Lattice Kinetic Schemes in Fusion Plasmas

Lattice Boltzmann Methods (LBM) are an established alternative approach for numerical simulations of a large spectrum of physical processes. They are based on a mesoscopic anal-ysis of the underlying physics through a velocity distribution function f (x,ξ, t) which obeys the Boltzmann Equation (BE). Furthermore, it has been argued that a number of macroscopic processes can be modeled through a mesoscopic evolution equation similar to BE appropriately tuned to recover the desired macroscopic behavior while retaining the multi-scale characteristics of LBM. Such an approach can be utilized to analyze a magnetohydrodynamic (MHD) system via lattice kinetic schemes [1,2]. All macroscopic quantities are given as moments of f and the algorithm is seen to solve consistently the hydrodynamic and magnetic induction dissipative equations in generalized 3D geometry [3]. We examine the potential of such an algorithm for large-scale fusion simulations. Initial conditions may be provided by the Integrated Tokamak Modelling (ITM) mdsplus server in ENEA frascati, Rome, for ITER related scenarios [4]. The case considered is the evolution of continuous shear Alfvén waves in a plasma [5]. Formulation The numerical implementation of the method is based on the discretized Boltzmann equation with a BGK formulation of the collision term (LBGK), which takes the form ∂t fi +ξ i · ∇ fi =− 1 τ ( fi − f (0)i) (1) where fi = f (x,ξ i, t), with x the spatial vector, ξ i the microscopic velocity set chosen, t the time and τ the relaxation time, all in dimensionless quantities. For the isothermal case, the equilibrium distribution function is given by a low Mach number series expansion of the Maxwellian a