbf{i}^q \; \prod _{j=1}^q j!^{i_j}$ and

$$\begin{aligned} P_\nu (\xi ) = 1+\sum _{q=1}^\nu \left( \begin{array}{c} \nu \\ q\end{array} \right) \sum _\mathbf{i} {\tilde{C}}_\mathbf{i}^{q} \, \left( \xi ^{|\mathbf{i}|} + |\mathbf{i}| \, \xi ^{|\mathbf{i}|-1}\right) = \left. \frac{d^\nu }{d x^\nu } \left( \frac{ e^{x+\frac{\xi x}{1-x}} }{1-x} \right) \right| _{x=0}. \end{aligned}$$

Considering now the second part of $\partial _\theta ^\nu (\Lambda (\varphi )_\theta -\Lambda ({\tilde{\varphi }})_\theta )$ and denoting $F=\langle \partial _u \varphi \rangle ^{-1} \langle f \circ \varphi \rangle$ and ${\tilde{F}}=\langle \partial _u {\tilde{\varphi }} \rangle ^{-1} \langle f \circ {\tilde{\varphi }} \rangle$, we have for $\nu \ge 1$

$$\begin{aligned} \left\| \partial _\theta ^\nu (\partial _u \varphi \, F - \partial _u {\tilde{\varphi }} \, {\tilde{F}})\right\| _{\rho -\delta }&\le \left\| \partial _\theta ^\nu \partial _u \varphi \right\| _{\rho -\delta } \left\| F-{\tilde{F}} \right\| _{\rho -\delta } \\&\quad + \left\| \partial _\theta ^\nu \partial _u \varphi - \partial _\theta ^\nu \partial _u {\tilde{\varphi }} \right\| _{\rho -\delta } \left\| {\tilde{F}} \right\| _{\rho -\delta } \\&\le \frac{8 M \varepsilon }{\delta } \left\| F-{\tilde{F}} \right\| _{\rho -\delta } + \frac{2 M}{\delta } \Vert \partial _\theta ^\nu (\varphi - {\tilde{\varphi }})\Vert _{\rho } \end{aligned}$$

Denoting $A_\theta (u)=\partial _u \varphi _{\theta }(u)$ and ${\tilde{A}}_\theta (u)=\partial _u {\tilde{\varphi }}_{\theta }(u)$, we have

$$\begin{aligned} \Vert F-{\tilde{F}}\Vert _{\rho -\delta }&\le \Vert \langle A\rangle ^{-1}\Vert _{\rho -\delta } \; \Vert \langle f \circ \varphi - f \circ {\tilde{\varphi }}\rangle \Vert _{\rho -\delta } \\&+\Vert \langle A\rangle ^{-1}-\langle {\tilde{A}}\rangle ^{-1}\Vert _{\rho -\delta } \; \Vert \langle f \circ {\tilde{\varphi }}\rangle \Vert _{\rho -\delta } \\&\le \frac{8M}{\delta } \; \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho } \end{aligned}$$

where we have used the bound in (a) $\Vert \langle A\rangle ^{-1}\Vert _{\rho -\delta } \le 2$, $\Vert \langle {\tilde{A}}\rangle ^{-1}\Vert _{\rho -\delta } \le 2$ and the inequality

$$\begin{aligned} \Vert \langle A\rangle ^{-1}-\langle {\tilde{A}}\rangle ^{-1}\Vert _{\rho -\delta } \le \Vert \langle A\rangle ^{-1}\Vert _{\rho -\delta } \Vert \langle {\tilde{A}}\rangle ^{-1}\Vert _{\rho -\delta } \Vert A-{\tilde{A}}\Vert _{\rho -\delta } \le \frac{4}{\delta } \Vert \varphi - {\tilde{\varphi }}\Vert _\rho . \end{aligned}$$

Finally, we obtain

$$\begin{aligned} \left\| \partial _\theta ^\nu (\partial _u \varphi \, F - \partial _u {\tilde{\varphi }} \, {\tilde{F}})\right\| _{\rho -\delta }&\le \frac{64 M^2 \varepsilon }{\delta ^2} \; \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho }+ \frac{2 M}{\delta } \Vert \partial _\theta ^\nu \varphi -\partial _\theta ^\nu {\tilde{\varphi }}\Vert _{\rho }. \end{aligned}$$

(A.3)

Collecting all terms, we finally have

$$\begin{aligned} \Vert \partial _\theta ^\nu (\Lambda (\varphi )-\Lambda ({\tilde{\varphi }}))\Vert _{\rho -\delta } \le \frac{2M}{\delta } Q_\nu \left( \frac{16 M \varepsilon }{\delta }\right) \Vert \varphi - {\tilde{\varphi }}\Vert _{\rho ,\nu } \end{aligned}$$

with $Q_\nu (\xi ) = 1+2 \xi + P_\nu (\xi )$. By a Cauchy estimate, it holds that

$$\begin{aligned} \forall \xi \ge 0, \quad P_\nu (\xi ) \le 2^{\nu +1} \sqrt{e} \, e^{\xi } \, \nu ! \quad \text{ and } \quad Q_\nu (0) \le Q_\nu (\xi ) \le 2^{\nu +1} \sqrt{e} \, e^{\xi } \, \nu ! + 2 \xi + 1. \end{aligned}$$

The corresponding bound for $\Gamma ^\varepsilon$ is then obtained straightforwardly. $\square$