A particle method for the homogeneous Landau equation

Abstract

We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7: 100066, 2020] using the random batch strategy. The collisions only take place inside the small but randomly selected batches so that the computational cost is reduced to O(N ) per time step. Meanwhile, our methods can preserve the conservation of mass, momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods.

Distribution dependent SDEs driven by additive continuous noise

“…This class includes drifts not satisfying the linear growth assumption 3.8, but for which uniqueness still holds. Let us mention that the special cases b(x) = −λ|x| γ−1 x for λ, γ > 0 are also the drift terms considered in [52,26,13]; the setting of these works however, also includes a non-linear diffusion term, associated to kernels of the form a(x) = |x| γ−1 (|x| 2 I d − x ⊗ x).…”

Multi-fidelity methods for uncertainty propagation in kinetic equations

“…The Boltzmann equation is used to model a very large number of different phenomena ranging from rarefied gas flows such as those found in hypersonic aerodynamics, gases in vacuum technologies, or fluids inside microelectromechanical devices [15,16], to the description of social and biological phenomena [64,74]. For these reasons, the development of efficient and accurate numerical methods for solving kinetic equations and in particular the Boltzmann equation has experienced great commitment in the past to which contributed many researchers working in different fields [9,13,31,42,43,65,70,72,104]. We refer to [20,22,71,72,88] for recent monographs, collections and surveys.…”

Structure-preserving marker-particle discretizations of Coulomb collisions for particle-in-cell codes

“…No algebraic proof of entropy dissipation exists either. Regarding this matter, we are now to thank Carrillo et al [1] for demonstrating how a deterministic particle-based discretization for the homogenous Landau collision operator, admitting arbitrary marker weights, can be constructed. Although Ref.…”

“…Although Ref. [1] didn't present a scheme that would guarantee algebraic discrete-time energy conservation and entropy dissipation, continuous-time structure-preservation was obtained and the idea of interpreting Coulomb collisions as compressible flow driven by an entropy functional enabled a natural introduction of marker particles with arbitrary weights. This was a major improvement and lays the path forward to structure-preserving discrete-time formalism.…”

“…In case of the standard Landau operator, the scheme preserves also the discretetime kinetic momentum. The improvements, the extensions of the structure-preserving discretizations in [1,2] to discrete time, are made possible by exploiting the underlying metriplectic structure of the collision operators involved and the so-called discretegradient integrators.…”