Abstract. Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L 2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L 2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.1. Introduction. Accurate numerical simulations of wave propagation through complex media are becoming increasingly important in seismology, especially as modern computational resources make the use of high fidelity subsurface models feasible for seismic imaging and full waveform inversion. A host of different numerical methods are currently in use, the most popular of which are high order finite difference methods [1]. While finite difference methods tend to perform excellently for simple geometries and smoothly varying data, their accuracy is degraded for heterogeneous media with interfaces or sharp gradients [2].In order to address these issues, high order finite element methods for wave propagation have been considered as alternatives to finite difference methods. A drawback of using continuous finite elements for time-domain simulations using explicit timestepping is the inversion of a global mass matrix system at each timestep. Spectral Element Methods (SEM) [3] address this issue by diagonalizing this mass matrix system through the use of mass-lumping, which co-locates interpolation nodes for Lagrange basis functions and Gauss-Legendre-Lobatto quadrature points. Since SEM is limited to unstructured hexahedral meshes, which are less geometrically flexible than tetrahedral meshes, triangular and tetrahedral mass-lumped spectral element methods have been investigated as alternatives [4,5,6]. However, due to a mismatch in the number of natural quadrature nodes and the dimension of polynomial approximation spaces on simplices, these methods necessitate adding additional nodes in the interior of the element to construct sufficiently accurate nodal points suitable for mass-lumping. Additionally, mass-lumpable nodal points on tetrahedra have only been determined for polynomial bases of degree four or less [4].High order discontinuous Galerkin (DG) methods have been considered as an alternative to Spectral Element Methods for seismic wave propagation [7,8,9,10]. Instead of using mass-lumping to arrive at a diagonal mass matrix, DG methods naturally induce a block diagonal mass matrix through the use of arbitrary-order approximation spaces which are disconti...

Abstract. This paper presents SunPy (version 0.5), a community-developed Python package for solar physics. Python, a free, cross-platform, general-purpose, highlevel programming language, has seen widespread adoption among the scientific community, resulting in the availability of a large number of software packages,

Magnetohydrodynamic turbulence is thought to be responsible for producing complex, multiscale magnetic field distributions in solar active regions. Here we explore the multiscale properties of a number of evolving active regions using magnetograms from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO). The multifractal spectrum was obtained by using a modified box-counting method to study the relationship between magnetic-field multifractality and region evolution and activity. The initial emergence of each active region was found to be accompanied by characteristic changes in the multifractal spectrum. Specifically, the range of multifractal structures (D div ) was found to increase during emergence, as was their significance or support (C div ). Following this, a decrease in the range in multifractal structures occurred as the regions evolved to become large-scale, coherent structures. From the small sample considered, evidence was found for a direct relationship between the multifractal properties of the flaring regions and their flaring rate.

Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for curvilinear meshes which reduce storage costs while retaining energy stability. A priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh, which are illustrated through convergence tests with different sequences of meshes. Numerical and computational experiments verify the accuracy and performance of WADG for a model problem on curved domains. arXiv:1608.03836v1 [math.NA] 12 Aug 2016 topologically (vertex, edges, faces, interior); however, the number of nodes exceeds the cardinality of natural approximation spaces on simplices. Additionally, such nodal sets have only been constructed up to degree 4 for tetrahedra.A similar approach is taken for flux reconstruction schemes on simplices, which are closely related to filtered nodal DG methods [23,24] where nodes are taken to be unisolvent quadrature points [25,26]. Unlike nodal sets for mass-lumped simplices, these quadrature points do not contain nodes which lie on the boundary, necessitating an additional interpolation step in the computation of numerical fluxes. However, numerical evidence indicates that co-locating nodes and quadrature points reduces instabilities resulting from the aliasing of spatially varying Jacobians [27], though an analysis of high order convergence and energy stability for curvilinear simplices are open problems.Krivodonova and Berger introduced an inexpensive treatment of curved boundaries for two-dimensional flow problems by modifying the DG formulation on affine triangles [28]. This was extended to wave propagation problems by Zhang in [8], and by Zhang and Tan for elements with non-boundary curved faces in [7]. A theoretical stability and convergence analysis remains to be shown, though numerical results suggest that each of these approaches preserves stability and high order accuracy on curvilinear meshes under the condition that curved triangles are well-approximated by planar triangles. However, sufficiently large differences between curved and planar triangles still result in unstable schemes [8].An alternative treatment addressing increased storage costs of curvilinear DG was addressed by Warburton using the Low-Storage Curvilinear DG (LSC-DG) method [29,30]. Under LSC-DG, the spatial variation of the Jacobian is incorporated into the physical basis functions over each element, resulting in identical mass matrices over each element. Work in [30] also includes a priori estimates for projection errors under the LSC-DG basis, and gives sufficient conditions under which convergence is guaranteed. Furthermore, the DG variational formulation is constructed to be a priori stable for surface quadratures with positive weights, allowing for stable under-integration of high order integrands present for curvilinear elements.In [31], the weight-adjusted DG (WADG)...

Empirical, three-dimensional electron-density maps of the solar corona can be tomographically reconstructed using polarized-brightness images measured from groundand space-based observatories. Current methods for computing these reconstructions require the assumption that the structure of the corona is unchanging with time. We present the first global reconstructions that do away with this static assumption and, as a result, allow for a more accurate empirical determination of the dynamic solar corona. We compare the new dynamic reconstructions of the coronal density during February 2008 to a sequence of static reconstructions. We find that the new dynamic reconstructions are less prone to certain computational artifacts that may plague the static reconstructions. In addition, these benefits come without a significant increase in computational cost.