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

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.     

Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

This paper studies the asymptotic behavior of coexistence steady-states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni [18] derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on [10] implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This paper gives a rigorous proof of his observation as far as the stationary problem. As a key ingredient of the proof, we establish a uniform $L^{\infty }$ estimate for all steady-states. Thanks to this a priori estimate, we show that the asymptotic profile of coexistence steady-states can be characterized by a solution of the limiting system.     

Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler’s equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.     

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.     

Porous convection and thermal oscillations

A theory is developed for thermal convection in a fluid saturated porous material when the temperature may propagate as a thermal wave. In particular, we are interested in the mechanism of thermal oscillation and so allow for Guyer?CKrumhansl effects but employ a heat flux equation developed by Christov and Morro. The instability mechanism is investigated in complete detail and it is shown that stationary convection is likely to prevail under normal terrestrial conditions, but if the thermal relaxation time is sufficiently large there is a possible parameter range which allows for oscillatory convection. However, the presence of the Guyer?CKrumhansl terms has the effect of damping the oscillatory convection and returning the instability mechanism to one of stationary convection.     

Convergence properties of dynamic string-averaging projection methods in the presence of perturbations

Assuming that the absence of perturbations guarantees weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string-averaging projection methods, which we establish here as well. Moreover, we show how this result can be applied to the superiorization methodology.     

Stability properties of steady-states for a network of ferromagnetic nanowires

We investigate the problem of describing the possible stationary configurations of the magnetic moment in a network of ferromagnetic nanowires with length L connected by semiconductor devices, or equivalently, of its possible L-periodic stationary configurations in an infinite nanowire. The dynamical model that we use is based on the one-dimensional Landau–Lifshitz equation of micromagnetism. We compute all L-periodic steady-states of that system, define an associated energy functional, and these steady-states share a quantification property in the sense that their energy can only take some precise discrete values. Then, based on a precise spectral study of the linearized system, we investigate the stability properties of the steady-states.     

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time‐dependent singularly perturbed parabolic convection–diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two‐point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme.     

A nonlocal dispersal equation arising from a selection–migration model in genetics

This paper is concerned with the existence, uniqueness and asymptotic stability of positive steady-states for a nonlocal dispersal equation arising from selection–migration models in genetics. Due to the lack of compactness and regularity of the nonlocal operators, many classical methods cannot be used directly to the nonlocal dispersal problems. This motivates us to find new techniques. We first establish a criterion on the stability and instability of steady-states. This result is effective to get a necessary condition to guarantee a positive steady-state, it also gives the uniqueness. Then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle. Finally, we consider the dynamic behavior of the initial value problem.     

Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

In this article, we prove the convergence of a discrete duality finite volume scheme for a system of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection‐diffusion‐dispersion equation on the concentration. We first establish some a priori estimates satisfied by the sequences of approximate solutions. Then, it yields the compactness of these sequences. Passing to the limit in the numerical scheme, we finally obtain that the limit of the sequence of approximate solutions is a weak solution to the problem under study. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 723–760, 2015     

Stability and convergence properties of Bergman kernel methods for numerical conformal mapping

Summary In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conformal mapping of simply and doubly-connected domains. In particular, by using certain wellknown results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map.     

Convergence and error bound analysis for the space‐time CESE method

In this work, we study the convergence behavior of a recently developed space‐time conservation element and solution element method for solving conservation laws. In particular, we apply the method to a one‐dimensional time‐dependent convection‐diffusion equation possibly with high Peclet number. We prove that the scheme converges and we obtain an error bound. This method performs well even for strong convection dominance over diffusion with good long‐time accuracy. Numerical simulations are performed to verify the results. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 64–78, 2001     

Compact mixed methods for convection/diffusion type problems

This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we search a fourth-order compact approximation of the primary solution field and flux. Based on Taylor series expansion, the proposed compact mixed formulation generalizes the work of Carey and Spotz [G.F. Carey, W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Meth. Eng. 13 (1997)]. We show that their fourth-order formulation corresponds to a particular case of our presented scheme, and we extend their work to variable diffusion and convection coefficients. Some numerical experiments are performed to demonstrate the fourth-order effective convergence rate.     

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

We study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for a class of Nicholson’s blowflies model with patch structure and multiple linear harvesting terms. Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples and numerical simulations to illustrate our main results.     

The positive almost periodic solution for Nicholson-type delay systems with linear harvesting terms

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for a class of Nicholson-type delay system with linear harvesting terms. Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution. Moreover, we give an example to illustrate our main results.     

Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model

In this paper we consider a competitor-competitor-mutualist model with cross-diffusion. We prove some existence and non-existence results concerning non-constant positive steady-states (patterns). In particular, we demonstrate that the cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

Martingale approach to limit theorems for jump processes

We consider the weak convergence of laws of càdiàg processes determined by a sequence of operators with singularly perturbed terms. We study the problem in the martingale approach, which was formulated to establish weak limit theorems for continuous processes by Papanicolaou, Stroock and Varadhan. However, in this paper, limit processes are not necessarily continuous but càdiàg. In particular, we consider a homogenization problem of càdiàg processes in the framework of martingale problem.     

The convergence of two numerical schemes for the Navier-Stokes equations

Summary This paper deals with some convergence/stability results concerning two numerical methods for solving the incompressible nonstationary Navier-Stokes equations. The algorithms are of a particular kind in what regards time discretization (more precisely, of the Peaceman-Rachford and the Strang type resp.), and have been obtained by modifying slightly the numerical treatment of the nonlinear terms in other schemes due to Glowinski et al. (1980). We first describe the full discretization of the homogeneous Dirichlet problem using a (general) external approximation of the spatial functional spaces involved (a particular and simple choice of such an approximation is the standardP ₂-Lagrange finite element for the velocity field when the fluid is bidimensional). Then we establish and prove convergence and stability and make some comments on the numerical treatment of other (generally nonhomogeneous) boundary conditions. The theoretical results show that the schemes are (at least) conditionally stable and convergent, which justifies the success of Glowinski's methods.