112 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
A Physical-Constraint-Preserving Finite Volume WENO Method for Special Relativistic Hydrodynamics on Unstructured Meshes
This paper presents a highly robust third-order accurate finite volume
weighted essentially non-oscillatory (WENO) method for special relativistic
hydrodynamics on unstructured triangular meshes. We rigorously prove that the
proposed method is physical-constraint-preserving (PCP), namely, always
preserves the positivity of the pressure and the rest-mass density as well as
the subluminal constraint on the fluid velocity. The method is built on a
highly efficient compact WENO reconstruction on unstructured meshes, a simple
PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux, and
third-order strong-stability-preserving time discretization. Due to the
relativistic effects, the primitive variables (namely, the rest-mass density,
velocity, and pressure) are highly nonlinear implicit functions in terms of the
conservative variables, making the design and analysis of our method
nontrivial. To address the difficulties arising from the strong nonlinearity,
we adopt a novel quasilinear technique for the theoretical proof of the PCP
property. Three provable convergence-guaranteed iterative algorithms are also
introduced for the robust recovery of primitive quantities from admissible
conservative variables. We also propose a slight modification to an existing
WENO reconstruction to ensure the scaling invariance of the nonlinear weights
and thus to accommodate the homogeneity of the evolution operator, leading to
the advantages of the modified WENO reconstruction in resolving multi-scale
wave structures. Extensive numerical examples are presented to demonstrate the
robustness, expected accuracy, and high resolution of the proposed method.Comment: 56 pages, 18 figure
Simulating magnetized neutron stars with discontinuous Galerkin methods
Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod/ limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized TOV stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method
Is the Classic Convex Decomposition Optimal for Bound-Preserving Schemes in Multiple Dimensions?
Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120,
2010], the Zhang--Shu framework has attracted extensive attention and motivated
many bound-preserving (BP) high-order discontinuous Galerkin and finite volume
schemes for various hyperbolic equations. A key ingredient in the framework is
the decomposition of the cell averages of the numerical solution into a convex
combination of the solution values at certain quadrature points, which helps to
rewrite high-order schemes as convex combinations of formally first-order
schemes. The classic convex decomposition originally proposed by Zhang and Shu
has been widely used over the past decade. It was verified, only for the 1D
quadratic and cubic polynomial spaces, that the classic decomposition is
optimal in the sense of achieving the mildest BP CFL condition. Yet, it
remained unclear whether the classic decomposition is optimal in multiple
dimensions. In this paper, we find that the classic multidimensional
decomposition based on the tensor product of Gauss--Lobatto and Gauss
quadratures is generally not optimal, and we discover a novel alternative
decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and
3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP
time step-size than the classic one, and moreover, it is rigorously proved to
be optimal to attain the mildest BP CFL condition, yet requires much fewer
nodes. The discovery of such an optimal convex decomposition is highly
nontrivial yet meaningful, as it may lead to an improvement of high-order BP
schemes for a large class of hyperbolic or convection-dominated equations, at
the cost of only a slight and local modification to the implementation code.
Several numerical examples are provided to further validate the advantages of
using our optimal decomposition over the classic one in terms of efficiency
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