54 research outputs found
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity
The aim of this paper is to compare a hyperelastic with a hypoelastic model
describing the Eulerian dynamics of solids in the context of non-linear
elastoplastic deformations. Specifically, we consider the well-known
hypoelastic Wilkins model, which is compared against a hyperelastic model based
on the work of Godunov and Romenski. First, we discuss some general conceptual
differences between the two approaches. Second, a detailed study of both models
is proposed, where differences are made evident at the aid of deriving a
hypoelastic-type model corresponding to the hyperelastic model and a particular
equation of state used in this paper. Third, using the same high order ADER
Finite Volume and Discontinuous Galerkin methods on fixed and moving
unstructured meshes for both models, a wide range of numerical benchmark test
problems has been solved. The numerical solutions obtained for the two
different models are directly compared with each other. For small elastic
deformations, the two models produce very similar solutions that are close to
each other. However, if large elastic or elastoplastic deformations occur, the
solutions present larger differences.Comment: 14 figure
Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes
We propose a new high order accurate nodal discontinuous Galerkin (DG) method
for the solution of nonlinear hyperbolic systems of partial differential
equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using
classical polynomials of degree N inside each element, in our new approach the
discrete solution is represented by piecewise continuous polynomials of degree
N within each Voronoi element, using a continuous finite element basis defined
on a subgrid inside each polygon. We call the resulting subgrid basis an
agglomerated finite element (AFE) basis for the DG method on general polygons,
since it is obtained by the agglomeration of the finite element basis functions
associated with the subgrid triangles. The basis functions on each sub-triangle
are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles
once and for all in a pre-processing stage for the reference element only.
Consequently, the construction of an efficient quadrature-free algorithm is
possible, despite the unstructured nature of the computational grid. High order
of accuracy in time is achieved thanks to the ADER approach, making use of an
element-local space-time Galerkin finite element predictor.
The novel schemes are carefully validated against a set of typical benchmark
problems for the compressible Euler and Navier-Stokes equations. The numerical
results have been checked with reference solutions available in literature and
also systematically compared, in terms of computational efficiency and
accuracy, with those obtained by the corresponding modal DG version of the
scheme
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
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