19,114 research outputs found
Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids
We present and discuss three discontinuous Galerkin (dG) discretizations for
the anisotropic heat conduction equation on non-aligned cylindrical grids. Our
most favourable scheme relies on a self-adjoint local dG (LDG) discretization
of the elliptic operator. It conserves the energy exactly and converges with
arbitrary order. The pollution by numerical perpendicular heat fluxes degrades
with superconvergence rates. We compare this scheme with aligned schemes that
are based on the flux-coordinate independent approach for the discretization of
parallel derivatives. Here, the dG method provides the necessary interpolation.
The first aligned discretization can be used in an explicit time-integrator.
However, the scheme violates conservation of energy and shows up stagnating
convergence rates for very high resolutions. We overcome this partly by using
the adjoint of the parallel derivative operator to construct a second
self-adjoint aligned scheme. This scheme preserves energy, but reveals
unphysical oscillations in the numerical tests, which result in a decreased
order of convergence. Both aligned schemes exhibit low numerical heat fluxes
into the perpendicular direction. We build our argumentation on various
numerical experiments on all three schemes for a general axisymmetric magnetic
field, which is closed by a comparison to the aligned finite difference (FD)
schemes of References [1,2
Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients
This is the post-print version of the final paper published in Computers & Mathematics with Applications. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2012 Elsevier B.V.This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional diffusion problems with variable coefficients. The methods use either a specially constructed parametrix (Levi function) or the standard fundamental solution for the Laplace equation to reduce the boundary-value problem (BVP) to a boundary–domain integral equation (BDIE) or boundary–domain integro-differential equation (BDIDE). The radial integration method (RIM) is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Furthermore, a subdomain decomposition technique (SDBDIE) is proposed, which leads to a sparse system of linear equations, thus avoiding the need to calculate a large number of domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed approaches
Non equilibrium thermodynamics and cosmological pancakes formation
We investigate the influence of non equilibrium thermodynamics on
cosmological structure formation. In this paper, we consider the collapse of
planar perturbations usually called "Zel'dovich pancakes". We have developed
for that purpose a new two fluids (gas and dark matter) hydrodynamical code,
with three different thermodynamical species: electrons, ions and neutral
particles (T_e\ne T_i \ne T_n). We describe in details the complex structure of
accretion shock waves. We include several relevant processes for a low density,
high temperature, collisional plasma such as non-equilibrium chemical
reactions, cooling, shock heating, thermal energy equipartition between
electrons, ions and neutral particles and electronic conduction. We find two
different regions in the pancake structure: a thermal precursor ahead of the
compression front and an equipartition wave after the compression front where
electrons and ions temperatures differ significantly. This complex structure
may have two interesting consequences: pre-heating of unshocked regions in the
vicinity of massive X-ray clusters and ions and electrons temperatures
differences in the outer regions of X-rays clusters.Comment: 30 pages, including 8 figures, accepted for publication in The
Astrophysical Journa
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