An Eulerian Vlasov-Fokker-Planck Algorithm for Spherical Implosion Simulations of Inertial Confinement Fusion Capsules

TL;DR

This paper introduces a novel Eulerian Vlasov-Fokker-Planck simulation algorithm with adaptive phase-space mesh refinement for spherical ICF capsule implosions, ensuring conservation and stability in complex implosion dynamics.

Contribution

The paper develops a new phase-space adaptive Eulerian VFP algorithm with a nonlinear stabilization strategy for spherical implosion simulations, including a novel pressure-driven boundary condition.

Findings

Successfully simulated Guderley and Van-Dyke implosions with VFP model

Developed a conservation-preserving, mesh-adaptive numerical scheme

Validated the approach with multiple implosion test problems

Abstract

We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov-Fokker-Planck (VFP) simulation of an inertial confinement fusion (ICF) capsule implosion. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma's evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: 1) an elastic moving wall boundary; 2) a time-dependent Maxwellian Dirichlet boundary; and 3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems --the first-ever reported using a Vlasov-Fokker-Planck model.