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

Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Sumgayit State University, Baku Str., 1, Sumgayit AZ5008, Azerbaijan Azerbaijan University, J. Hajibeyli, 71, Baku AZ1007, Azerbaijan Baku State University,Institute for Physical Problems, Z.Khalilov, 23, AZ1148, Baku, Azerbaijan Institute of Mathematics and Mechanics ANAS, B.Vahabzade, 9, Baku AZ1148, Azerbaijan


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. e ADE scheme is documented by Saul'yev in [8] and is the abbreviation of the phrase "alternating direction explicit" method. e Saul'yev scheme is explicit; therefore, it needs less computational works in comparison with implicit schemes. It is unconditionally stable and first-order accurate.
e 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 advectiondiffusion problems. ey 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.
e 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]. e 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. e 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. e contents of this paper are the following: in the next section, we introduce a nonlinear partial differential equation. In Section 3, we apply the Saul'yev scheme for the nonlinear problem. e 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.

Nonlinear Partial Differential Equation
Consider the following nonlinear partial differential equation: where σ 0 and a are the given points, with initial and Dirichlet boundary conditions, where E is a known value. Function Ψ(A) is the solution of the following nonlinear ordinary differential equation: e 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, eorem 1 plays an important role in the consistency of the scheme in Section 4. Moreover, the solution of equation (3) is defined in eorem 2, which will be used in Section 5.

Theorem 1. Let g(A) � AΨ(A). en, g(A)
is a continuously differentiable function at A � 0 and satisfies where R is the real numbers set, and A 2 ≃ 9.58, Proof. see [28].

Theorem 2. e unique solution of equation (3) is defined as
Proof. See [29]. Grid of mesh points is defined by where x i � ih, and τ n � nk. Notation u n i is used for u(x i , τ n ) and approximated the solution of our fully nonlinear PDE in x i and τ n . Here, we use the forward finite difference approximation for the first-order time derivative as Also, we introduce two operators (Δ, central, and δ, Saul'yev) for approximation of the second-order derivative (z 2 u/zx 2 ) as Substituting approximations (8), (9), and (10) into equation (1) leads to the following difference equation: In function Ψ, we approximate the second-order spatial derivative by Δ n i . If we consider δ n (11), the scheme can only be solved by a nonlinear iteration in each time step which is quite timeconsuming. By denoting (σ 2 0 /2)(1 + Ψ(a 2 x 2 i Δ n i ))x 2 i � α n i and s � (k/h 2 ), we get the difference equation: for i � 1, 2, . . . , M − 1. Although the approximation (12) does not appear explicit because u n+1 i and u n+1 i−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 u n+1 i is unknown, and the scheme will be explicit.
According to eorem 2, the range of function Ψ is the interval (−1, +∞); therefore, 1 + Ψ > 0, and hence α n i > 0, for all n and i. On the other hand, it is obvious that s > 0; therefore, in (13), we have (1 + sα n i ) ≠ 0. e next section contains an analysis of the scheme described above.

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 L(u) � 0, and let F(u n i ) � 0 represent the finite difference scheme (11). Hence, It will be shown that T n i (u) ⟶ 0 as h ⟶ 0 and k ⟶ 0. By means of Taylor series, we have in which In the same state, in which in which Mathematical Problems in Engineering 3 Hence, by (15) and (17), (1). It should be noted that the use of Δ in (9) will not cause any ambiguity here. We see As we introduce the function g in eorem 1, it follows that Applying the mean value theorem for the function g, we will have erefore, For the truncation error, we will see Hence, where the upper bound of E n i (2), E n i (3), and |g ′ | are introduced in (19), (22), and eorem 1, respectively.

