Structure-preserving numerical methods for collision operators in plasma physics

Abstract

In this thesis, we present two new families of structure-preserving particle methods for collision operators in plasma physics. Many advances have been made in recent years for modelling ideal kinetics, such as the Vlasov--Poisson or Vlasov--Maxwell systems, in a structure preserving manner. The problem of treating dissipation, or collisions specifically, in this way has been more recently studied, since collision operators do not fit straightforwardly into a geometric framework in the same manner as the ideal dynamics. Though there has now been some work on this topic, many of the existing methods are grid-based approaches and do not lend themselves to being coupled to particle based methods which solve the ideal problem. Thus, it is of interest to consider structure-preserving particle methods for such operators. The two operators considered in this thesis are the Lenard-Bernstein and Landau collision operators.

A deterministic particle method approach is taken to discretise a conservative version of the Lenard-Bernstein operator. An L2 projection is used to construct a second representation of the distribution function, which has sufficient regularity for computation of the collision operator. The obtained semi-discretisation is shown to preserve mass, momentum and energy automatically. The implicit midpoint method is shown to preserve these properties in time as well. The method is implemented in the one-dimensional case with a cubic spline basis, and various examples are constructed to demonstrate the method's behaviour and conservation properties. Preservation of the invariants is demonstrated to machine precision, and entropy is shown to be monotonically dissipated in time as well.

The Landau operator is discretised using the metriplectic approach of Morrison (1986). Two separate discretisations are constructed, and both make use of the projection approach introduced in the Lenard-Bernstein case. The first method is obtained by directly discretising the metric bracket for the Landau operator by the particle distribution function, with the projected representation being used for evaluation of the entropy only. The method is shown to preserve mass, momentum, and energy, and the method of discrete gradients is shown to preserve these quantities in time as well. However, the method is shown to have a computational cost of O(N^2), where N is the number of particles. A second discretisation is constructed also via the metriplectic approach, but where the metric bracket is discretised via a two-step approach. First, the bracket is restricted to finite-element type functions. Then, the bracket is further restricted to those finite-element functions that are obtained by the projection of a particle solution. This leads to a method with a much improved cost of O(NM^2), where M is the number of basis functions for projection. There are, however, some unresolved challenges in implementing these methods, and the work is concluded with a discussion of these.