# An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation

Authors: Somayeh Pourghanbar; Jalil Manafian; Mojtaba Ranjbar; Aynura Aliyeva; Yusif S. Gasimov
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, the Saul’yev finite difference scheme for a fully nonlinear partial differential equation with initial and boundary conditions is analyzed. The main advantage of this scheme is that it is unconditionally stable and explicit. Consistency and monotonicity of the scheme are discussed. Several finite difference schemes are used to compare the Saul’yev scheme with them. Numerical illustrations are given to demonstrate the efficiency and robustness of the scheme. In each case, it is found that the elapsed time for the Saul’yev scheme is shortest, and the solution by the Saul’yev scheme is nearest to the Crank–Nicolson method.


---

## Body

## 1. Introduction

Many problems in financial derivatives [1], option pricing [2], chemical diffusion [3], computational fluid dynamics [4], hydrodynamics [5], and control theory [6] can be modeled using partial differential equations (PDEs). In recent years, a lot of attention has been devoted to the study of nonlinear PDEs and methods for numerical solutions of nonlinear problems. Our aim in this paper is to approximate the solution of a nonlinear PDE with initial and boundary conditions by using an efficient numerical method so-called the Saul’yev finite difference scheme.As said in [7], a noticeable feature of the explicit finite difference methods is the restriction of the size of the time step due to stability requirements. For most problems, these are impractical methods. On the other hand, it is widely known that implicit schemes are stable but must be solved in a matrix system at each time level. Hence, it would be nice if we could find stable schemes which are explicit too.The ADE scheme is documented by Saul’yev in [8] and is the abbreviation of the phrase “alternating direction explicit” method. The Saul’yev scheme is explicit; therefore, it needs less computational works in comparison with implicit schemes. It is unconditionally stable and first-order accurate. The Saul’yev method has been applied to many problems in literature. Roberts and Weiss [9] applied the Saul’yev scheme to first-order hyperbolic equations. Dehghan [7] obtained the numerical solution to an inverse problem by the Saul’yev scheme. In [10], Campbell and Yin applied the Saul’yev scheme to some quasilinear one-dimensional advection-diffusion problems. They derive the stability conditions for combinations of diffusion and advection schemes. Authors of [11] presented simple revisions to the Saul’yev scheme that make it more accurate without a significant loss of computation efficiency. Saul’yev scheme was used to solve telegraph equation in [12]. Sun [13] used Saul’yev’s scheme to formulate certain approximation schemes. In addition, the first article discusses the application of the Saul’yev scheme to one-factor option pricing problems [14]. Soheili and co-authors [15] used the Saul’yev scheme in order to approximate the solution of stochastic partial differential equations. Moreover, stochastic alternating direction explicit (SADE) finite difference schemes for solving stochastic time-dependent advection-diffusion and diffusion equations are applied in [16]. Abbasbandy and Shirzadi [5] used the Saul’yev scheme for solving one-dimensional equations of conservation law form.The Saul’yev method was used for the digital simulation under derivative boundary conditions [17, 18]. Stability criteria, a modified Saul’yev’s explicit method, for parabolic PDEs were derived and compared with those of classical Saul’yev’s explicit method and Crank–Nicholson’s implicit method by Towler and Yang [19]. Ali and Abdullah discussed the analysis and implementation of the explicit Saul’yev difference scheme for the solution of 2D time fractional subdiffusion equations [20]. A new group explicit method for solution of diffusion equation was presented by Tavakoli and Davami [21]. Saul’yev-type asymmetric schemes had been widely utilized in solving diffusion and advection equations, so that Saul’yev type schemes are derived from the exponential splitting of the semidiscretized equation [22]. The Saul’yev finite difference algorithms were utilized for the electrochemical kinetic simulations of mixed diffusion/homogeneous reaction problems [23, 24]. A combined mixed finite element ADI scheme was employed to solve Richards’ equation with mixed derivatives on irregular grids [25].In this paper, we use the Saul’yev scheme for a fully nonlinear partial differential equation with initial and boundary conditions, in which this scheme not only is explicit but also is unconditionally stable. As we shall see, the scheme is consistent and monotone too. We will study the numerical solution of the problem by means of a particular kind of the finite difference schemes. The Saul’yev scheme is an explicit method for solving partial differential equations. To the best of our knowledge, such an approach has not been previously employed to solve this kind of nonlinear problem.The contents of this paper are the following: in the next section, we introduce a nonlinear partial differential equation. In Section3, we apply the Saul’yev scheme for the nonlinear problem. The consistency, monotonicity, and stability of the scheme are shown in Section 4. Numerical experiments are performed in Section 5. Finally, we sum up the conclusions in Section 6.


