High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs [1,2]. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices [3,4,5,6]. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.The focus of this work is the construction of high order polynomial DG methods which satisfy a discrete analogue of the conservation of entropy (5) and the dissipation of entropy (7). Discrete differential operators and quadrature-based matrices Mathematical assumptions and notationsWe begin with a d-dimensional reference element D with boundary ∂ D. We denote the ith component of the outward normal vector on the boundary of the reference element ∂ D as n i . For this work, we assume

To determine independent prognostic factors for the survival of patients with endometrial stromal sarcoma (ESS), data were abstracted from the Surveillance, Epidemiology, and End Results (SEER) database of the National Cancer Institute from 1988 to 2003. Kaplan -Meier and Cox proportional hazards models were used for analyses. Of 831 women diagnosed with ESS, the median age was 52 years (range: 17 -96 years). In total, 59.9% had stage I, 5.1% stage II, 14.9% stage III, and 20.1% had stage IV disease. Overall, 13.0, 36.1, and 34.7% presented with grades 1, 2, and 3, respectively. Patients with stage I-II vs III-IV disease had 5 years DSS of 89.3% vs 50.3% (Po0.001) and those with grades 1, 2, and 3 cancers had survivals of 91.4, 95.4, and 42.1% (Po0.001). In multivariate analysis, older patients, black race, advanced stage, higher grade, lack of primary surgery, and nodal metastasis were independent prognostic factors for poorer survival. In younger women (o50 years) with stage I -II disease, ovarian-sparing procedures did not adversely impact survival (91.9 vs 96.2%; P ¼ 0.1). Age, race, primary surgery, stage, and grade are important prognostic factors for ESS. Excellent survival in patients with grade 1 and 2 disease of all stages supports the concept that these tumors are significantly different from grade 3 tumors. Ovarian-sparing surgeries may be considered in younger patients with early-stage disease.

The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points [1,2,3] and their equivalence with summation-by-parts (SBP) finite difference operators [4]. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

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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...

We construct entropy conservative and entropy stable high order accurate discontinuous Galerkin (DG) discretizations for time-dependent nonlinear hyperbolic conservation laws on curvilinear meshes. The resulting schemes preserve a semi-discrete quadrature approximation of a continuous global entropy inequality. The proof requires the satisfaction of a discrete geometric conservation law, which we enforce through an appropriate polynomial approximation. We extend the construction of entropy conservative and entropy stable DG schemes to the case when high order accurate curvilinear mass matrices are approximated using low-storage weight-adjusted approximations, and describe how to retain global conservation properties under such an approximation. The theoretical results are verified through numerical experiments for the compressible Euler equations on triangular and tetrahedral meshes.

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