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NASA NTRS1999

Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method

The efficiency of the quadrature-free form of the dis- continuous Galerkin method in two dimensions, and briefly in three dimensions, is examined. Most of the work for constant-coefficient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the flux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simplified physics and the most efficient element types be used over most of the domain.

Lockard, David P.↗

NASA NTRS2008

Fast Plasma Instrument for MMS: Simulation Results

Magnetospheric Multiscale (MMS) mission will study small-scale reconnection structures and their rapid motions from closely spaced platforms using instruments capable of high angular, energy, and time resolution measurements. The Dual Electron Spectrometer (DES) of the Fast Plasma Instrument (FPI) for MMS meets these demanding requirements by acquiring the electron velocity distribution functions (VDFs) for the full sky with high-resolution angular measurements every 30 ms. This will provide unprecedented access to electron scale dynamics within the reconnection diffusion region. The DES consists of eight half-top-hat energy analyzers. Each analyzer has a 6 deg. x 11.25 deg. Full-sky coverage is achieved by electrostatically stepping the FOV of each of the eight sensors through four discrete deflection look directions. Data compression and burst memory management will provide approximately 30 minutes of high time resolution data during each orbit of the four MMS spacecraft. Each spacecraft will intelligently downlink the data sequences that contain the greatest amount of temporal structure. Here we present the results of a simulation of the DES analyzer measurements, data compression and decompression, as well as ground-based analysis using as a seed re-processed Cluster/PEACE electron measurements. The Cluster/PEACE electron measurements have been reprocessed through virtual DES analyzers with their proper geometrical, energy, and timing scale factors and re-mapped via interpolation to the DES angular and energy phase-space sampling measurements. The results of the simulated DES measurements are analyzed and the full moments of the simulated VDFs are compared with those obtained from the Cluster/PEACE spectrometer using a standard quadrature moment, a newly implemented spectral spherical harmonic method, and a singular value decomposition method. Our preliminary moment calculations show a remarkable agreement within the uncertainties of the measurements, with the results obtained by the Cluster/PEACE electron spectrometers. The data analyzed was selected because it represented a potential reconnection event as currently published.

Figueroa-Vinas, Adolfo↗

NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗

NASA NTRS2011

The 2011 February 15 Coronal Mass Ejection: Reconciling SOHO and STEREO Observations in Quadrature

The Large-Angle and Spectrometric Coronagraph (LASCO) on board SOHO observed a fast halo coronal mass ejection on 2011 February 15. The STEREO spacecraft were in quadrature with SOHO (STEREO-A ahead of Earth by 87 degrees and STEREO-B 94 degrees behind Earth), enabling CME measurement using the three spacecraft. The sky-plane speed measured by SOHO/LASCO is closely related to the expansion speed of the CME, while the radial speed was measured by STEREO-A and STEREO-B. In addition, STEREO-A and STEREO-B images measured the width of the CME, which is unknown from Earth view. From the SOHO and STEREO measurements, we confirm the relationship between the expansion speed (V(sub exp) ) and radial speed (V(sub rad)) derived previously from geometrical considerations (Gopalswamy et al. 2009): = V(sub rad) = 1/2 (1 + cot w) V(sub exp), where w is the half width of the CME. We can also measure the Earthward speed of the CME directly from the STEREO measurements. The travel time to Earth predicted from the Earthward speed using the Empirical Shock Arrival model is approximately 12 hours shorter than the actual travel time obtained from in situ measurements at Ll. The primary reason for this discrepancy seems to be the interaction with the two preceding CMEs that slowed down the CME in question. The CME interaction is also confirmed from the radio enhancement observed by Wind/WAVES and STEREO WAVES experiments.

Gopalswamy, Natchimuthuk↗

NASA NTRS2000

3D Higher Order Modeling in the BEM/FEM Hybrid Formulation

Higher order divergence- and curl-conforming bases have been shown to provide significant benefits, in both convergence rate and accuracy, in the 2D hybrid finite element/boundary element formulation (P. Fink and D. Wilton, National Radio Science Meeting, Boulder, CO, Jan. 2000). A critical issue in achieving the potential for accuracy of the approach is the accurate evaluation of all matrix elements. These involve products of high order polynomials and, in some instances, singular Green's functions. In the 2D formulation, the use of a generalized Gaussian quadrature method was found to greatly facilitate the computation and to improve the accuracy of the boundary integral equation self-terms. In this paper, a 3D, hybrid electric field formulation employing higher order bases and higher order elements is presented. The improvements in convergence rate and accuracy, compared to those resulting from lower order modeling, are established. Techniques developed to facilitate the computation of the boundary integral self-terms are also shown to improve the accuracy of these terms. Finally, simple preconditioning techniques are used in conjunction with iterative solution procedures to solve the resulting linear system efficiently. In order to handle the boundary integral singularities in the 3D formulation, the parent element- either a triangle or rectangle-is subdivided into a set of sub-triangles with a common vertex at the singularity. The contribution to the integral from each of the sub-triangles is computed using the Duffy transformation to remove the singularity. This method is shown to greatly facilitate t'pe self-term computation when the bases are of higher order. In addition, the sub-triangles can be further divided to achieve near arbitrary accuracy in the self-term computation. An efficient method for subdividing the parent element is presented. The accuracy obtained using higher order bases is compared to that obtained using lower order bases when the number of unknowns is approximately equal. Also, convergence rates obtained using higher order bases are compared to those obtained with lower order bases for selected sample

Fink, P. W.↗