Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients.     

Compact finite difference method for the fractional diffusion equation

High-order compact finite difference scheme for solving one-dimensional fractional diffusion equation is considered in this paper. After approximating the second-order derivative with respect to space by the compact finite difference, we use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the local truncation error and discuss the stability using the Fourier method, then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.     

Quantum Langevin equation and input-output fields for arbitrary linear response

A retarded quantum Langevin equation is derived for a small subsystem coupled to an arbitrary number of large reservoirs by treating the small back-action on the reservoir within linear response theory. Interpreting the coupling to the reservoirs as input to the small subsyste, and using the advanced quantum Langevin equation to define the corresponding output emitted into the reservoirs, causally connected input and output variables are constructed which are used to set up anS-matrix formalism relating input and output variables in a unitary and causal way. An application to squeezing by subharmonic generation with arbitrary linear response is given.     

A nominally second-order accurate finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids

In this paper, we describe a second-order accurate cell-centered finite volume method for solving anisotropic diffusion on two-dimensional unstructured grids. The resulting numerical scheme, named CCLAD (Cell-Centered LAgrangian Diffusion), is characterized by a local stencil and cell-centered unknowns. It is devoted to the resolution of diffusion equation on distorted grids in the context of Lagrangian hydrodynamics wherein a strong coupling occurs between gas dynamics and diffusion. The space discretization relies on the introduction of two half-edge normal fluxes and two half-edge temperatures per cell interface using the partition of each cell into sub-cells. For each cell, the two half-edge normal fluxes attached to a node are expressed in terms of the half-edge temperatures impinging at this node and the cell-centered temperature. This local flux approximation can be derived through the use of either a sub-cell variational formulation or a finite difference approximation, leading to the two variants CCLADS and CCLADNS. The elimination of the half-edge temperatures is performed locally at each node by solving a small linear system which is obtained by enforcing the continuity condition of the normal heat flux across sub-cell interface impinging at the node. The accuracy and the robustness of the present scheme is assessed by means of various numerical test cases.     

On the impact of triangle shapes for boundary layer problems using high-order finite element discretization

The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. Large angles are also found to be not problematic for the conditioning of the linear systems arising from the discretizations. Further, when choosing preconditioning strategies, coupling strengths among elements rather than element angle sizes should be taken into account. With an appropriate preconditioner, solutions on meshes with and without large angles can be achieved within a comparable time.     

Improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme

New approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact eigendata, i.e., the exact natural frequencies and mode shapes, of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that for the same number of elements, the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model. Alternatively, the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model.     

A diffusion model for the Dirac equation

In previous work the author was able to derive the Schrödinger equation by an analytical approach built around a physical model that featured a special diffusion process in an ensemble of particles. In the present work, this approach is extended to include the derivation of the Dirac equation. To do this, the physical model has to be modified to make provision for intrinsic electric and magnetic dipoles to be associated with each ensemble particle.     

A lattice Boltzmann equation for diffusion

The formulation of lattice gas automata (LGA) for given partial differential equations is not straightforward and still requires some sort of magic. Lattice Boltzmann equation (LBE) models are much more flexible than LGA because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements. Here a LBE is presented for diffusion in an arbitrary number of dimensions. The model is probably the simplest LBE which can be formulated. It is shown that the resulting algorithm with relaxation parameter =1 is identical to an explicit finite-difference (EFD) formulation at its stability limit. Underrelaxation (0<<1) allows stable integration beyond the stability limit of EFD. The time step of the explicit LBE integration is limited by accuracy and not by stability requirements.     

Hirota method for the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential

In this paper, a Hirota method is developed for applying to the nonlinear Schrödinger equation with an arbitrary time-dependent linear potential which denotes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensation. The nonlinear Schrödinger equation is decoupled to two equations carefully. With a reasonable assumption the one- and two-soliton solutions are constructed analytically in the presence of an arbitrary time-dependent linear potential.     

A diffractive optical element for shaping arbitrary beams

A design method is presented for an optical element that shapes an arbitrary collimated beam. The optical element consists of a pair of diffractive optical elements (DOEs). The outgoing beam is also collimated, and can have any desired intensity profile. The phase functions of the DOEs are computed by minimizing an appropriate cost function under an energy conservation constraint.     

Invariant circles for the piecewise linear standard map

We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.     

A linear equation for Wilson loops

The Makeenko–Migdal loop equation is non-linear and first order in the area derivative, but we show that for simple loops in QCD₂ it is possible to reformulate this equation as a linear equation with second order derivatives. This equation is a bound state Schrödinger equation with a three-dimensional Coulomb potential. Thus, loop dynamics leads to a surprising new picture of confinement, where this phenomenon is due to a (bound state) localization in loop space, with the Wilson loops decaying exponentially outside a characteristic radius.     

An improved monotone finite volume scheme for diffusion equation on polygonal meshes

We construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes. The new scheme uses the cell-edge unknowns instead of cell-vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes. Specially, numerical results show that the new scheme is robust, and more accurate than the existing monotone scheme on some kinds of meshes.     

A piecewise linear model for the zones of instability of an area-preserving map

In this note we study the global behavior of the piecewise linear area-preserving transformation x₁ = 1 − y₀ + |x₀|, y₁ = x₀, of the plane. We show that there are infinitely many invariant polygons surrounding an elliptic fixed point. The regions between these invariant polygons serve as models for the “zones of instability” in the corresponding smooth case. For our model we show that some of these annular zones contain only finitely many elliptic islands. The map is hyperbolic on the complement of these islands and hence exhibits stochastic behavior in this region. Unstable periodic points are dense in this region.     

A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging

During the last few years a quite large number of fluorescence molecular imaging applications have been reported in the literature, as one of the most challenging aspects in medical imaging is to “see” a tumor embedded into tissue, which is a turbid medium, by using fluorescent probes for tumor labeling. However, the forward solvers, required for the successful convergence of the inverse problem, are still lacking accuracy and time feasibility. Moreover, initialization of these solvers may be proven even more difficult than solving the inverse problem itself. This paper describes in depth a coupled radiative transfer equation and diffusion approximation model for solving the forward problem in fluorescence imaging. The theoretical confrontation of these solvers comprises the model deployment, its Galerkin finite elements approximation and the domain discretization scheme. Finally, a new optical properties mapping algorithm, based on super-ellipsoid models, is implemented, providing a fully automated simulation target construction within feasible time.     

Surface exchange reactions and diffusion in composite materials: A finite element approach

A two-dimensional square grain model has been applied to model oxygen exchange processes between a gas phase and a ceramic composite consisting of two randomly distributed phases of equal grain size (side length of squares). Both average diffusion profiles for thin films and the time dependence of the total amount of exchanged oxygen (relaxation curves) have been calculated numerically by means of the finite element method. The boundary conditions refer to an instantaneous change of the oxygen partial pressure in the surrounding gas phase, which gives rise to surface exchange reactions as well as to diffusion in the composite. Both local equilibrium at the interface between different phases (host phase and inclusions) and blocking heterophase boundaries have been taken into account. The numerical results are compared with the analytical solution for diffusion in a homogeneous medium introducing effective diffusion and surface exchange coefficients. When the relaxation time for effective medium diffusion is considerably shorter than that for the transport process from the host phase into the inclusions, relaxation curves with two separate time constants are predicted. Based on analytical approximations, relaxation times for various limiting cases are given.     

A vector raytracing equation for hologram lenses of arbitrary shape

A single vector equation is given, including the effect of wavelength change, for tracing rays in the geometrical optics approximation.