An explicit multiscale pseudo orbit-averaging time integration algorithm

Abstract

We present an explicit multiscale algorithm for solving differential equations for problems with high-frequency modes that can be averaged over by separating and scaling the fast and slow dynamics within a single equation. We introduce a phased time integrator for cases where the boundaries of dynamical scales are known: one phase solves the unmodified equation, while the other freezes part of phase-space and slows down the evolution of the fast dynamics. This algorithm is applied to reduced kinetic models of plasmas in magnetic mirrors, which feature a distinct boundary between a region dominated by rapid particle transit and a region characterized by slow collisions. Two representative model problems are presented that decompose the dynamics of the magnetic mirror into a simpler, computationally inexpensive form. The model problems demonstrate a speedup by a factor of order $ω/ ν_c$, where $ω$ is the fast oscillation frequency and $ν_c$ is the slow damping rate. This is a 30,000$\times$ speedup for a case of practical interest.

Figures (14)

• Figure 1: An illustration of the differences between trajectories in phase space for particles on transit trajectories versus orbit trajectories. Orbiting trajectories feature a periodic motion, while transiting trajectories do not and will be lost at a boundary in $z$. Here, $z$ is a spatial coordinate and $v_\parallel$ is a 1D velocity, $\dot z$.
• Figure 2: Analytical solutions to equation \ref{['eq: 1D toy model equation dfdt']} showing the distribution function profile for parameters $a = 0.9$, $S= 1.0$, $D=1.0$ with varying parallel streaming rates $\nu_\parallel$.
• Figure 3: Comparison of steady-state solutions obtained using the POA algorithm (orange dots) versus a direct RK4 method (blue exes) in linear (a) and logarithmic (b) scales. Both simulations evolved to $t=20 L_v^2/D$. The analytic solution (black dashed line) is provided for reference. Parameters $D = 1$, $\nu_\parallel = 10^5$, $a = 0.9$, $L_v = 1.0$, $S = 1$, $\alpha=2D/(\nu_\parallel(\Delta v_{\rm center} )^2) = 2\times 10^{-3}$, $\tau_{\rm FDP} \nu_\parallel= 5$, $\tau_{\rm OAP} D / L_v^2 = 5$, $\Delta v_{\rm orbit} = 1/10$, and $\Delta v_{\rm transit}=1/300$.
• Figure 4: Time evolution of the distribution function using the RK4 (solid blue) and the POA algorithm (dashed orange) inside (a: $v = 0.45$) and outside (b: $v = 0.912$) the orbit/transit separatrix. Parameters are identical to figure \ref{['fig:1D_toy_model_steady_state_comparison']}. The inset shows the first FDP after a long OAP. The inset is shown in a linear x-y scale where the time axis starts at $\tau_{\rm OAP} - 0.6\tau_{\rm FDP}$ and extends to $\tau_{\rm OAP} + 1.6\tau_{\rm FDP}$. The POA RK4 line in the inset is shown with dots on each time step to show the rapidness of evolution during the FDP.
• Figure 5: Schematic representation of the phase space diagram in a magnetic mirror. $f_{o1}$, $f_{o2}$, and $f_{o3}$ are orbiting regions, while $f_{t1}$ and $f_{t2}$ are transiting with an exhaust.