A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

The porous medium equation (PME) $$ {{\text{u}}_{\text{t}}} = \Delta ({{\text{u}}^{\text{m}}}),\,{\text{m>1}} $$ is one of the simplest models of nonlinear diffusion equations. It arises naturally in the study of a number of problems describing the evolution of a continuous quantity subject to a nonlinear diffusion mechanism, which we can instance explain as caused by a diffusion coefficient of the form $$ {\text{c(u) = m}}{{\text{u}}^{{{\text{m - 1}}}}} $$ (1.1) if we write the PME as ut = div(c(u) ∇u). Among the applications of the PME have (i) Percolation of gas through porous media, where m ≥ 2 [M], (ii) Radiative heat transfer in ionized plasmas, where m ≃ 6 [ZR], (iii) Thin liquid films spreading under gravity, where m = 4 [Bu], (iv) Crowd-avoiding population spreading, where m>1 [GM].

A Superconducting QUantum Interference Device (SQUID) modulated by a fast oscillating magnetic flux can be used as a parametric amplifier, providing gain with very little added noise. Here, we develop linearized models to describe the parametrically flux-pumped SQUID in terms of an impedance. An unpumped SQUID acts as an inductance, the Josephson inductance, whereas a flux-pumped SQUID develops an additional, parallel element which we have coined the ``pumpistor.'' Parametric gain can be understood as a result of a negative resistance of the pumpistor. In the degenerate case, the gain is sensitive to the relative phase between the pump and signal. In the nondegenerate case, gain is independent of this phase. We develop our models first for degenerate parametric pumping in the three-wave and four-wave cases, where the pump frequency is either twice or equal to the signal frequency, respectively. We then derive expressions for the nondegenerate case where the pump frequency is not a multiple of the signal frequency, where it becomes necessary to consider idler tones which develop. For the nondegenerate three-wave case, we present an intuitive picture for a parametric amplifier containing a flux-pumped SQUID where current at the signal frequency depends upon the load impedance at an idler frequency. This understanding provides insight and readily testable predictions of circuits containing flux-pumped SQUIDs.

New contributions to the solution of transport equations in porous media

We obtain an analytical expression for the probability density of the Gaussian vector with a degenerate covariance matrix. The expression generalizes a similar expression for the nondegenerate case. A definition of the Gaussian vector on the basis of a probability density, which is common for the degenerate and nondegenerate cases, is given. Examples of using the obtained probability density for the synthesis of optimal signal-processing algorithms are presented.

In this paper we consider a geometric viewpoint to analyze the behaviour of the Conjugate Gradient (CG) method, for the solution of a symmetric linear system, when at current step a pivot breakdown possibly occurs (degenerate case). As well known this can occur when the system matrix is indefinite or singular. In the latter case the CG gets stuck, since the steplength along the current search direction cannot be computed. We show here that a simple geometric interpretation can be provided for the degenerate case, as long as some basics on projective geometry in the Euclidean space are considered.

We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.

This paper deals with multichannel time series, and extends the autocorrelation method for fitting the time series from the nondegenerate case to the general case. We shall show that the AR model obtained by solving the normal equations is always stable even in the degenerate case. The multichannel Levinson algorithm for solving the normal equations is extended from the nondegenerate case to the general case. In the nondegenerate case, the first n+1 values of the auto-correlation sequence of the obtained AR model of order n are identical with those of the original data. The next considered is the extrapolation problem of an autocorrelation sequence, that is, given data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{R_{0}, R_{1},..., R_{n}}</tex> , it is to construct a model whose first n+1 values of the autocorrelation sequence are identical with these data. We shall show that, even in the degenerate case, when the given data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{R_{0}, R_{1},..., R_{n}}</tex> are taken from the autocorrelation sequence of a purely nondeterministic time series, they are identical with the first n+1 values of the autocorrelation sequence of the AR model of order n.

Accurate numerical simulation of moisture front in porous material

Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.

A. In this paper, we apply a weak Galerkin method for solving system of non linear time Fractional Reaction-advection-diffusion Equation in Porous Media(FRDEs).Based on a conservation form for nonlinear term and some of the technical derivational.Theorticly, the convergence for discrete time weak Galerkin finite element schemes is proved.Numerically, the accuracy and effectiveness of the weak Galerkin finite element method (WG-FEM) are illustrated by using Numerical example with the lower order Raviart-Thomas element RT k for discrete weak derivative space.

Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63---94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137---A160, 2012; SIAM J. Numer. Anal. 45(5):2098---2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246---1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315---343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.

Tomography vs quantum control for a three-level atom

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.

Asymptotic spreading of a time periodic diffusion equation with degenerate monostable nonlinearity

The compartmentalization of distinct templates in protocells and the exchange of templates between them (migration) are key elements of a modern scenario for prebiotic evolution. Here we use the diffusion approximation of population genetics to study analytically the steady-state properties of such a prebiotic scenario. The coexistence of distinct template types inside a protocell is achieved by a selective pressure at the protocell level (group selection) favoring protocells with a mixed template composition. In the degenerate case, where the templates have the same replication rate, we find that a vanishingly small migration rate suffices to eliminate the segregation effect of random drift and so to promote coexistence. In the nondegenerate case, a small migration rate greatly boosts coexistence as compared with the situation where there is no migration. However, increase of the migration rate beyond a critical value leads to the complete dominance of the more efficient template type (homogeneous regime). In this case, we find a continuous phase transition separating the homogeneous and the coexistence regimes, with the order parameter vanishing linearly with the distance to the transition point.