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     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

Wave simulations of Gray‐Scott reaction‐diffusion system

In this work, we study the numerical simulation of the one‐dimensional reaction‐diffusion system known as the Gray‐Scott model. This model is responsible for the spatial pattern formation, which we often meet in nature as the result of some chemical reactions. We have used the trigonometric quartic B‐spline (T4B) functions for space discretization with the Crank‐Nicolson method for time integration to integrate the nonlinear reaction‐diffusion equation into a system of algebraic equations. The solutions of the Gray‐Scott model are presented with different wave simulations. Test problems are chosen from the literature to illustrate the stationary waves, pulse‐splitting waves, and self‐replicating waves.     

Existence and non‐existence of steady states to a cross‐diffusion system arising in a Leslie predator–prey model

This paper is concerned with a cross‐diffusion system arising in a Leslie predator–prey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non‐existence of non‐constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross‐diffusion coefficients are fixed, then under some conditions there exists non‐constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross‐diffusion coefficients are small enough, then there exists no non‐constant positive solution. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

An efficient high‐order algorithm for solving systems of reaction‐diffusion equations

An efficient higher‐order finite difference algorithm is presented in this article for solving systems of two‐dimensional reaction‐diffusion equations with nonlinear reaction terms. The method is fourth‐order accurate in both the temporal and spatial dimensions. It requires only a regular five‐point difference stencil similar to that used in the standard second‐order algorithm, such as the Crank‐Nicolson algorithm. The Padé approximation and Richardson extrapolation are used to achieve high‐order accuracy in the spatial and temporal dimensions, respectively. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 340–354, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10012     

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.     

Two‐grid methods for mixed finite‐element solution of coupled reaction‐diffusion systems

We develop 2‐grid schemes for solving nonlinear reaction‐diffusion systems: where p = (p, q) is an unknown vector‐valued function. The schemes use discretizations based on a mixed finite‐element method. The 2‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction‐diffusion equations. An application to prepattern formation in mathematical biology illustrates the method's effectiveness. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 589–604, 1999     

Semi‐implicit spectral collocation methods for reaction‐diffusion equations on annuli

In this article, we develop numerical schemes for solving stiff reaction‐diffusion equations on annuli based on Chebyshev and Fourier spectral spatial discretizations and integrating factor methods for temporal discretizations. Stiffness is resolved by treating the linear diffusion through the use of integrating factors and the nonlinear reaction term implicitly. Root locus curves provide a succinct analysis of the A‐stability of these schemes. By utilizing spectral collocation methods, we avoid the use of potentially expensive transforms between the physical and spectral spaces. Numerical experiments are presented to illustrate the accuracy and efficiency of these schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1113–1129, 2011     

An IMEX‐BDF2 compact scheme for pricing options under regime‐switching jump‐diffusion models

In this paper, an implicit‐explicit two‐step backward differentiation formula (IMEX‐BDF2) together with finite difference compact scheme is developed for the numerical pricing of European and American options whose asset price dynamics follow the regime‐switching jump‐diffusion process. It is shown that IMEX‐BDF2 method for solving this system of coupled partial integro‐differential equations is stable with the second‐order accuracy in time. On the basis of IMEX‐BDF2 time semi‐discrete method, we derive a fourth‐order compact (FOC) finite difference scheme for spatial discretization. Since the payoff function of the option at the strike price is not differentiable, the results show only second‐order accuracy in space. To remedy this, a local mesh refinement strategy is used near the strike price so that the accuracy achieves fourth order. Numerical results illustrate the effectiveness of the proposed method for European and American options under regime‐switching jump‐diffusion models.     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

We study a special case of Shigesada–Kawasaki–Teramoto (SKT) model for two competing species with the Dirichlet boundary conditions. In our case, one of the species is not influenced by self‐diffusion or cross‐diffusion. We specify the explicit range of parameters by contradiction such that there are no coexisting steady‐state solutions to the model.     

Dynamical transition for S‐K‐T biological competing model with cross‐diffusion

The main objective of this paper is to study the dynamical transition for the S‐K‐T biological competition system with cross‐diffusion. Based on the spectral analysis, the principle of exchange of stabilities conditions for eigenvalues are obtained. By using the dynamical transition theory, 2 different types of dynamical transition for the S‐K‐T model are also derived. In addition, an example is given to illustrate our main results.     

Global synchronization of fractional‐order and integer‐order N component reaction diffusion systems: Application to biochemical models

This study considers the problem of control and synchronization between fractional‐order and integer‐order, N‐components reaction‐diffusion systems with nonidentical coefficients and different nonlinear parts. The control scheme is designed using the Lyapunov direct method. The results are exemplified by two significant biochemical models, namely, the fractional‐order Lengyel‐Epstein model and the Gray‐Scott model. To illustrate the effectiveness of the proposed scheme, numerical simulations are performed in one and two space dimensions using Homotopy Analysis Method (HAM).     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

High‐order approximation of 2D convection‐diffusion equation on hexagonal grids

We derive a fourth‐order finite difference scheme for the two‐dimensional convection‐diffusion equation on an hexagonal grid. The difference scheme is defined on a single regular hexagon of size h over a seven‐point stencil. Numerical experiments are conducted to verify the high accuracy of the derived scheme, and to compare it with the standard second‐order central difference scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Dissipativity and synchronization control of fractional‐order memristive neural networks with reaction‐diffusion terms

This paper deals with the dissipativity and synchronization control of fractional‐order memristive neural networks (FOMNNs) with reaction‐diffusion terms. By means of fractional Halanay inequality, Wirtinger inequality, and Lyapunov functional, some sufficient conditions are provided to ensure global dissipativity and exponential synchronization of FOMNNs with reaction‐diffusion terms, respectively. The underlying model and the obtained results are more general since the reaction‐diffusion terms are first introduced into FOMNNs. The given conditions are easy to be checked, and the correctness of the obtained results is confirmed by a living example.     

Numerical approximation of two‐dimensional convection‐diffusion equations with boundary layers

Our aim in this article is to show how one can improve the numerical solution of singularity perturbed problems involving boundary layers. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. In this article we discuss convection‐diffusion equations in the two‐dimensional space with a homogeneous Dirichlet boundary condition and a mixed boundary condition. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second‐order time evolution

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016