## 2. Nonlinear Partial Differential Equation

Consider the following nonlinear partial differential equation:(1)∂u∂τ=σ0221+Ψa2x2∂2u∂x2x2∂2u∂x2,x∈0,∞,τ∈0,T,where σ0 and a are the given points, with initial and Dirichlet boundary conditions,(2)ux,0=maxx−E,0,u0,τ=0,limx⟶∞ux,τ=x,where E is a known value. Function ΨA is the solution of the following nonlinear ordinary differential equation:(3)Ψ′A=ΨA+12AΨA−A,A≠0,Ψ0=0.The famous work of [26] was one of the first to call broad attention to the transaction costs in finance and presents the theoretical discussion of function Ψ. In contrast to [27] that the ordinary differential equation (3) is solved with the ode45 function in MATLAB, we use the exact solution of (3) in this paper. It is worth noting that the ode45 function in MATLAB is based on the Runge–Kutta 4,5 one-step solver. Some other similar equations arise in computational finance in various forms, and it is important to know that these PDEs do not have analytic computable solution, and we must resort to numerical methods in order to find an approximate solution.To study the consistency of the Saul’yev scheme, we need to bound the approximation of the nonlinear term in the equation. Hence, Theorem1 plays an important role in the consistency of the scheme in Section 4. Moreover, the solution of equation (3) is defined in Theorem 2, which will be used in Section 5.Theorem 1.
LetgA=AΨA. Then, gA is a continuously differentiable function at A=0 and satisfies(4)g′A≤maxG,2AΨ′A2+d2,A∈ℛ,where ℛ is the real numbers set, and(5)A2≃9.58,d2=ΨA2−Ψ′A2A2≃2.62,G=maxg′A;A1≤A≤A2,A1=−4π−33236.Proof.
see [28].Theorem 2.
The unique solution of equation (3) is defined as(6)A=−arcsinhΨΨ+1+Ψ2,Ψ>0,A=−arcsinh−ΨΨ+1−−Ψ2,−1<Ψ<0.Proof.
See [29].


## 3. Saul’yev Scheme for the Nonlinear Problem

The problem (1) is described in an infinite domain 0,∞×0,T. We consider the truncated numerical domain 0,B×0,T and divide it into an M×N mesh (M and N are integers) with the spatial step size h=B/M in direction x and the time step size k=T/N in direction τ, respectively.Grid of mesh points is defined by(7)x,τ=xi,τn,i=0,1,…,M,n=0,1,…,N,where xi=ih, and τn=nk. Notation uin is used for uxi,τn and approximated the solution of our fully nonlinear PDE in xi and τn. Here, we use the forward finite difference approximation for the first-order time derivative as(8)∂u∂τxi,τn≃uin+1−uink.Also, we introduce two operators (Δ, central, and δ, Saul’yev) for approximation of the second-order derivative ∂2u/∂x2 as(9)∂2u∂x2xi,τn≃ui+1n−2uin+ui−1nh2=Δin,(10)∂2u∂x2xi,τn≃ui+1n−uin−uin+1+ui−1n+1h2=δin.Substituting approximations (8), (9), and (10) into equation (1) leads to the following difference equation:(11)uin+1−uink−σ0221+Ψa2xi2Δinxi2δin=0.In functionΨ, we approximate the second-order spatial derivative by Δin. If we consider δin instead of Δin in 1+Ψa2xi2Δin in (11), the scheme can only be solved by a nonlinear iteration in each time step which is quite time-consuming. By denoting σ02/21+Ψa2xi2Δinxi2=αin and s=k/h2, we get the difference equation:(12)1+sαinuin+1−sαinui−1n+1=sαinui+1n+1−sαinuin,for i=1,2,…,M−1. Although the approximation (12) does not appear explicit because uin+1 and ui−1n+1 are on the left-hand side, a suitable use of the equation makes it explicit. If we write (12) in the form of (13) and begin the calculation at the left boundary, then move to the right, so that only the single value uin+1 is unknown, and the scheme will be explicit.(13)uin+1=sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n.According to Theorem2, the range of function Ψ is the interval −1,+∞; therefore, 1+Ψ>0, and hence αin>0, for all n and i. On the other hand, it is obvious that s>0; therefore, in (13), we have 1+sαin≠0. The next section contains an analysis of the scheme described above.


