Hyperbolic conservation laws model phenomena characterized by waves propagating at finite speeds; examples include the wave equation for acoustic wave propagation, the compressible Euler equations for gas dynamics, Maxwell’s equations for electromagnetic wave propagation. The Vlasov-Landau equation is a kinetic equation that is used to describe plasma dynamics such as in a Tokamak fusion reactor. The Vlasov equation by itself models the time evolution of how the plasma as it moves through space and velocity and the effect of the plasma’s velocity from a force term such as the electromagnetic field. On the other hand, the Landau equation models the time evolution of particles, such as ions and electrons, colliding with each other.
It is possible to obtain a structure-preserving method by modifying an existing spatial discretization. In the case of hyperbolic conservation laws that satisfy an entropy inequality, a standard discontinuous Galerkin method alone may not be entropy stable. However, adding a Laplace artificial viscosity term and discretizing with the Bassi Rebay (BR-1) or the local discontinuous Galerkin
(LDG) method results in an entropy-stable method at the semi discrete level.
In the case of the Landau equation, the solution conserves mass, momentum, energy, and satisfies the Boltzmann entropy inequality. Furthermore, the steady-state solution is a Maxwellian. Particle methods can be used on a regularized Landau equation. If the equation is regularized by regularizing the entropy functional, at the semi-discrete level, the numerical method conserves mass, momentum, energy, satisfies the Boltzmann entropy inequality, and the steady-state solution is a Maxwellian. The choice in temporal discretization can lead to the fully discrete numerical solution to preserve some or all of the structure. If the discrete gradient method is used the numerical solution conserves mass, momentum, energy, and satisfies the Boltzmann entropy inequality. This
compared to the Forward Euler method which only conserves mass and momentum.

BIO: Sam Van Fleet is a Postdoctoral Associate in the Department of Applied Mathematics and Operations Research at Rice University. Previously, he was a Person Fellow and Acting Instructor in the Department of Applied Mathematics at the University of Washington, where he was awarded the Pacific Institute for the Mathematical Sciences (PIMS) Simons Postdoctoral Award in 2023. He received his Ph.D. in Applied Mathematics from Iowa State University in 2022. His research interests lie broadly in numerical analysis and scientific computing, with a focus on the development, implementation, and analysis of numerical methods for partial differential equations and kinetic equations.