DuFort–Frankel averaging is a tactic to stabilize Richardson’s unstable three-level leapfrog timestepping scheme. By including the next time level in the right-hand-side evaluation, it is implicit, but it can be rearranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection–diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Finally, the rabies problem is revisited and a stable method is found for a restricted range of parameter values, although no prescriptive recipe is known which selects this particular choice.

The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013J. Solid State Electr.17, 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numeric schemes.

In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, ∂2u/∂n2 and ∂4u/∂n4 are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods.

A numerical study is carried out to study the effects of variable fluid properties on the boundary layer flow and heat transfer of a nanofluid at a flat sheet. The effects of Brownian motion, thermophoresis and viscous dissipation due to frictional heating are also considered. The temperature-dependent variable fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function and a linear function of temperature. Using a similarity transformation, the governing non-linear partial differential equations of the model problem are transformed into coupled non-linear ordinary differential equations and these equations are solved numerically by Keller-Box method. Velocity, temperature, and nanoparticles volume fraction profiles are presented and analyzed for several sets of values of the governing parameters; namely, variable fluid viscosity, variable thermal conductivity, Brownaian motion, thermophoresis and plate-velocity parameters with changes in the Prandtl and Schmidt numbers. It is observed that there is an increase in the skin friction in the upstream movement of the plate: But quite the opposite is true in the downstream movement of the plate. Also, the effect of the Schmidt number and the Brownian motion parameter is to reduce the Sherwood number, where as the effect of thermophoresis parameter is to enhance it.

The problem of stagnation point flow with heat transfer of an electrically conducting fluid impinging normally on a permeable axisymmetric surface in the presence of a uniform transverse magnetic field is analysed. The governing nonlinear differential equations and their associated boundary conditions are reduced to dimensionless form using suitable similarity transformations. Comparison with previously published work shows good agreement. Effects of the injection–suction parameter, magnetic parameter and Prandtl number on the flow and thermal fields are presented. The investigations show that the wall shear stress and heat transfer rate from the surface increase with increased applied magnetic field. An increase in the velocity and thermal boundary layer thicknesses is observed with an increase in the wall injection, while the velocity and thermal boundary layers become thinner when increasing the wall suction and applied magnetic field.

Un modèle numérique permettant d’étudier l’influence du comportement élastiquenon-linéaire des conduites dans les écoulements transitoires avec cavitation de vapeur estprésenté. L’application des lois de conservation de la masse et de la quantité demouvement aboutit à un système de deux équations aux dérivées partielles de typehyperbolique qui se résout numériquement par un schéma prédicteur-correcteur auxdifférences finies conservatif. Par analogie avec la loi de Henry pour le dégazage del’air dissous, une quantité de vapeur se dégage dès que la pression devient inférieure àla pression de vapeur du liquide. Pour mettre en évidence l’interaction fluide-structure,une relation liant la pression à la section de la conduite a été introduite. Les résultatsobtenus permettent d’étudier l’influence de la déformation de la paroi de la conduite surl’évolution des phénomènes de coup de bélier et de cavitation de vapeur. La formulationmathématique présentée est validée en comparant ces résultats à ceux expérimentaux tirésde la littérature.

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H1 norm are derived.

This paper analyses the implementation of the generalizedfinite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani,SIAM J. Numer. Anal.41 (2003) 1008–1021]. The computation of coefficients needs tosolve at each point of the grid (and for each control)a linear programming problem.We show here that, for two dimensional problems, this linear programming problem can be solved in O(p max) operations, where p max is the size of the stencil. The method is based on a walk on the Stern-Brocot tree,and on the related filling of the set of positive semidefinite matrices of size two.

About two years ago, Gobbino [21]gave a proof of a De Giorgi's conjectureon the approximation of the Mumford-Shah energy by means offinite-differences based non-local functionals.In this work, we introduce a discretized version of De Giorgi'sapproximation, that may be seen as a generalization ofBlake and Zisserman's “weak membrane” energy(first introduced in the image segmentation framework).A simple adaptation of Gobbino's results allows us tocompute the Γ-limit of this discrete functional asthe discretization step goes to zero; this generalizes a previouswork by the author on the “weak membrane” model [10].We deduce how to design in a systematic way discreteimage segmentation functionals with “less anisotropy” thanBlake and Zisserman's original energy, and we show insome numerical experiments how it improves the method.