Abstract: The Vlasov-Maxwell equations describe the coupled evolution of collisionless plasma particle distribution function (PDF) and the electromagnetic field. The system is exceedingly multiscale in both space and time; it supports a reduced order description like the quasi-neutrality in certain asymptotic limiting scenarios; it is heavily constrained (e.g., conservation properties, positivity, Gauss-law, ) and is high-dimensional (6D+time). As such, it is incredibly challenging to credibly obtain numerical solutions to realistic multiscale problems such as in fusion devices, and the equations have attracted broad interest from both physicists and mathematicians that has led to an extensive literature to tackle the different numerical challenges associated with it.
In this talk, we discuss a novel multiscale framework for solving the kinetic plasma system based on the concept of conditional distribution from probability theory. The underlying kinetic equation is consistently modified to evolve the conditional distribution function (CDF) rather than the PDF by invoking a series of coordinate transformations and a transformation of variables. The CDF does not directly contain information on mass, momentum, and energy density (i.e., the first three velocity moments are local invariants), and associated moment prior equations are evolved for it instead with self-consistent closure provided from the CDF. In these moment equations, the various multiscale challenges pertaining to discrete conservation, asymptotic preservation, involution constraints, and fast time scales are exposed and dealt with effectively. We use the Vlasov-Ampere equation as an example to demonstrate the power of conditional kinetics on a challenging multiscale collisionless ion-acoustic shock wave problem. Using the new formulation, we develop a numerical scheme that, for the first time, simultaneously 1) conserves mass, momentum, and energy, 2) preserves the positivity of the distribution function, 3) enforces the Gauss' law involution constraint, and 4) recover the quasi-neutrality asymptotic.
I will give a two-part talk, and in the second half, I will also discuss our recent work on hybridizing low-rank tensor decomposition schemes like the tensor train decomposition with adaptive mesh refinement techniques for the Boltzmann equation to deal with the curse of dimensionality on complex engineering problems. I will briefly describe how the new conditional distribution formulation could also make the kinetic equation further amenable to low-rank decomposition schemes by leveraging coordinate transformations that reduce the solution's inherent rank.
Speaker’s Bio: Will Taitano earned his Ph.D. in Nuclear Engineering from the University of New Mexico in 2014. Currently, he works as an R&D staff scientist in the Applied Mathematics and Plasma Physics group in the theoretical division at the Los Alamos National Laboratory (LANL). His work includes the development of algorithms for multiscale and multiphysics applications in both inertial and magnetic confinement fusion systems. These techniques include phase-space grid adaptivity techniques, asymptotic preserving discretizations, conservative discretization schemes, and hierarchical iterative solvers such as the high-order low-order method.
These algorithms are implemented into the LANL multiphysics code, iFP, in which he is one of the lead developers and solves the coupled kinetic plasma and radiation equations to study the impacts of kinetic effects in inertial confinement fusion experiments at the Omega and the National Ignition facilities.
Last Updated: May 20, 2024 - 7:24 am