Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas

Abstract

The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes (Filbet and Rodrigues in SIAM J. Numer. Anal. 54(2):1120–1146, 2016) for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-in-Cell methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.

Appendix: Convergence analysis for oscillatory ODEs

As announced in the introduction, we conclude with abstract considerations on the numerical analysis of oscillatory ODEs. Though insufficient to prove the relevant results, these considerations provide enlightening insights supporting correct educated guesses on the final outcomes.

Let us discuss a system of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} ({\varvec{a}}_\varepsilon +\varepsilon ^{r_t}\,{{\mathbf {G}}}_t^\varepsilon (\cdot ,{\varvec{a}}_\varepsilon ,{\varvec{b}}_\varepsilon ))'(t)=\displaystyle {{\mathbf {F}}}_{\varvec{a}}^\varepsilon (t,{\varvec{a}}_\varepsilon (t))+\varepsilon ^{r_x}\,{{\mathbf {G}}}_x^\varepsilon (t,{\varvec{a}}_\varepsilon (t),{\varvec{b}}_\varepsilon (t))\,, \\ \displaystyle {\varvec{b}}_\varepsilon '(t)=-\frac{1}{\varepsilon ^2}{{\mathbf {J}}}\,{\varvec{b}}_\varepsilon (t)+{{\mathbf {F}}}_{\varvec{b}}^\varepsilon (t, {\varvec{a}}_\varepsilon (t),{\varvec{b}}_\varepsilon (t))\,, \end{array}\right. } \end{aligned}$$

(A.1)

(with ${{\mathbf {F}}}_{\varvec{a}}^\varepsilon$, ${{\mathbf {F}}}_{\varvec{b}}^\varepsilon$, ${{\mathbf {G}}}_t^\varepsilon$, ${{\mathbf {G}}}_x^\epsilon$ uniformly smooth) and try to guess what may be expected on the numerical computation of the slow variable ${\varvec{a}}_\varepsilon$. Expanding the first equation of the system with the second suggests that, with such a goal in mind, a direct convergence analysis of a discretization of (A.1) could be carried out by working with the vector $({\varvec{a}}_\varepsilon ,\varepsilon ^{\min (r_t-2,r_x)}{\varvec{b}}_\varepsilon )$, and, arguing recursively, that its $(m+1)$th derivative is bounded by a multiple of $\max (\varepsilon ^{-(2m-\min (r_t-2,r_x))_+},\varepsilon ^{-(2(m+1)-\min (r_t-2,r_x))})$. As a consequence, a direct convergence analysis of a scheme of order m that would be unconditionally stable would result for the numerical approximation of $({\varvec{a}}_\varepsilon ,\varepsilon ^{\min (r_t-2,r_x)}{\varvec{b}}_\varepsilon )$, thus also of ${\varvec{a}}_\varepsilon$, into a bound on numerical error by a multiple of

$$\begin{aligned} (\Delta t)^m\times \max \left( \frac{1}{\varepsilon ^{(2m-\min (r_t-2,r_x))_+}}, \frac{1}{\varepsilon ^{(2(m+1)-\min (r_t-2,r_x))}}\right) \,. \end{aligned}$$

For concreteness note that when analyzing the computation of the guiding center, $r_t=3$ and $r_x=2$ so that the bound is $\Delta t^m/\varepsilon ^{(2m+1)}$. The bound is somewhat optimal in the prediction of the computational error for $\varepsilon ^{\min (r_t-2,r_x)}{\varvec{b}}_\varepsilon$. With this respect note that even if by a particularly clever method, for instance through stroboscopic averaging, one is able to improve the computation of a fast variable at particularly well-chosen set of discrete times, this extra precision will be lost when recovering by interpolation from these discrete times an approximation of ${\varvec{b}}_\varepsilon$ on the whole continuous time interval.

System (A.1) suggests that the variable ${\varvec{a}}_\varepsilon$ is actually ${{\mathcal {O}}}(\varepsilon ^{\min (r_t,r_x)})$-close as $\varepsilon \rightarrow 0$ to a solution ${\varvec{a}}$ of the uncoupled non-stiff equation

$$\begin{aligned} {\varvec{a}}_\varepsilon '(t)=\displaystyle {{\mathbf {F}}}_{\varvec{a}}^\varepsilon (t,{\varvec{a}}(t))\,. \end{aligned}$$

(A.2)

For a scheme of order m consistent with the foregoing asymptotic and unconditionally stable this suggests a bound of the numerical error in the approximation of ${\varvec{a}}_\varepsilon$ by a multiple of

$$\begin{aligned} \varepsilon ^{\min (r_t,r_x)}+\Delta t^m\,. \end{aligned}$$

Note that to conclude to the latter bound it is sufficient to know that the ${\varvec{a}}$-part of the solution of the discrete scheme for (A.1) converge as $\varepsilon \rightarrow 0$ to a solution of a scheme of order m for (A.2) with rate ${{\mathcal {O}}}(\varepsilon ^{\min (r_t,r_x)}+\varepsilon \,\Delta t^{m\,\frac{\min (r_t,r_x)-1}{\min (r_t,r_x)}})$, leaving room for some depreciation of the continuous rate ${{\mathcal {O}}}(\varepsilon ^{\min (r_t,r_x)})$.

This provides a final bound of the numerical error for the variable ${\varvec{a}}_\varepsilon$ by a multiple of

$$\begin{aligned} \min \left( \varepsilon ^{\min (r_t,r_x)}+\Delta t^m,(\Delta t)^m\times \max \left( \frac{1}{\varepsilon ^{(2m-\min (r_t-2,r_x))_+}},\frac{1}{\varepsilon ^{(2(m+1)-\min (r_t-2,r_x))}}\right) \right) \,. \end{aligned}$$

In the present paper, we have turned the foregoing formal discussion into rigorous convergence analysis for some of the schemes introduced in [14] and a few extensions.