## 4. Consistency, Monotonicity, and Stability Analysis

### 4.1. Consistency

As said in [30], dealing with reliable numerical schemes, the consistency of the difference scheme with the equation is a necessary requirement because this means that the exact theoretical solution of the partial differential equation approximates well to the solution of the difference equation as the step sizes tend to zero. In other words, the problem of consistency is the problem of finding the condition for which a discrete problem is an approximation of the corresponding continuous problem. Let us consider equation (1) with exact solution u as Lu=0, and let Fuin=0 represent the finite difference scheme (11). Hence,(14)Tinu=Fuin−Luin=uin+1−uink−σ0221+Ψa2xi2△inxi2δin−uτ−σ0221+Ψa2x2uxxx2uxxin.It will be shown thatTinu⟶0 as h⟶0 and k⟶0. By means of Taylor series, we have(15)δin=uxx|in−khEin1,in which(16)Ein1=∂2u∂x∂τ|η1n,η1∈0,B,Ein1≤uin1max=maxτ∈0,T∂2u∂τ∂x;0≤x≤B.In the same state,(17)Δin=uxx|in+h2Ein2,in which(18)Ein2=112∂4u∂x4|η2n,η2∈0,B,(19)Ein2≤112uin2max=112maxτ∈0,T∂4u∂x4;0≤x≤B.Also,(20)uin+1−uink=uτ|in+kEin3,in which(21)Ein3=12∂2u∂τ2|iξ3,ξ3∈0,T,(22)Ein3≤12uin3max=12maxτ∈0,T∂2u∂τ2;0≤x≤B.Hence, by (15) and (17),(23)1+Ψa2xi2△inδin−1+Ψa2xi2uxx|inuxx|in=1+Ψa2xi2uxx|in+a2xi2h2Ein2uxx|in−kh1+Ψa2xi2uxx|in+a2xi2h2Ein2Ein1−1+Ψa2xi2uxx|inuxx|in.Let us denoteAin=a2xi2uxx|in, ΔAin=a2xi2h2Ein2 and ΔBin=a2xi2Ein1. It should be noted that the use of Δ in (9) will not cause any ambiguity here. We see(24)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in=1+ΨAin+ΔAinAin−kh1+ΨAin+ΔAinΔBin−1+ΨAinAin1a2xi2<1+ΨAin+ΔAinAin+1+ΨAin+ΔAinΔAin−1+ΨAinAin1a2xi2=1+ΨAin+ΔAinAin+ΔAin−1+ΨAinAin1a2xi2.As we introduce the functiong in Theorem 1, it follows that(25)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in≤gAin+ΔAin−gAin+ΔAina−2xi−2.Applying the mean value theorem for the functiong, we will have(26)gAin+ΔAin−gAin=g′ηΔAin.Therefore,(27)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in≤ΔAin1+g′ηa−2xi−2.For the truncation error, we will see(28)Tinu=−σ022xi21+Ψa2xi2△inδin−1+Ψa2xi2uxx|inuxx|in+kEin3.Hence,(29)Tin<−σ022h21+g′ηEin2xi2+kEin3,where the upper bound of Ein2, Ein3, and g′ are introduced in (19), (22), and Theorem 1, respectively.


