Halvtidsseminarium - Junjie Wen: “Finite Element Approximations for the Vlasov Equations”

Jag har glädjen att bjuda in er till mitt halvtidsseminarium med titeln ”Finite Element Approximations for the Vlasov Equations”. Seminariet kommer att hållas på engelska. / Hälsningar Junjie Wen

Extern granskare: Stefano Markidis (KTH Royal Institute of Technology)

Abstract: The Vlasov equation governs collisionless plasma dynamics and has wide applications across many fields. However, because it is an advection-dominated and high-dimensional equation, solving it numerically can be challenging. In additions, it is also crucial that numerical solutions preserve key physical invariants, such as mass conservation.

In this seminar, I will present two manuscripts on finite element approximations for the Vlasov equations. The first work introduces a high-order continuous finite element discretization for the Vlasov–Poisson system, combined with a novel anisotropic, nonlinear artificial-viscosity (shock-capturing) stabilization. The viscosity is tensor-valued and aligned with the directional convection field, effectively suppressing spurious oscillations without compromising the scheme’s formal accuracy. The method is explicit in time (using Runge–Kutta schemes) and achieves optimal convergence with respect to both polynomial degree and time integration. Numerical benchmarks—including Landau damping, two-stream instability, and bump-on-tail tests in 2D phase space—demonstrate both robustness and high-order accuracy.

The second paper extends these ideas to a structure-preserving finite element framework for the Vlasov–Maxwell system. Here, the Vlasov equation is discretized using tensor-product continuous polynomial spaces in space–velocity, while Maxwell’s equations employ curl- and divergence-conforming finite elements (Nédélec / Raviart–Thomas) on Cartesian grids. A consistent residual-based, nonlinear anisotropic viscosity is combined with compatible polynomial spaces to preserve divergence constraints and key invariants, yielding a conservative, stable, and high-order scheme suitable for long-time electromagnetic kinetic simulations.