6,098 research outputs found
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
Finite volume methods for problems involving second order operators with full
diffusion matrix can be used thanks to the definition of a discrete gradient
for piecewise constant functions on unstructured meshes satisfying an
orthogonality condition. This discrete gradient is shown to satisfy a strong
convergence property on the interpolation of regular functions, and a weak one
on functions bounded for a discrete norm. To highlight the importance of
both properties, the convergence of the finite volume scheme on a homogeneous
Dirichlet problem with full diffusion matrix is proven, and an error estimate
is provided. Numerical tests show the actual accuracy of the method
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
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