### 4.2. Monotonicity

In the numerical scheme(30)Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n=0,we say it is monotonicity preserving, if any positive perturbation to any of ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n produces a positive perturbation of uin+1. If Fi is differentiable, then this is equivalent to(31)∂uin+1∂ui−1n+1=−∂Fi/∂ui−1n+1∂Fi/∂uin+1≥0,∂uin+1∂ui+1n+1=−∂Fi/∂ui+1n+1∂Fi/∂uin+1≥0,∂uin+1∂uin=−∂Fi/∂uin∂Fi/∂uin+1≥0,∂uin+1∂ui−1n=−∂Fi/∂ui−1n∂Fi/∂uin+1≥0,∂uin+1∂ui+1n=−∂Fi/∂ui+1n∂Fi/∂uin+1≥0.In the case of nondifferentiableFi, we will use the following definition of monotonicity [31]. A discretization of form (30) is monotone if either ∀i(32)Fiuin+1,ui−1n+1+εi−1n+1,ui+1n+1+εi+1n+1,uin+εin,ui−1n+εi−1n,ui+1n+εi+1n≥Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,Fiuin+1+εin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n≤Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,or ∀i(33)Fiuin+1,ui−1n+1+εi−1n+1,ui+1n+1+εi+1n+1,uin+εin,ui−1n+εi−1n,ui+1n+εi+1n≤Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,Fiuin+1+εin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n≥Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n.In many problems such as financial applications of PDEs, finite difference schemes should produce positive values, but not all schemes have this property. In the next theorem, we will see the monotonicity of the Saul’yev scheme.Theorem 3.
If the time step in numerical scheme (13) is selected, such that(34)k≤h2αin,0≤i≤M,0≤n≤N,then the scheme is monotone.Proof.
We consider perturbingui−1n+1 by an amount ε>0. Because sαin/1+sαin>0,(35)Fiuin+1,ui−1n+1+ε,uin,ui+1n=−uin+1+sαin1+sαinui−1n+1+ε+1−sαin1+sαinuin+sαin1+sαinui+1n>−uin+1+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n=Fiuin+1,ui−1n+1,uin,ui+1n.
Also, we perturbui+1n by an amount ε>0, the difficulty in verifying these relations for discrete (13) comes from the nonlinear term αin. Consider αi′n as follows:(36)αi′n=12σ2xi21+Ψa2xi2ui+1n+ε−2uin+ui−1nh2.
BecauseΨ is an increasing function [29], hence(37)Ψa2xi2ui+1n+ε−2uin+ui−1nh2≥Ψa2xi2ui+1n−2uin+ui−1nh2,and then, αi′n≥αin>0. Therefore,(38)sαi′n1+sαi′n≥sαin1+sαin,hence sαi′n/1+sαi′nui+1n+ε>sαin/1+sαinui+1n, and finally,(39)Fiuin+1,ui−1n+1,uin,ui+1n+ε>Fiuin+1,ui−1n+1,uin,ui+1n.
In the same manner, we perturbuin+1 by ε>0. Then, we have(40)Fiuin+1+ε,ui−1n+1,uin,ui+1n=−uin+1−ε+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n<−uin+1+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n=Fiuin+1,ui−1n+1+ε,uin,ui+1n.
In the case of perturbinguin by ε>0, we consider αi″n as follows:(41)αi″n=12σ2xi21+Ψa2xi2ui+1n−2uin+ε+ui−1nh2.
As−2uin+ε<−2uin and because Ψ is an increasing function,(42)Ψa2xi2ui+1n−2uin+ε+ui−1nh2≤Ψa2xi2ui+1n−2uin+ui−1nh2,and then, αi″n≤αin. By supposing k≤h2/αin and then sαin≤1, we have(43)1−sαi″n1+sαi″n≥1−sαin1+sαin.
Therefore,1−sαi″n/1+sαi″nuin+ε>1−sαin/1+sαinuin, and finally,(44)Fiuin+1,ui−1n+1,uin+ε,ui+1n>Fiuin+1,ui−1n+1,uin,ui+1n.


### 4.3. Stability

Another important feature of a finite difference equation is its stability. It is a property concerned with the behavior of errors produced in the finite difference solution due to errors introduced in a previously calculated solution. In this subsection, we show that the Saul’yev scheme for the nonlinear equation (1) is unconditionally stable by using the matrix method.Equation (12) can be written in the matrix form(45)Anun+1=Bnun,where(46)An=1+sα0n0000…0−sα1n1+sα1n000…00−sα2n1+sα2n00…000−sα3n1+sα3n0…0⋮⋮⋮⋮⋮⋱⋮00000−sαMn1+sαMn,Bn=1−sα0nsα0n000…001−sα1nsα1n00…0001−sα2nsα2n0…0⋮⋮⋮⋮⋮⋱⋮000001−sαM−1nsαM−1n0000001−sαMn.Consideringuexactin−uappin=ein, we will have(47)En=e1n,e2n,…,eM−1n,En+1=e1n+1,e2n+1,…,eM−1n+1.For the scheme given by (45), the error equation at the n+1th level can be written as En+1=An−1BnEn. Eigenvalues of matrix M=An−1Bn are as(48)λM=λBnλAn=1−sα0n1+sα0n,1−sα1n1+sα1n,…,1−sαMn1+sαMn.By Theorem2, 1+Ψ>0, and hence, αin>0, for all n and i. On the other hand because s>0, hence we have(49)1−sαin<1+sαin,i=0,…,M.Therefore, the spectral radiusρM=max1−sαin/1+sαin,i=0,…,M<1, and the Saul’yev finite difference scheme is unconditionally stable.


