Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.     

Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron

This article is devoted to the study of a mathematical model arising in the mathematical modeling of pulse propagation in nerve fibers. A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations. When considered as part of a particular initial boundary value problem the equation models the electrical activity in a neuron. A small perturbation parameter ε is introduced to the highest order derivative term. The parameter if decreased, speeds up the fast variables of the model equations whereas it does not affect the slow variables. In order to formally reduce the problem to a discussion of the moment of fronts and backs we take the limit ε → 0. This limit is singular and is therefore the solution tends to a slowly moving solution of the limiting equation. This leads to the boundary layers located in the neighborhoods of the boundary of the domain where the solution has very steep gradient. Most of the classical methods are incapable of providing helpful information about this limiting solution. To this effort a parameter robust numerical method is constructed on a piecewise uniform fitted mesh. The method consists of standard upwind finite difference operator. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives. A parameter uniform error estimate for the numerical scheme so constructed is established in the maximum norm. It is then proven that the numerical method is unconditionally stable and provides a solution that converges to the solution of the differential equation. A set of numerical experiment is carried out in support of the predicted theory, which validates computationally the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Multiple time scale formalism and its application to long water waves

In the present work, by employing the multiple time scaling method, we studied the nonlinear waves in shallow-water problem and obtained a set of Korteweg–deVries equations governing the various order terms in the perturbation expansion. By seeking a travelling wave type of solutions for the evolution equations, we have obtained a set of wave speeds associated with each time parameter. It is shown that the speed corresponding to the lowest order time parameter given the wave speed of the conventional reductive perturbation method, whereas the wave speeds corresponding to the higher order time parameters give the speed correction terms. The result obtained here is exactly the same with that of Demiray [H. Demiray, Modified reductive perturbation method as applied to long water waves: Korteweg–deVries hierarchy, Int. J. Nonlinear Sci. 6 (2008) 11–20] who employed the modified reductive perturbation method.     

Modulation equations and parabolic limits of reaction random-walk systems

Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

一类广义非线性强阻尼扰动发展方程的行波解

研究了一类非线性强阻尼广义扰动发展方程问题.它们在数学、力学、物理学等领域中广泛出现.首先,引入一个行波变换,把相应的偏微分方程问题转化为行波方程问题并求出原典型问题的精确解.再用小参数方法和引入伸长变量构造了问题的渐近解.最后, 用泛函分析的不动点理论证明了原非线性强阻尼广义扰动发展方程初值问题渐近行波解的存在性,并证明渐近解具有较高的精度和一致有效性.该文求得的渐近解是一个解析展开式, 所以它还可继续进行解析运算, 而单纯用数值模拟的方法是不行的.     

A singular reaction–diffusion system modelling prey–predator interactions: Invasion and co-extinction waves

We consider a singular reaction–diffusion system arising in modelling prey–predator interactions in a fragile environment. Since the underlying ODEs system exhibits a complex dynamics including possible finite time quenching, one first provides a suitable notion of global travelling wave weak solution. Then our study focusses on the existence of travelling waves solutions for predator invasion in such environments. We devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co-extinction tail for both species. Under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere, such a property arising for every admissible wave speeds.     

Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation

This paper deals with the numerical analysis of time dependent parabolic partial differential equation. The equation has bistable nonlinearity and models electrical activity in a neuron. A qualitative analysis of the model is performed by means of a singular perturbation theory. A small parameter is introduced in the highest order derivative term. This small parameter is known as singular perturbation parameter. Boundary layers occur in the solution of singularly perturbed problems when the singular perturbation parameter tend to zero. These boundary layers are located in neighbourhoods of the boundary of the domain, where the solution has a very steep gradient. Most of the conventional methods fails to capture this effect. A numerical scheme is constructed to overcome this discrepancy in literature. A rigorous analysis is carried out to obtain a-priori estimates on the solution of the problem and its derivatives. It is then proven that the numerical method is unconditionally stable. Convergence and stability analysis is carried out. A set of numerical experiment is carried out and it is observed that the scheme faithfully mimics the dynamics of the model.     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

BVPs on infinite intervals: A test problem,a nonstandard finite difference scheme and a posteriori error estimator

As far as the numerical solution of boundary value problems defined on an infinite interval is concerned, in this paper, we present a test problem for which the exact solution is known. Then we study an a posteriori estimator for the global error of a nonstandard finite difference scheme previously introduced by the authors. In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from 2 nested quasi‐uniform grids. We observe that if the grids are sufficiently fine, the Richardson error estimate gives an upper bound of the global error.     

12参双参数矩形板元的误差估计

双参数方法是构造高阶问题有限元的有效方法．以此方法构造的双参数元是一种非标准元，以往文献中只证明了它的收敛性．此文针对具体12参双参数矩形板元给出它的误差估计式，并分析了节点参数的扰动量．文中的分析方法也适合于其它双参数矩形板元的误差估计．     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.     

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.     

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.     

弹性直杆中一类动力屈曲问题的渐近解

对于弹性杆受刚性块轴向撞击的动力屈曲问题而言,由于轴向载荷形式较为复杂,问题将归结为关于非线性偏微分方程组解的讨论,至今仍未能得到一个理论上的解析解,为此,讨论了有限长理想弹性直杆的此类动力屈曲问题,采用小参数的摄动展开和变分法,成功地得到了这一问题的一个理论上的近似解,并给出了相应的算例,从中得到了一些有益的结论.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

The motive of the current study is to derive pointwise error estimates for the three-step Taylor Galerkin finite element method for singularly perturbed problems. Pointwise error estimates have not been derived so far for the said method in the finite element framework. Singularly perturbed problems represent a class of problems containing a very sharp boundary layer in their solution. A small parameter called singular perturbation parameter is multiplied with the highest order derivative terms. When this parameter becomes smaller and smaller, a boundary layer occurs and the solution changes very abruptly in a very small portion of the domain. Because of this sudden change in the nature of the solution, it becomes very difficult for the numerical methods to capture the solution accurately specially in the boundary layer region. In the present study finite element analysis has been carried out for such one-dimensional singularly perturbed time dependent convection-diffusion equations. Exponentially fitted splines have been used for the three-step Taylor Galerkin finite element method to converge. Pointwise error estimates have been derived for the method and it is shown that the method is conditionally convergent of first order accurate in space and third order accurate in time. Numerical results have been presented for both the linear and nonlinear problems.     

Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations

Periodic travelling waves (wavetrains) are an important solution type for many partial differential equations. In this paper I review the use of numerical continuation for studying these solutions. I discuss the calculation of the form and stability of a given periodic travelling wave, and the calculation of boundaries in a two-dimensional parameter plane for wave existence and stability. I also describe the automated implementation of these numerical continuation procedures via the software package wavetrain (http://www.ma.hw.ac.uk/wavetrain). I conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves.     

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.