Abstract
The Vlasov equation is a fundamental model in plasma physics and kinetic theory, describing the evolution of particle distribution functions under self-consistent electromagnetic fields. It plays a critical role in capturing collisionless plasma dynamics, but its numerical simulation is particularly challenging due to many factors, such as its high dimensionality.
In this talk, I will present our recent work on finite element approximations of the Vlasov equation. Since it is advection-dominated, standard finite element methods tend to produce spurious oscillations—a challenge common to many numerical schemes. To address this, we employ a residual-based artificial viscosity method, which introduces diffusion terms constructed from the PDE residual. These terms vanish in smooth regions, preserving accuracy while providing stabilization near discontinuities. Numerical results demonstrate that the method achieves high-order accuracy for smooth problems and remains stable in the presence of strong shocks. For the Vlasov–Maxwell system, we adopt finite element exterior calculus to discretize Maxwell’s equations, ensuring that Gauss’s law—a key physical property—is maintained. Another challenge in solving the Vlasov equation is its high-dimensional nature. To address this, we employ tensor-product constructions, enabling efficient application of these methods in higher dimensions.