P2A12 P2A13

Octagonal Lattices in Thermal Lattice Boltzmann & MHD Simulations

L. Vahala Old Dominion University, Norfi,lk, VA 23529

P. Pavlo , lnstirure of’Plusmu Physics, Prugue, Czech Rep.

G. V. A. Macnab College oj‘ Willium & Mory, Williumshurg, VA 23187

Lattice Boltzmann methods are an extremely efficient and highly parallelizeable and vectorizeable algorithm for mesoscopic representation of nonlinear macroscopic problems. The major hurdle facing the extensive use of TLBM is its numerical instability when wide parameter regimes are considered. Considerable research is underway to obviate this, but the root of the problem is clear: if one introduces discrete phase space velocity lattices one is forced to consider relaxation distribution functions that must be non-Maxwellian. The number of constraints needed to be enforced on the relaxed distribution function is reduced as one moves to higher isotropy lattice. We are investigating the use of octagonal lattices in 2D - and there generalization to 3D. However, since the octagonal lattice is no longer space filling the spatial grid is necessarily uncoupled from the velocity lattice. This uncoupling requires an extra step to be incorporated into the TLBM algorithm an interpolation procedure that couples the free-streaming with the nodes of the chosen spatial gridl3. Even if one employed lower symmetry space-filling lattices, it would still be necessary to introduce interpolation if non-uniform spatial grids are employed (e.g., for wall-bounded flows..). We are currently looking into employing temperature-dependent velocity lattices, and will present some 2D jet flow simulations for Mach number flows up to 0.5. We are also investigating the use of octagonal lattices in MHD, rather than the customary square or hexagonal lattices. Besides aiming for higher numerical stability, one will test the accuracy to which div B = 0 can be enforced in this non-vector potential representation. A reason for our continued interest in TLBM is its possible role in studying the scrape-of-layer in a tokamak.

Reduced Modeling of MHD Instabilities

M. A. Pinsky, V. Machin Universiry of Nevudu-Reno

Mathematical models of plasma instabilities, such as dense z-pinches, are often described by systems of nonlinear PDEs, which involve numerous uncertainties in equation parameters, boundary and initial conditions. Instabilities and high dimension of these models may amplify uncertainties and result in unpredictable simulations, which mirrors the unpredictability of real systems. This has two important aspects: one is that underling dynamics may exhibit extreme sensitivity to the variation of their parameters, initial and boundary conditions, which has been studied in the context of bifurcation phenomena and deterministic chaos; the second is the combinatorial complexity of evaluating all model combinations that arise from possible variations in assumptions, parameters and initial data which prohibits direct evaluation of model uncertainties. Thus, it is important to understand and quantify the limits of predictability for full system simulation in terms of the uncertainties; inherent structure of the model and its components; the length of the observation interval; and to develop computational approaches minimizing the effect of uncertainties and reducing simulation time while preserving and controlling the accuracy of obtained results. In this paper we outline a computational approach leading to the derivation of a hierarchy of reduced models of initial complex systems. Each of these simplified models intend to provide a certain degree of inner averaging for individual elaborated simulations of the initial system and present more robust and practically significant results than individual computation events. Evaluation of simplified models significantly reduces simulation time and lead to more accurate and profound classification of complex unstable phenomena.

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