Monotonicity. In the numerical scheme
we say it is monotonicity preserving, if any positive perturbation to any of u n+1 i−1 , u n+1 i+1 , u n i , u n i−1 , u n i+1 produces a positive perturbation of u n+1 i . If F i is differentiable, then this is equivalent to 4 Mathematical Problems in Engineering In the case of nondifferentiable F i , we will use the following definition of monotonicity [31]. A discretization of form (30) is monotone if either ∀i or ∀i 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. (13) is selected, such that

Theorem 3. If the time step in numerical scheme
then the scheme is monotone.
Proof. We consider perturbing u n+1 i−1 by an amount ε > 0. Because (sα n i /(1 + sα n i )) > 0, Also, we perturb u n i+1 by an amount ε > 0, the difficulty in verifying these relations for discrete (13) comes from the nonlinear term α n i . Consider α ′n i as follows: Because Ψ is an increasing function [29], hence and then, α ′n i ≥ α n i > 0. erefore, hence (sα ′n i /1 + sα ′n i )(u n i+1 + ε) > (sα n i /1 + sα n i )u n i+1 , and finally, In the case of perturbing u n i by ε > 0, we consider α ″n i as follows: As −2(u n i + ε) < − 2u n i and because Ψ is an increasing function, and then, α ″n i ≤ α n i . By supposing k ≤ (h 2 /α n i ) and then sα n i ≤ 1, we have 1 − sα ″n 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 Considering u(exact) n i − u(app) n i � e n i , we will have E n � e n 1 , e n 2 , . . . , e n M−1 , E n+1 � e n+1 1 , e n+1 2 , . . . , e n+1 M−1 . (47) By eorem 2, 1 + Ψ > 0, and hence, α n i > 0, for all n and i. On the other hand because s > 0, hence we have erefore, the spectral radius ρ(M) � max | (1 − sα n i / 1 + sα n i )|, i � 0, . . . , M} < 1, and the Saul'yev finite difference scheme is unconditionally stable.

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. e scheme was implemented in MATLAB R2013a.
Comparing an explicit method, we consider the classical forward time central space (FTCS) scheme [32]. We solve nonlinear equation (1) with (2), (3) in the matrix form u n+1 � B n u n , and 0 ≤ n ≤ N − 1, in which matrix B n can be written as 6 Mathematical Problems in Engineering B n 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, and T � 0.5 on interval (0, 100) are used and shown in Figure 1. Elapsed time is 0.274005 seconds.
In comparison by implicit schemes, first we consider the classical backward time central space (BTCS) scheme [32]. e difference equation for (1) by BTCS can be written as   Mathematical Problems in Engineering 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, e max � 9.52e − 2 and e mean � 1.4e − 3 , in which We note that max|u Saul′yev − u BTCS | returns a row vector E containing the maximum element from each column matrix |u Saul′yev − u BTCS |; then, (52) returns the largest element in E. In the same state, we have mean instead of max in (52) for e mean . Also, the error of BTCS and Saul'yev schemes for M � 40, N � 80, E � 50, a � 0.005, σ � 0.005, and T � 0.5 on (0, 100) is plotted in Figure 4. e 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) for each i � 1, . . . , M − 1. is scheme is unconditionally stable for all s > 0 like the BTCS scheme. is procedure is applied for the parameters M � 120, N � 120, σ � 0.005, E � 50, a � 0.005, and T � 0.5 on interval (0, 100) and is shown in Figure 5. Here, elapsed time is 1.608705 seconds. We applied omas algorithm in the CN program as we used it in the BTCS program.
On the other hand, we compare Saul'yev and CN in Figure 6 for parameters M � 40, N � 80, σ � 0.001, E � 50, a � 0.001, and T � 1. Elapsed time for CN is 0.192018 seconds, and elapsed time for Saul'yev is 0.128141 seconds. e difference between two solutions by Saul'yev and CN is shown in Figure 7. In this computation, we see e max � 3.6e − 3 and e mean � 4.4455e − 5 .
Among four schemes FTCS, BTCS, CN, and Saul'yev, only Saul'yev is both explicit and unconditionally stable. As can be seen in Table 1, 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, and T � 1 on (0, 100).
We computed e max and e mean of BTCS and CN for some a and are shown in Table 2 for parameters M � 40, N � 100, E � 40, T � 0.5, andσ 0 � 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.

Conclusions
is 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. e 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.
Data Availability e datasets supporting the conclusions of this article are included within the article and its additional file.

Conflicts of Interest
e authors declare that they have no conflicts of interest.