22,325 research outputs found
Finite-difference schemes for anisotropic diffusion
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10 to the 12 th times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretisation schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.</p
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
The non-locality of Markov chain approximations to two-dimensional diffusions
In this short paper, we consider discrete-time Markov chains on lattices as
approximations to continuous-time diffusion processes. The approximations can
be interpreted as finite difference schemes for the generator of the process.
We derive conditions on the diffusion coefficients which permit transition
probabilities to match locally first and second moments. We derive a novel
formula which expresses how the matching becomes more difficult for larger
(absolute) correlations and strongly anisotropic processes, such that
instantaneous moves to more distant neighbours on the lattice have to be
allowed. Roughly speaking, for non-zero correlations, the distance covered in
one timestep is proportional to the ratio of volatilities in the two
directions. We discuss the implications to Markov decision processes and the
convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in
the Barles-Souganidis framework.Comment: Corrected two errata from previous and journal version: definition of
R in (5) and summations in (7
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
Modeling anisotropic diffusion using a departure from isotropy approach
There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method
VAGO method for the solution of elliptic second-order boundary value problems
Mathematical physics problems are often formulated using differential
oprators of vector analysis - invariant operators of first order, namely,
divergence, gradient and rotor operators. In approximate solution of such
problems it is natural to employ similar operator formulations for grid
problems, too. The VAGO (Vector Analysis Grid Operators) method is based on
such a methodology. In this paper the vector analysis difference operators are
constructed using the Delaunay triangulation and the Voronoi diagrams. Further
the VAGO method is used to solve approximately boundary value problems for the
general elliptic equation of second order. In the convection-diffusion-reaction
equation the diffusion coefficient is a symmetric tensor of second order
Sobolev gradients and image interpolation
We present here a new image inpainting algorithm based on the Sobolev
gradient method in conjunction with the Navier-Stokes model. The original model
of Bertalmio et al is reformulated as a variational principle based on the
minimization of a well chosen functional by a steepest descent method. This
provides an alternative of the direct solving of a high-order partial
differential equation and, consequently, allows to avoid complicated numerical
schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze
our algorithm in an infinite dimensional setting using an evolution equation
and obtain global existence and uniqueness results as well as the existence of
an -limit. Using a finite difference implementation, we demonstrate
using various examples that the Sobolev gradient flow, due to its smoothing and
preconditioning properties, is an effective tool for use in the image
inpainting problem
Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions. I. Pre-Collapse Evolution
The evolution of a spherical single-mass star cluster is followed in detail
up to core collapse by numerically solving the orbit-averaged two-dimensional
Fokker-Planck equation in energy-angular momentum space. Velocity anisotropy is
allowed in the two-dimensional Fokker-Planck model. Using improved numerical
codes, the evolution has been followed until the central density increased by a
factor of with high numerical accuracy. The numerical results clearly
show self-similar evolution of the core during the late stages of the core
collapse. When Plummer's model is chosen as the initial condition, the collapse
time is about 17.6 times the initial half-mass relaxation time. This is longer
than the collapse time for the isotropic model by about 13%. As the result of
strong relaxation in the core, the halo becomes to be dominated by radial
orbits. The degree of anisotropy monotonically increases as the radius
increases.Comment: 19 pages, latex, 7 postscript figures, uuencoded gzipped tar file.
Submitted to PAS
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