This work focuses on the numerical analysis of one-dimensional nonlinear diffusion equations involving a convolution product. First, homogeneous friction equations are considered. Algorithms follow recent ideas on mass transportation methods and lead to simple schemes which can be proved to be stable, to decrease entropy, and to converge toward the unique solution of the continuous problem. In particular, for the first time, homogeneous cooling states are displayed numerically. Further, we present results on the more delicate fourth-order thin-film equation for which a nonnegativity-preserving scheme is derived. The dead core phenomenon is presented for the Hele–Shaw cell.

This work is concerned with the semiclassical approximation of the Schro ̈dinger-Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a N-body quantum problem, then K-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for Paulis exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects

The objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one-dimensional filtration equations. It is based on an alternative formulation of the problem using the pseudoinverse of the density's repartition function. In particular, the numerical approximations can be proven to satisfy a contraction property for a Wasserstein metric. Various numerical results illustrate the ability of this numerical process to capture the time-asymptotic decay towards self-similar solutions even for fast-diffusion equations.

Abstract. One-dimensional electronic conduction is investigated in a special case usually referred to as the harmonic crystal, meaning essentially that atoms are assumed to move like coupled harmonic oscillators within the Born–Oppenheimer approximation. We recall their dispersion relation and derive a WKB system approximately satisfied by any electron's wavefunction inside a given energy band. This is then numerically solved according to the method of K-branch solutions. Numerical results are presented in the case where atoms move with one- or two-modes vibrations; finally, we include the case where the Poisson self-interaction potential also influences the electrons' dynamics.

We consider the Cauchy problem for the 2 x 2 nonstrictly hyperbolic system [...] For possiblylarge, discontinuous and resonant data, the generalized solution to the Riemann problem is introduced, interaction estimates are carried out using an original change of variables and the convergence of Godunov approximations is shown. Uniqueness is addressed relying on a suitable extension of Kruz?kov's techniques

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrodinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.

We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow theWell-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type Dt <= O(Dx^2) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.

We present a computational approach for the WKB approximation of the wavefunction of an electron moving in a periodic one-dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the Bloch spectrum of the Schrodinger operator. This second part focuses on the handling of the source terms which originate from adding a slowly varying exterior potential. Physically, relevant examples are the occurrence of Bloch oscillations in case it is linear, a quadratic one modelling a confining field and the harmonic Coulomb term resulting from the inclusion of a ''donor impurity'' inside an otherwise perfectly homogeneous lattice.

We derive and study Well-Balanced schemes for quasimonotone discrete kinetic models. By means of a rigorous localization procedure, we reformulate the collision terms as nonconservative products and solve the resulting Riemann problem whose solution is self-similar. The construction of an Asymptotic Preserving (AP) Godunov scheme is straightforward and various compactness properties are established within different scalings. At last, some computational results are supplied to show that this approach is realizable and efficient on concrete $2 \times 2$ models.

Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schr\"odinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

This paper is devoted to a numerical simulation of the classical WKB system arising in geometric optics expansions. It contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous. We address the problem of treating it in such a way superimposed signals can be reproduced by means of the kinetic formulation of ``multibranch solutions'' originally due to Brenier and Corrias. Some existence and uniqueness results are given together with computational test-cases of increasing difficulty displaying up to five multivaluations.

We propose here a well-balanced numerical scheme for the one-dimensional Goldstein-Taylor system which is endowed with all the stability properties inherent to the continuous problem and works in both rarefied and diffusive regimes.

This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

We consider the Cauchy problem for $n\times n$ strictly hyperbolic systems of nonresonant balance laws $$ \left\{\begin{array}{c} u_t+f(u)_x=g(x,u), \qquad x \in \reali, t>0\\ u(0,.)=u_o \in \L1 \cap \BV(\reali; \reali^n), \\ | \la_i(u)| \geq c > 0 \mbox{ for all } i\in \{1,\ldots,n\}, \\ |g(.,u)|+\norma{\nabla_u g(.,u)}\leq \om \in \L1\cap\L\infty(\reali), \\ \end{array}\right. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that $\|\om\|_{\L1(\reali)}$ and $\|u_o\|_{\BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.