## 4.1. Consistency

As said in [30], dealing with reliable numerical schemes, the consistency of the difference scheme with the equation is a necessary requirement because this means that the exact theoretical solution of the partial differential equation approximates well to the solution of the difference equation as the step sizes tend to zero. In other words, the problem of consistency is the problem of finding the condition for which a discrete problem is an approximation of the corresponding continuous problem. Let us consider equation (1) with exact solution u as Lu=0, and let Fuin=0 represent the finite difference scheme (11). Hence,(14)Tinu=Fuin−Luin=uin+1−uink−σ0221+Ψa2xi2△inxi2δin−uτ−σ0221+Ψa2x2uxxx2uxxin.It will be shown thatTinu⟶0 as h⟶0 and k⟶0. By means of Taylor series, we have(15)δin=uxx|in−khEin1,in which(16)Ein1=∂2u∂x∂τ|η1n,η1∈0,B,Ein1≤uin1max=maxτ∈0,T∂2u∂τ∂x;0≤x≤B.In the same state,(17)Δin=uxx|in+h2Ein2,in which(18)Ein2=112∂4u∂x4|η2n,η2∈0,B,(19)Ein2≤112uin2max=112maxτ∈0,T∂4u∂x4;0≤x≤B.Also,(20)uin+1−uink=uτ|in+kEin3,in which(21)Ein3=12∂2u∂τ2|iξ3,ξ3∈0,T,(22)Ein3≤12uin3max=12maxτ∈0,T∂2u∂τ2;0≤x≤B.Hence, by (15) and (17),(23)1+Ψa2xi2△inδin−1+Ψa2xi2uxx|inuxx|in=1+Ψa2xi2uxx|in+a2xi2h2Ein2uxx|in−kh1+Ψa2xi2uxx|in+a2xi2h2Ein2Ein1−1+Ψa2xi2uxx|inuxx|in.Let us denoteAin=a2xi2uxx|in, ΔAin=a2xi2h2Ein2 and ΔBin=a2xi2Ein1. It should be noted that the use of Δ in (9) will not cause any ambiguity here. We see(24)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in=1+ΨAin+ΔAinAin−kh1+ΨAin+ΔAinΔBin−1+ΨAinAin1a2xi2<1+ΨAin+ΔAinAin+1+ΨAin+ΔAinΔAin−1+ΨAinAin1a2xi2=1+ΨAin+ΔAinAin+ΔAin−1+ΨAinAin1a2xi2.As we introduce the functiong in Theorem 1, it follows that(25)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in≤gAin+ΔAin−gAin+ΔAina−2xi−2.Applying the mean value theorem for the functiong, we will have(26)gAin+ΔAin−gAin=g′ηΔAin.Therefore,(27)1+Ψa2xi2Δinδin−1+Ψa2xi2uxx|inuxx|in≤ΔAin1+g′ηa−2xi−2.For the truncation error, we will see(28)Tinu=−σ022xi21+Ψa2xi2△inδin−1+Ψa2xi2uxx|inuxx|in+kEin3.Hence,(29)Tin<−σ022h21+g′ηEin2xi2+kEin3,where the upper bound of Ein2, Ein3, and g′ are introduced in (19), (22), and Theorem 1, respectively.


