DIPLODOCUS Framework

DIPLODOCUS

Distribution In PLateaux methODOlogy for the compUtation of transport equationS.

Diplodocus is a novel framework for evolving the a set of particle distribution functions $f(\mathit{x},\mathit{p})$ through the seven dimensions of phase space: one time, three space and three momentum. Evolution includes advection of particles, continuous forcing and discrete interactions between particles.

Transport Equations

Transport equations refers to a set of equations that dictate the evolution of particles through phase space. In Diplodocus, that takes the form of an integrated Boltzmann equation:

$${\int }_{\partial Q}f(\mathit{x},\mathit{p})\mathit{\omega }={\int }_Q\mathit{C}(\mathit{x},\mathit{p}).$$

Terms on the left-hand-side dictate the continuous transport of particle through phase space including advection through space and forcing through momentum. The right-hand-side, described termination and beginning of particle worldlines in a volume of phase space due to discrete interaction between particles.

Distribution-In-Plateaux

To solve the evolution described by the transport equations computationally, a method called "Distribution-In-Plateaux" is used to discretise particle distribution functions. In brief, a continuous distribution over a surface in phase space is divided into a number of plateaux over sub-areas of the surface. The "height" of this plateaux is then taken to be the average value of the continuous distribution over that sub-area.

INSERT IMAGE OF CONTINIOUS SURFACE TO GRID

This discretisation allows the effects of discrete interactions, collisions, to be pre-computed while simultaniously providing a flexable and conservative numbical scheme for the transport of distribution function through phase space.

Pre-computation of collision terms is implemented in the DiplodocusCollisions.jl package.

Transport of the distribution functions is handled by the DiplodocusTransport.jl package.

Plotting of results is split into a separate package DiplodocusPlots.jl.