A new linearized compact multisplitting scheme for the nonlinear convection–reaction–diffusion equations with delay (original) (raw)

Introduction

Delay partial differential equations (DPDEs) such as∂w(x,t)∂t=Lw(x,t)+fx,t,w(x,t),w(x,t-s),∂w(x,t)∂x,∂w(x,t-s)∂x,where L is a linear operator, play an important role in the simulation of many real-world systems, such as biological systems, epidemiology, medicine, engineering control systems, climate models, and many others, (c.f. [1], [2], [3], [4]). DPDEs (1.1) are believed to provide a powerful tool in describing a variety of natural phenomena and behaviors in social sciences, where the quantity w(x,t) depends not only on the solution at a present stage but also on the solution at some past stage. For a survey of early results, we refer the reader to the books [11], [12], [13] and references therein.

In the past decades, DPDEs have been getting more and more attention and theoretical analysis on DPDEs has been widely developed (see e.g., [5], [6], [7], [8], [9], [10]). As we know, analytical solutions of most of DPDEs are unavailable (c.f. [12], [13]). Even for some simple and specific equations, their theoretical solutions often are piecewise continuous or even can not be expressed in analytical expression.

Recently, many scholars are dedicated to the numerical investigation on DPDEs. For instance, Marzban and his collaborator investigated a hybrid approximation method for solving Hutchinson’s equation [19], Jackiewicz et al. [20] simulated traveling wave solutions in a drift paradox inspired diffusive delay population model using the embedded pair of continuous Runge–Kutta methods of order four and three, Sun et al. constructed a Crank–Nicolson scheme and a linearized compact difference method for the nonlinear delay reaction–diffusion equations [22], [23], Jackiewicz and Zubik-Kowal [24] considered Chebyshev spectral collocation and waveform relaxation methods for nonlinear DPDEs, Li et al. [25] proposed a LDG method for reaction–diffusion dynamical systems with time delay, Zhang et al. [26] concentrated on a compact difference scheme combined with extrapolation techniques for a class of neutral DPDEs and J. Ruiz-Ramírez et al. proposed a skew symmetry-preserving, finite-difference scheme for a time-delayed advection–diffusion–reaction equation [27]. The numerical research focused on stability analysis can be referred in [28], [29], [30].

Our interest in this paper is to develop a new linearized compact multisplitting scheme to solve the nonlinear delay convection-reaction–diffusion equations as follows∂w∂t+α∂w∂x-β∂2w∂x2=g(w(x,t),w(x,t-s),x,t),(x,t)∈(a,b)×[0,T],where s>0,α is the constant speed of convection, β>0 is the diffusion coefficient. The initial condition associated with (1.2) is given byw(x,t)=φ(x,t),x∈[a,b],t∈[-s,0]and the boundary conditions are given byw(a,t)=wa(t),w(b,t)=wb(t),t>0.Special cases of Eq. (1.2) are frequently encountered in a vast array of fields. For example, taking g(w(x,t),w(x,t-s),x,t)=0, Eq. (1.2) reduces to unsteady convection–diffusion equation which has been widely used to model various processes in science and engineering, (c.f. [14], [15]). When α=0,β>0, Eq. (1.2) becomes a nonlinear delay reaction–diffusion equation, which can be used to character many model problems. For instance, diffusive Nicholson’s blowflies equation is applied to model laboratory fly population [16], [17], [18] and the diffusive Hutchinson’s equation or delayed Fisher equation can be used to measure how efficiently the particles of a single species disperse from a high to a low density [19], [21].

In the paper [23], Sun et al. proposed a linearized compact difference scheme for delay parabolic differential equations based on the Crank–Nicholson method in temporal direction. Enlightened by the idea, we proposed a new linearized compact difference scheme for convection-reaction–diffusion equations with delay, where temporal direction is discreted by second-order backward differentiation formulas (c.f. [31]). For the sake of brevity, we use the name multisplitting scheme in our paper. Compared with the scheme in [23], the construction of the multisplitting scheme is more direct, the nonlinear term is linearized by a simpler way, and the accuracy of the scheme is also competitive to some extent. To improve the computational accuracy in temporal dimension, a kind of Richardson extrapolation technique is utilized and the numerical experiments show that a fourth-order accuracy in both temporal and spatial dimensions is obtained.