## 4.2. Monotonicity

In the numerical scheme(30)Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n=0,we say it is monotonicity preserving, if any positive perturbation to any of ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n produces a positive perturbation of uin+1. If Fi is differentiable, then this is equivalent to(31)∂uin+1∂ui−1n+1=−∂Fi/∂ui−1n+1∂Fi/∂uin+1≥0,∂uin+1∂ui+1n+1=−∂Fi/∂ui+1n+1∂Fi/∂uin+1≥0,∂uin+1∂uin=−∂Fi/∂uin∂Fi/∂uin+1≥0,∂uin+1∂ui−1n=−∂Fi/∂ui−1n∂Fi/∂uin+1≥0,∂uin+1∂ui+1n=−∂Fi/∂ui+1n∂Fi/∂uin+1≥0.In the case of nondifferentiableFi, we will use the following definition of monotonicity [31]. A discretization of form (30) is monotone if either ∀i(32)Fiuin+1,ui−1n+1+εi−1n+1,ui+1n+1+εi+1n+1,uin+εin,ui−1n+εi−1n,ui+1n+εi+1n≥Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,Fiuin+1+εin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n≤Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,or ∀i(33)Fiuin+1,ui−1n+1+εi−1n+1,ui+1n+1+εi+1n+1,uin+εin,ui−1n+εi−1n,ui+1n+εi+1n≤Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n,Fiuin+1+εin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n≥Fiuin+1,ui−1n+1,ui+1n+1,uin,ui−1n,ui+1n.In many problems such as financial applications of PDEs, finite difference schemes should produce positive values, but not all schemes have this property. In the next theorem, we will see the monotonicity of the Saul’yev scheme.Theorem 3.
If the time step in numerical scheme (13) is selected, such that(34)k≤h2αin,0≤i≤M,0≤n≤N,then the scheme is monotone.Proof.
We consider perturbingui−1n+1 by an amount ε>0. Because sαin/1+sαin>0,(35)Fiuin+1,ui−1n+1+ε,uin,ui+1n=−uin+1+sαin1+sαinui−1n+1+ε+1−sαin1+sαinuin+sαin1+sαinui+1n>−uin+1+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n=Fiuin+1,ui−1n+1,uin,ui+1n.
Also, we perturbui+1n by an amount ε>0, the difficulty in verifying these relations for discrete (13) comes from the nonlinear term αin. Consider αi′n as follows:(36)αi′n=12σ2xi21+Ψa2xi2ui+1n+ε−2uin+ui−1nh2.
BecauseΨ is an increasing function [29], hence(37)Ψa2xi2ui+1n+ε−2uin+ui−1nh2≥Ψa2xi2ui+1n−2uin+ui−1nh2,and then, αi′n≥αin>0. Therefore,(38)sαi′n1+sαi′n≥sαin1+sαin,hence sαi′n/1+sαi′nui+1n+ε>sαin/1+sαinui+1n, and finally,(39)Fiuin+1,ui−1n+1,uin,ui+1n+ε>Fiuin+1,ui−1n+1,uin,ui+1n.
In the same manner, we perturbuin+1 by ε>0. Then, we have(40)Fiuin+1+ε,ui−1n+1,uin,ui+1n=−uin+1−ε+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n<−uin+1+sαin1+sαinui−1n+1+1−sαin1+sαinuin+sαin1+sαinui+1n=Fiuin+1,ui−1n+1+ε,uin,ui+1n.
In the case of perturbinguin by ε>0, we consider αi″n as follows:(41)αi″n=12σ2xi21+Ψa2xi2ui+1n−2uin+ε+ui−1nh2.
As−2uin+ε<−2uin and because Ψ is an increasing function,(42)Ψa2xi2ui+1n−2uin+ε+ui−1nh2≤Ψa2xi2ui+1n−2uin+ui−1nh2,and then, αi″n≤αin. By supposing k≤h2/αin and then sαin≤1, we have(43)1−sαi″n1+sαi″n≥1−sαin1+sαin.
Therefore,1−sαi″n/1+sαi″nuin+ε>1−sαin/1+sαinuin, and finally,(44)Fiuin+1,ui−1n+1,uin+ε,ui+1n>Fiuin+1,ui−1n+1,uin,ui+1n.


