244 research outputs found
Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes
We have developed the formalism necessary to employ the
discontinuous-Galerkin approach in general-relativistic hydrodynamics. The
formalism is firstly presented in a general 4-dimensional setting and then
specialized to the case of spherical symmetry within a 3+1 splitting of
spacetime. As a direct application, we have constructed a one-dimensional code,
EDGES, which has been used to asses the viability of these methods via a series
of tests involving highly relativistic flows in strong gravity. Our results
show that discontinuous Galerkin methods are able not only to handle strong
relativistic shock waves but, at the same time, to attain very high orders of
accuracy and exponential convergence rates in smooth regions of the flow. Given
these promising prospects and their affinity with a pseudospectral solution of
the Einstein equations, discontinuous Galerkin methods could represent a new
paradigm for the accurate numerical modelling in relativistic astrophysics.Comment: 24 pages, 19 figures. Small changes; matches version to appear in PR
SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics
We introduce a new relativistic astrophysics code, SpECTRE, that combines a
discontinuous Galerkin method with a task-based parallelism model. SpECTRE's
goal is to achieve more accurate solutions for challenging relativistic
astrophysics problems such as core-collapse supernovae and binary neutron star
mergers. The robustness of the discontinuous Galerkin method allows for the use
of high-resolution shock capturing methods in regions where (relativistic)
shocks are found, while exploiting high-order accuracy in smooth regions. A
task-based parallelism model allows efficient use of the largest supercomputers
for problems with a heterogeneous workload over disparate spatial and temporal
scales. We argue that the locality and algorithmic structure of discontinuous
Galerkin methods will exhibit good scalability within a task-based parallelism
framework. We demonstrate the code on a wide variety of challenging benchmark
problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the
code's scalability including its strong scaling on the NCSA Blue Waters
supercomputer up to the machine's full capacity of 22,380 nodes using 671,400
threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains
simulation input file
Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acousticwaves
We introduce a space-time discretization for elastic and acoustic waves using a
discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. For the
dG method, the upwind flux is evaluated by explicitly solving a Riemann problem. Then we
show well-posedness and convergence of the discrete system. Based on goal-oriented dualweighted
error estimation an adaptive strategy is introduced. The full space-time linear system
is solved with a parallel multilevel preconditioner. Numerical experiments for acoustic and
elastic waves underline the efficiency of the overall adaptive solution process
An implementation of the relativistic hydrodynamic equations in conservative form using Dogpack
Let be the metric associated with a stationary spacetime. In the splitting
of spacetime, this allows us to cast the relativistic hydrodynamic equations as a balance law of
the form q_{,t} + \nabla\cdot\vect{F}(q) = \vect{S}, which is
a system of hyperbolic partial differential equations. These hyperbolic equations admit shocks
and rarefactions in their weak solutions. Because of this, we employ a Runge-Kutta Discontinuous
Galerkin method in both Minkowski and Schwarzschild spacetimes through the use of the
Discontinuous Galerkin Package. In this thesis, we give a quick
background on topics in general relativity necessary to implement the method, as well as details
on the DG method itself. We present tests of the method in the form of shock tube tests and
smooth flow into a black hole to show its versatility
A high-order shock capturing discontinuous Galerkin-finite-difference hybrid method for GRMHD
We present a discontinuous Galerkin-finite-difference hybrid scheme that
allows high-order shock capturing with the discontinuous Galerkin method for
general relativistic magnetohydrodynamics. The hybrid method is conceptually
quite simple. An unlimited discontinuous Galerkin candidate solution is
computed for the next time step. If the candidate solution is inadmissible, the
time step is retaken using robust finite-difference methods. Because of its a
posteriori nature, the hybrid scheme inherits the best properties of both
methods. It is high-order with exponential convergence in smooth regions, while
robustly handling discontinuities. We give a detailed description of how we
transfer the solution between the discontinuous Galerkin and finite-difference
solvers, and the troubled-cell indicators necessary to robustly handle
slow-moving discontinuities and simulate magnetized neutron stars. We
demonstrate the efficacy of the proposed method using a suite of standard and
very challenging 1d, 2d, and 3d relativistic magnetohydrodynamics test
problems. The hybrid scheme is designed from the ground up to efficiently
simulate astrophysical problems such as the inspiral, coalescence, and merger
of two neutron stars.Comment: Matches published version (sorry for delay reposting). 45 pages, 14
figures. Showed 2 more Riemann problems, added rotating NS tes
Accelerating The Discontinuous Galerkin Cell-Vertex Scheme (Dg-Cvs) Solver On Cpu-Gpu Heterogeneous Systems