The rest of the article is organized as follows. In Section 2, by a class of new exponential transformation, Eqs. (1.2), (1.3), (1.4) are deduced to a kind of nonlinear delay reaction–diffusion equations. Section 3 is devoted to the construction of the compact multisplitting scheme and the detailed local truncation error is derived. In Section 4, the solvability, convergence and stability of compact multisplitting scheme are analyzed. In Section 5, extensive numerical examples are carried out and compared to verify the accuracy and the efficiency of the proposed scheme. Concluding remarks are given in Section 6.

Section snippets

The equivalent form of Eqs. (1.2)–(1.4)

In the paper [23], an efficient finite difference scheme was proposed to solve a one-dimensional nonlinear DPDEs. The method works well for the reaction and diffusion terms, but it does not work for the convection term. To solve the problem, a novel transformation is introduced to eliminate the convection term. Although the scheme developed in our paper aims to solve the nonlinear delay reaction–diffusion equation, it works for the nonlinear delay convection-reaction–diffusion equation as well.

Compact multisplitting scheme and local truncation error

Before the presentation of the compact multisplitting scheme, we introduce some notations used later. Firstly, we divide the region Ω×(0,T], where Ω=(a,b). Then, let h=b-aM with M being positive number. Constrained mesh is utilized (c.f. [28]), i.e., time step τ is a submultiple of delay s, τ=sn. Denote xi=a+ih,tk=kτ and define Ωhτ=Ωh×Ωτ, where Ωh={xi|0⩽i⩽M},Ωτ={tk|-n⩽k⩽N},N=Tτ,Uik=u(xi,tk),0⩽i⩽M,-n⩽k⩽N. Suppose W={vik|0⩽i⩽M,-n⩽k⩽N} is the grid function space defined on Ωhτ. We denoteδtvik+12=1τ

The solvability, convergence and stability of compact multisplitting scheme

This section is devoted to study the solvability, convergence and stability of the compact multisplitting scheme (3.12), (3.13), (3.14).

Firstly, we introduce some notations and lemmas, which play a very important role in proving the convergence and the stability. Let V={v|v=(v0,v1,…,vM),v0=vM=0} be the grid function space defined on Ωh. For any v,w∈V, define the inner products and corresponding norms as follows.(v,w)=h∑i=1M-1viwi,‖v‖=(v,v)=h∑i=1M-1(vi)2,|v|1=(δxv,δxv)=h∑i=1Mvi-vi-1h2,‖v‖∞=max0⩽i

Numerical experiment

In this section, we perform several numerical simulations to check the performance of our methods. For ease of comparison, the difference scheme presented in [23] for the Eqs. (2.4) in Section 2 is listed out as follows,Aδtuik+12-βδx2uik+12=Af32uik-12uik-1,12uik+1-n+12uik-n,xi,tk+12,1⩽i⩽M-1,0⩽k⩽N-1,uik=ψ(xi,tk),0⩽i⩽M,-n⩽k⩽0,u0k=ua(tk),uMk=ub(tk),1⩽k⩽N.

In the following, we compute the numerical solutions using scheme (5.1), (5.2), (5.3), the compact multisplitting scheme (3.12), (3.13), (3.14)

Concluding remarks

This paper constructs a linearized multisplitting scheme for the nonlinear delay convection–reaction–diffusion equations. A new transformation is introduced to convert the original equations into a class of nonlinear reaction–diffusion equations with delay. Strict theoretical analysis of the resulting difference scheme is given. It is shown that the accuracy of the proposed scheme is of the second-order in time and the fourth-order in space with respect to the discrete L∞-norm. Four numerical

Acknowledgement

The authors thank Dr. Xueqing Yan at Kansas State University for her many helpful discussions. Also, the authors are grateful to the anonymous referees for the valuable and constructive comments, which have greatly improved the article.

Cited by (50)

2015, Applied Mathematical Modelling
The unique solvability, convergence and stability of the method are proposed. Later, the scheme was further investigated for solving neutral parabolic differential equations with delay [3] and advection–diffusion equations with delay [4]. In spite of their interesting and instructive work, these studies do not pay much attention to the long time stability of the scheme.

2014, International Journal of Non-Linear Mechanics
Exact solutions to some non-linear delay PDEs (and systems of non-linear delay PDEs) other than (1) and (2) can be found, for example, in [12,42–44]. For numerical solution methods for non-linear coupled delay reaction–diffusion systems and other non-linear systems of delay PDEs as well as related difficulties, see [45–50]. The exact solutions presented below in Sections 3–9 may be used as test problems for independent verification of numerical methods for non-linear systems of delay PDEs. View all citing articles on Scopus

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