## 4.3. Stability

Another important feature of a finite difference equation is its stability. It is a property concerned with the behavior of errors produced in the finite difference solution due to errors introduced in a previously calculated solution. In this subsection, we show that the Saul’yev scheme for the nonlinear equation (1) is unconditionally stable by using the matrix method.Equation (12) can be written in the matrix form(45)Anun+1=Bnun,where(46)An=1+sα0n0000…0−sα1n1+sα1n000…00−sα2n1+sα2n00…000−sα3n1+sα3n0…0⋮⋮⋮⋮⋮⋱⋮00000−sαMn1+sαMn,Bn=1−sα0nsα0n000…001−sα1nsα1n00…0001−sα2nsα2n0…0⋮⋮⋮⋮⋮⋱⋮000001−sαM−1nsαM−1n0000001−sαMn.Consideringuexactin−uappin=ein, we will have(47)En=e1n,e2n,…,eM−1n,En+1=e1n+1,e2n+1,…,eM−1n+1.For the scheme given by (45), the error equation at the n+1th level can be written as En+1=An−1BnEn. Eigenvalues of matrix M=An−1Bn are as(48)λM=λBnλAn=1−sα0n1+sα0n,1−sα1n1+sα1n,…,1−sαMn1+sαMn.By Theorem2, 1+Ψ>0, and hence, αin>0, for all n and i. On the other hand because s>0, hence we have(49)1−sαin<1+sαin,i=0,…,M.Therefore, the spectral radiusρM=max1−sαin/1+sαin,i=0,…,M<1, and the Saul’yev finite difference scheme is unconditionally stable.


