a b s t r a c tWe present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourthorder accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge-Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.

Li-insertion-induced phase transformation in nanoscale olivine particles is studied by phase-field simulations in this paper. We show that the anisotropic growth morphology observed in experiments is thermodynamically controlled by the elastic energy arising from the misfit strain between the Li-rich and Li-poor olivine phases and kinetically influenced by the Li surface-reaction kinetics. The one-dimensional Li diffusivity inherent to the olivine structure is found to kinetically stabilize the phase boundary morphology after Li insertion termintates and facilitate ex-situ observation. Our calculations suggest that examination of the phase boundary morphology provides an effective approach to determine the limiting process of the Li intercalation kinetics in olivine nanoparticles.

Articles you may be interested inEffects of ion-ion collisions and inhomogeneity in two-dimensional kinetic ion simulations of stimulated Brillouin backscattering Phys. Plasmas 13, 022705 (2006); An analysis of the effects of ion trapping on ion acoustic waves excited by the stimulated Brillouin scattering of crossing intense laser beams is presented. Ion trapping alters the dispersion of ion acoustic waves by nonlinearly shifting the normal mode frequency and by reducing the ion Landau damping. This in turn can influence the energy transfer between two crossing laser beams in the presence of plasma flows such that stimulated Brillouin scattering ͑SBS͒ occurs. The same ion trapping physics can influence the saturation of SBS in other circumstances. A one-dimensional analytical model is presented along with reasonably successful comparisons of the theory to results from particle simulations and laboratory experiments. An analysis of the vulnerability of the National Ignition Facility Inertial Confinement Fusion point design ͓S. W. Haan et al., Fusion Sci. Technol. 41, 164 ͑2002͔͒ is also presented.

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Summary.In the h-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, the p-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing the h-version, p-version, and combined h-p-version for a one dimensional problem are presented.

Abstract. The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five dimensional (3d2v) gyrokinetic continuum edge-plasma code: (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells.Good agreement is found over a wide range of collisionality, confining potential, and mirror ratio; and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the banana regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL. (5) Our 5D gyrokinetic formulation yields a set of nonlinear electrostatic gyrokinetic equations that are applicable to both neoclassical and turbulence simulations.

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The purpose of this article is to present the approximation theory underlying the p-version of the finite element method. By exploiting the relationship between polynomial approximation and certain weighted Sobolev spaces, which are identified as the domains of positive real powers of the Legendre operator, this one-dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted Sobolev spaces on triangulated domains of R". It is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions. Inverse results are given for the special case of approximation in L_.