Low-noise particle algorithms for extended magnetohydrodynamic closure

Two new low-noise particle closure methods are developed and tested. Closure of a small set of moment equations is accomplished with first or second order moments computed from a delta-f particle in cell (δf PIC) distribution. Conservation laws are developed and in one case apply to the discrete system, showing that squared weights are part of the system energy and therefore bounded for all time. Implicit time differencing and orbit averaging techniques are developed and implemented. Low-order moment constraints are satisfied exactly by a new particle representation. Numerical tests for one dimension, k_{perpendicular}=0, and two dimension, k_parallel=0 show the successful application of both methods to damped waves and of the second order closure method to unstable gravitational modes. The methods described here are a natural and efficient way to close extended magnetohydrodynamic (MHD) equations to obtain a full kinetic description