Dg-Cvs (Discontinuous Galerkin Cell-Vertex Scheme) is an efficient, accurate and robust numerical solver for general hyperbolic conservation laws. It can solve a broad range of conservation laws such as the shallow water equation and Magnetohydrodynamics equations. Dg-Cvs is a Riemann-Solver-free high order space-time method for arbitrary space conservation laws. It fuses the discontinuous Galerkin (dg) method and the conservation element/solution element (ce/se) method to take advantage of the best features of both methods. Thanks to the ce/se method, the time derivative of the solution is treated as an independent unknown, which is amendable to gpu\u27s parallel execution. In this thesis, we use a cpu-gpu heterogeneous processor to accelerate Dg-Cvs to demonstrate that complex scientific applications can benefit from a heterogeneous computing system. There are challenges that such scientific program poses on the gpu architecture such as thread divergence and low kernel occupancy. We developed optimizations to address these concerns. Our proposed optimizations include thread remapping to minimize thread divergence and register pressure reduction to increase kernel occupancy. Our experiment results show that Dg-Cvs on gpu outperforms cpu by up to 57\% before optimization and 145\% afterwards. We also use Dg-Cvs as a real world application to explore the possibility of using shared virtual memory (svm) for tighter collaboration between cpu and gpu. However, svm did not help improve the performance due to the overhead of address translation and atomic operations. We developed a microbenchmark to better understand the performance impact of svm
Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows
In this work, we consider the discretization of nonlinear hyperbolic systems
in nonconservative form with the high-order discontinuous Galerkin spectral
element method (DGSEM) based on collocation of quadrature and interpolation
points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp.136--155; Carpenter
et al., SIAM J. Sci. Comput., 36 (2014), pp.~B835-B867). We present a general
framework for the design of such schemes that satisfy a semi-discrete entropy
inequality for a given convex entropy function at any approximation order. The
framework is closely related to the one introduced for conservation laws by
Chen and Shu (J. Comput. Phys., 345 (2017), pp.~427--461) and relies on the
modification of the integral over discretization elements where we replace the
physical fluxes by entropy conservative numerical fluxes from Castro et al.
(SIAM J. Numer. Anal., 51 (2013), pp.~1371--1391), while entropy stable
numerical fluxes are used at element interfaces. Time discretization is
performed with strong-stability preserving Runge-Kutta schemes. We use this
framework for the discretization of two systems in one space-dimension: a
system with a nonconservative product associated to a
linearly-degenerate field for which the DGSEM fails to capture the physically
relevant solution, and the isentropic Baer-Nunziato model. For the latter, we
derive conditions on the numerical parameters of the discrete scheme to further
keep positivity of the partial densities and a maximum principle on the void
fractions. Numerical experiments support the conclusions of the present
analysis and highlight stability and robustness of the present schemes
A statistical approach for fracture property realization and macroscopic failure analysis of brittle materials
Lacking the energy dissipative mechanics such as plastic deformation to rebalance localized stresses, similar to their ductile counterparts, brittle material fracture mechanics is associated with catastrophic failure of purely brittle and quasi-brittle materials at immeasurable and measurable deformation scales respectively. This failure, in the form macroscale sharp cracks, is highly dependent on the composition of the material microstructure. Further, the complexity of this relationship and the resulting crack patterns is exacerbated under highly dynamic loading conditions. A robust brittle material model must account for the multiscale inhomogeneity as well as the probabilistic distribution of the constituents which cause material heterogeneity and influence the complex mechanisms of dynamic fracture responses of the material. Continuum-based homogenization is carried out via finite element-based micromechanical analysis of a material neighbor which gives is geometrically described as a sampling windows (i.e., statistical volume elements). These volume elements are well-defined such that they are representative of the material while propagating material randomness from the inherent microscale defects. Homogenization yields spatially defined elastic and fracture related effective properties, utilized to statistically characterize the material in terms of these properties. This spatial characterization is made possible by performing homogenization at prescribed spatial locations which collectively comprise a non-uniform spatial grid which allows the mapping of each effective material properties to an associated spatial location. Through stochastic decomposition of the derived empirical covariance of the sampled effective material property, the Karhunen-Loeve method is used to generate realizations of a continuous and spatially-correlated random field approximation that preserve the statistics of the material from which it is derived. Aspects of modeling both isotropic and anisotropic brittle materials, from a statistical viewpoint, are investigated to determine how each influences the macroscale fracture response of these materials under highly dynamic conditions. The effects of modeling a material both explicitly by representations of discrete multiscale constituents and/or implicitly by continuum representation of material properties is studies to determine how each model influences the resulting material fracture response. For the implicit material representations, both a statistical white noise (i.e., Weibull-based spatially-uncorrelated) and colored noise (i.e., Karhunen-Loeve spatially-correlated model) random fields are employed herein
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