# Mathematics Colloquium

#### Implicit, manifold-preserving numerical representations and solvers for multiscale kinetic simulations of plasmas

**Speaker:**
Luis Chacon, Los Alamos National Lab

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, February 11, 2019, 3:45 p.m.

**Synopsis:**

First-principles models for plasma simulation (such as the Vlasov-Maxwell and Vlasov-Fokker-Planck models) are high-dimensional (6D+time), highly nonlinear, and exceedingly multiscale. However, their efficient and accurate solution is key for progress in many areas of applied plasma physics, including magnetic confinement and inertial confinement fusion. While effective solution methods have been developed in both collisional and collisionless regimes for specific applications, we argue by example that manifold-preserving numerical methods are key for asymptotic well-posedness in general. Such methods strictly respect continuum constraints such as positivity and conservation invariants, and must be implicit in time and adaptive to be able to bridge disparate temporal and spatial scales without drifting from physical solutions. This, in turn, demands effective nonlinear solver strategies. Here, we leverage low-dimensional (e.g., fluid moment) models and efficient, scalable multilevel inversion methods to produce very competitive iterative solution methods. In this talk, we will introduce recent breakthroughs in manifold-preserving implicit numerical representations and associated low-dimensional solvers for both collisionless Lagrangian and collisional Eulerian kinetic plasma models, and will demonstrate their effectiveness with various applications of interest.