Anisotropic Diffusion Operator Research Articles

Entropy-Controlled Artificial Anisotropic Diffusion for the Numerical Solution of Conservation Laws Based on Algorithms from Image Processing

In image processing are nonlinear anisotropic diffusion filters used to construct suitable filter algorithms for denoising, edge enhancement, and edge detection. We applied a nonlinear anisotropic diffusion operator in the context of the numerical solution of a scalar hyperbolic conservation law. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular we stabilize a central scheme, known for its oscillatory behavior, by the construction of a nonlinear diffusion term. This means constructing a diffusion matrix consisting of eigenvectors parallel and perpendicular to discontinuities and eigenvalues denoting the amount of dissipation depending on the local strength of the gradients. These directions are used to steer the amount of dissipation, which means suppressing diffusion across the shock front and using the perpendicular direction to enable the necessary diffusion to stabilize the underlying second-order scheme. This new approach allows a multidimensional view to the concept of artificial dissipation. We take the concept of entropy production in the vicinity of shock regions as an indicator for the steering of the diffusion matrix. In smooth regions, which obey an entropy equality instead of an entropy inequality, no information about the diffusion direction and strength is needed. So we add as a stabilizing term diffusion parallel to the characteristics, which turns out to be the Lax–Wendroff diffusion rate. In the case of unsteady regions we use the entropy production to blend between this diffusion term and an—still second-order—extra diffusion term with diffusion parallel to the gradient of the entropy production.

Image Processing for Numerical Approximations of Conservation Laws: Nonlinear Anisotropic Artificial Dissipation

We employ a nonlinear anisotropic diffusion operator like the ones used as a means of filtering and edge enhancement in image processing and in numerical methods for conservation laws. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher order dissipative terms. In particular, we stabilize the well-known Lax--Wendroff formula by means of a nonlinear diffusion term.

A spectral approach to ballooning theory. Part 1

This paper and a forthcoming one develop the spectral theory of ballooning transformations relevant to tokamak physics from first principles in a rigorous and yet intuitively clear manner. The power of the ballooning representation to throw light on the spectral characteristics of the plasma problems to which it is applicable is emphasized, and examples are given to illustrate the general notions. The ballooning representation is shown to be essentially a method to separate variables and reduce two-dimensional partial differential equations with periodic coefficients to infinite sets of soluble ordinary differential equations. This paper is concerned with an elementary approach to the techniques in the context of nearly exactly soluble problems involving the anisotropic diffusion operator in toroidal geometry. Two different perturbation methods are discussed. Applications to plasma instability problems and the subtleties involving the continuous spectra of ballooning operators will be taken up in Part 2.

Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems

We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in ℝ d . Our analysis is based on the theory of additive subspace correction methods and applies to finite element and prewavelet schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore, the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.

Tidal averaging and models for anisotropic dispersion in coastal waters

Several methods for the computation of long-term approximate solutions of the advection-diffusion equation in tidal waters are discussed. The time step used is a tidal period and the effect of tidal fluctuation is represented by a space-varying diffusion tensor. Further, a new method is presented that allows for more complicated dispersion patterns than can be obtained by the anisotropic diffusion operator. The method is applied to the Dutch Coastal waters, where horizontal and vertical density gradients cause anisotropic and irregular dispersion. Numerical results are compared with a solution, computed with small timesteps and tidal velocities. DOI: 10.1034/j.1600-0870.1994.t01-1-00006.x

The weighted particle method for convection-diffusion equations. II. The anisotropic case

This paper is devoted to the presentation and the analysis of a new particle method for convection-diffusion equations. The method has been presented in detail in the first part of this paper for an isotropic diffusion operator. This part is concerned with the extension of the method to anisotropic diffusion operators. The consistency and the accuracy of the method require much more complex conditions on the cutoff functions than in the isotropic case. After detailing these conditions, we give several examples of cutoff functions which can be used for practical computations. A detailed error analysis is then performed.

Approximate methods for the fast computation of EPR and ST-EPR spectra. V. Application of the perturbation approach to the problem of anisotropic motion

The modulation perturbation treatment of Galloway and Dalton is applied to the solution of the stochastic Liouville equation for the spin density matrix which incorporates an anisotropic rotational diffusion operator. Pseudosecular and saturation terms of the spin hamiltonian are explicitly considered as is the interaction of the electron spins with the applied Zeeman modulation field. The modulation perturbation treatment results in a factor of four improvement in computational speed relative to inversion of the full supermatrix with little or no loss of computational accuracy. The theoretical simulations of EPR and ST-EPR spectra are in nearly quantitative agreement with experimental spectra taken under high resolution conditions.

Relaxation effects and molecular interactions

The molecular rotational motion is one of the most important mechanisms determining relaxation of macroscopic, physical properties and spectral profiles in liquid crystals. Under these circumstances, the relaxation effects are interpreted in terms of a conditional probability P(Ω o; Ω, t) for the change of orientation from Ω o to Ω. When this is assumed to occur through small angular steps, P(Ω o; Ω, t) is the solution of a generalized diffusion equation in which the anisotropic pseudo-potential V(Ω) responsible for the liquid crystalline ordering is introduced [1]: ▪ where L is the operator generating infinitesimal rotations and D the diffusion tensor. In this way, information on the long-range interactions which give rise to the orientational ordering can be obtained from relaxation experiments. As and example, it is examined the effect of a `diffuse cone' or `tilted' rotation [2,3] in smectic-A mesophases on a variety of spectroscopical techniques, including NMR relaxation, ESR lineshapes, dielectric dispersion and neutron scattering. To describe the tilted rotation, the orientational potential is written as a sum of second and fourth-rank legendre polynomials: ▪ the parameters α and λ being adjusted to give selected order parametersP̄ 4. The validity of the diffusional model is theoretically supported by a general memory-function approach, which also provides additional information on the short-range frictional forces. The method is based on the three-variable expansion of the Mori equations [4, 5], adapted to anisotropic systems, where the generalized spherical harmonics are no longer independent variables. The Fourier transforms J MN(ω) of the correlation functions G MN( t) = δ d M( t, δ D N(O) x ) are found to be: ▪ where X and {λ} are eigenfunctions and eigenvalues of the anisotropic diffusion operator. The expansion parameters K 0, K 1 are related to mean-square angular velocity and (total) torque N, and l/γ to the torque relaxation time. The diffusion equation results are recovered under `strong anisotropic interaction limit' (SAIL) conditions [6], K 1,/ K 0 = N/kT ⪢ 1.