An adaptive, implicit, conservative, 1D-2V multi-species Vlasov–Fokker–Planck multi-scale solver in planar geometry

Abstract

We consider a 1D-2V Vlasov-Fokker-Planck multi-species ionic description coupled to fluid electrons. We address temporal stiffness with implicit time stepping, suitably preconditioned. To address temperature disparity in time and space, we extend the conservative adaptive velocity-space discretization scheme proposed in [Taitano et al., J. Comput. Phys., 318, 391-420, (2016)] to a spatially inhomogeneous system. In this approach, we normalize the velocity-space coordinate to a temporally and spatially varying local characteristic speed per species. We explicitly consider the resulting inertial terms in the Vlasov equation, and derive a discrete formulation that conserves mass, momentum, and energy up to a prescribed nonlinear tolerance upon convergence. Our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the Vlasov operator and the Fokker-Planck collision operator. Numerical examples of varying degrees of complexity, including shock-wave propagation, demonstrate the favorable efficiency and accuracy properties of the scheme.

An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system

“…As demonstrated in Ref. [6], failing to enforce conservation self-consistently for only the Vlasov portion of the system can lead to extremely large errors in the overall solution.…”

“…To study these systems, radiation-hydrodynamic models are typically used; however, to resolve the long mean-free-path effects it is necessary to employ a kinetic approach. Vlasov-Fokker-Planck codes, such as iFP [6] and FPion [7], have been developed with the goal of resolving ion kinetic effects in weakly collisional regimes with arbitrary Knudsen numbers. However, they continue to treat the electrons as a quasineutral, ambipolar fluid, including only an electron temperature equation.…”

“…Indeed, as we shall demonstrate later, the failure to ensure discrete charge conservation (i.e., enforcing the discrete Gauss's law) leads to catastrophic failure in simulations, with significant departure from the correct solution. Further, in the case of the Vlasov-Fokker-Planck equation, Taitano et al [6] showed that even neglecting to ensure discrete momentum and energy conservation relationships only in the Vlasov equation (while enforcing it in the Fokker-Planck collision operator) leads to extremely large numerical errors (∼ O (1)). Thus, discrete conservation is a key for achieving high fidelity and accuracy.…”

Ion species stratification within strong shocks in two-ion plasmas

“…iFP is a state-of-the-art code, which is mass, energy, and momentum conserving; it is also adaptive and well verified. [19][20][21][22][23] iFP treats ions fully kinetically, resolving all ion species within their own separate velocity-spaces, while simultaneously solving the quasi-neutral fluid equations for electrons. 24 Additionally, our multi-ion hydro code and theory are grounded in a multi-species generalization of the 1.…”

Fully kinetic simulations of strong steady-state collisional planar plasma shocks

“…Problem setup: To study this problem, we employ the Eulerian VFP code iFP [29][30][31][32], and the Lagrangian code VPIC [33][34][35]. The iFP code solves the coupled VFP equations for each of the plasma species in a 1D planar electrostatic approximation and the electric field is given by the 1D Ampère equation:…”