## 5. Numerical Experiments and Discussion

In this section, we use a 1.80 GHz Intel(R) Core(TM)i5−3337U with 4 GB memory for all computations. The scheme was implemented in MATLAB R2013a.Comparing an explicit method, we consider the classical forward time central spaceFTCS scheme [32]. We solve nonlinear equation (1) with (2), (3) in the matrix form un+1=Bnun, and 0≤n≤N−1, in which matrix Bn can be written as(50)Bn=1−2sα0nsα0n000…0sα1n1−2sα1nsα1n00…00sα2n1−2sα2nsα2n0…0⋮⋮⋮⋮⋮⋱⋮0000sαM−1n1−2sαM−1nsαM−1n00000sαMn1−2sαMn.Bn is a M+1×N+1 tri-diagonal matrix. Following choice of parameters σ=0.005,E=40,a=0.005,M=20,N=20,andT=0.5 on interval 0,100 are used and shown in Figure 1. Elapsed time is 0.274005 seconds.Figure 1
FTCS method byM = 20, N = 20, E = 40, σ=0.005, T = 0.5, and a = 0.005 on 0,100.In comparison by implicit schemes, first we consider the classical backward time central spaceBTCS scheme [32]. The difference equation for (1) by BTCS can be written as(51)−sαinui−1n+1+1+2sαinuin+1−sαinui+1n+1=uin,for 1≤i≤M−1 and 0≤n≤N−1. This scheme is unconditionally stable for all s>0. It is worth to note that we applied Thomas algorithm for solving systems in the BTCS program. Parameters σ=0.005,E=50,a=0.005,M=40,N=100,andT=0.5 are used in computational. Abovementioned parameters are plotted in Figure 1. Elapsed time for this computation is 2.283767 seconds. We compare the Saul’yev solution of (1) with the BTCS solution for the choice of parameters M=40,N=100,σ=0.005,E=50,a=0.005,andT=0.5. It is plotted in Figure 2. Here, the elapsed time for BTCS is 0.886669 seconds, and the elapsed time for Saul’yev is 0.064191 seconds. Absolute error of BTCS and Saul’yev schemes is plotted in Figure 3 too. Errors are computed, emax=9.52e−2 and emean=1.4e−3, in which(52)emax=maxmaxuSaul′yev−uBTCS.Figure 2
BTCS scheme forM = 40, N = 100, E = 50, σ=0.005, a = 0.005, and T = 0.5 on (0, 100).Figure 3
(a) BTCS. (b) Saul’yev schemes. (c) Error of BTCS and Saul’yev forM=40,N=100,E=50,a=0.005,σ=0.005,andT=0.5 on (0, 100).
(a)(b)(c)We note thatmaxuSaul′yev−uBTCS returns a row vector E containing the maximum element from each column matrix uSaul′yev−uBTCS; then, (52) returns the largest element in E. In the same state, we have mean instead of max in (52) for emean. Also, the error of BTCS and Saul’yev schemes for M=40,N=80,E=50,a=0.005,σ=0.005,andT=0.5 on (0, 100) is plotted in Figure 4.Figure 4
Error of BTCS and Saul’yev schemes forM = 40, N = 80, E = 50, a = 0.005, σ=0.005, and T = 0.5 on (0, 100).The second implicit method which we compare with the Saul’yev scheme is the Crank–Nicolson (CN) method [32]. We solve nonlinear equation (1) with (2) and (3) by CN as(53)−sαinui+1n+1+21+sαinuin+1−sαinui−1n+1=sαinui−1n+21−sαinuin+sαinui+1n,for each i=1,…,M−1. This scheme is unconditionally stable for all s>0 like the BTCS scheme. This procedure is applied for the parameters M=120,N=120,σ=0.005,E=50,a=0.005,andT=0.5 on interval 0,100 and is shown in Figure 5. Here, elapsed time is 1.608705 seconds. We applied Thomas algorithm in the CN program as we used it in the BTCS program.Figure 5
CN method,M = 120, N = 120, σ=0.005, E = 50, a = 0.005, and T = 0.5 on (0, 100).On the other hand, we compare Saul’yev and CN in Figure6 for parameters M=40,N=80,σ=0.001,E=50,a=0.001,andT=1. Elapsed time for CN is 0.192018 seconds, and elapsed time for Saul’yev is 0.128141 seconds. The difference between two solutions by Saul’yev and CN is shown in Figure 7. In this computation, we see emax=3.6e−3 and emean=4.4455e−5.Figure 6
(a) CN and (b) Saul’yev schemes,M = 40, N = 80, σ=0.001, E = 50, a = 0.001, and T = 1 on (0, 100).
(a)(b)Figure 7
Error of CN and Saul’yev,M = 40, N = 100, σ=0.001, E = 50, a = 0.001, and T = 1 on (0, 100).Among four schemes FTCS, BTCS, CN, and Saul’yev, only Saul’yev is both explicit and unconditionally stable. As can be seen in Table1, the elapsed time for the Saul’yev scheme is shortest than BTCS and CN, which are implicit. In these computations, the used parameters are M=40,N=80,σ=0.005,a=0.005,E=50,andT=1 on 0,100.Table 1
M = 40, N = 100, σ=0.005, a = 0.005, E = 50, and T = 1, on 0,100.
SchemeExplicit/ImplicitElapsed time (seconds)FTCSExplicit0.116025BTCSImplicit0.886669CNImplicit0.171459Saul’yevExplicit0.125903We computedemax and emean of BTCS and CN for some a and are shown in Table 2 for parameters M=40,N=100,E=40,T=0.5,andσ0=0.005 on 0,100. We note if a=0, partial differential equation (1) will be a quasilinear equation instead of the nonlinear equation. It is widely known that the accuracy of CN is higher than BTCS. Table 2 shows that Saul’yev solutions are closer to CN solutions.Table 2
Implicit schemes in comparison of Saul’yev by parametersM = 30, N = 60, E = 40, T = 0.5, and σ0=0.005 on 0,100.
aemaxBTCSemeanBTCSemaxCNemeanCN0.021.315e−12.5e−36.71e−21.2e−30.0151.066e−11.9e−35.42e−29.5952e−40.018.84e−21.5e−34.48e−27.6978e−40.0057.72e−21.3e−33.91e−26.5879e−407.35e−21.3e−33.72e−26.2228e−4


## 6. Conclusions

This article has outlined an approach for the study of a nonlinear partial differential equation with initial and boundary conditions. We studied the numerical solution of the problem by means of a particular kind of the finite difference schemes. Saul’yev scheme is an explicit method for solving partial differential equations. An advantage of this scheme is the little amount of CPU times used in determining the solution compared to the implicit schemes. In contrast to more explicit schemes, the Saul’yev finite difference scheme is unconditionally stable. Hence, there is no restriction of the size of the time step. Several numerical methods have been considered and compared for computing the solution of the nonlinear partial differential equation. The theoretical analysis carried out in this paper shows that the method is consistent and monotone too. Practical implications of the scheme demonstrated the robustness of the scheme. We employed the Saul’yev scheme for a fully nonlinear partial differential equation with initial and boundary conditions, in which this scheme not only is explicit but also is unconditionally stable. As we shall see, the scheme is consistent. We studied the numerical solution of the problem by means of a particular kind of the finite difference schemes. For future works, this approach can be employed to solve these kinds of nonlinear problems.



---

*Source: 9647416-2020-11-24.xml*