Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross‐diffusion system possesses a formal gradient‐flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

Young measure solutions of some nonlinear mixed‐type equations

This contribution deals with measure‐valued solutions to two types of nonlinear partial differential equations for which, in general, the results on the existence of classical or weak solutions fail. These are the potential equation for transonic flow and the associated unsteady problem (forward–backward diffusion equation). The solutions are constructed by an iteration scheme (Katchanov method) and additional time discretization (Rothe method) in the second case. The existence is proved in the sense of spatial gradient Young measures. Copyright © 2010 John Wiley & Sons, Ltd.     

A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A control volume technique based on integrated RBFNs for the convection–diffusion equation

This article reports a new high‐order control‐volume discretization for the convection–diffusion equation in one and two dimensions. Diffusive fluxes at the faces of a control volume and other terms embracing the unknown field variable are all approximated using one‐dimensional integrated radial‐basis‐function networks; line integrals involving these fluxes and other integrals are evaluated using a high‐order numerical integration scheme. The accuracy of the proposed technique is investigated numerically through the solution of several linear and nonlinear test problems, including a benchmark thermally driven cavity flow. High‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.     

A fourth‐order compact algorithm for nonlinear reaction‐diffusion equations with Neumann boundary conditions

In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction‐diffusion equations is fourth‐order accurate in both the temporal and spatial dimensions. We also prove that the standard second‐order approximation to zero Neumann boundary conditions provides fourth‐order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

In this paper a reaction–diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L²‐stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability–instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.     

On multiple soliton similariton‐pair solutions,conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.     

A series of new exact solutions of nonlinear evolution equations

With the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Numerical analysis of a finite volume scheme for a seawater intrusion model with cross‐diffusion in an unconfined aquifer

We consider a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion. We propose and analyze a finite volume scheme based on two‐point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges toward a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.     

New asymptotic estimates of solutions for generalized Rosenau equations

This paper presents new methods widely applicable to expand solutions for wave equations with damping terms such as Rosenau‐type equations. Some of them have the diffusion structure that appears strongly in the low‐frequency region, and some detailed analysis on diffusion waves is seen in this report. In the high‐frequency region, difficulties arising from the regularity‐loss type are overcome by a new discovery of suitable asymptotic profiles and expanding techniques of solutions even if regularity assumptions on the initial data are not imposed. It is also shown that stronger regularity assumptions on the initial data give better asymptotic estimates.     

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

In this paper, we develop a practical numerical method to approximate a fractional diffusion equation with Dirichlet and fractional boundary conditions. An approach based on the classical Crank–Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second‐order accurate numerical estimates. The solvability, stability, and convergence of the proposed numerical scheme are proved via the Gershgorin theorem. Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014