Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation
The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.
In this article, the nonlocal boundary value problem for the Schrödinger equation in a Hilbert space with the self-adjoint operator is considered. The Schrödinger equation plays an important role in the modeling of many phenomena. Methods of solutions for the Schrödinger equation have been studied extensively by many researchers (see, e.g., [1–9] and the references given therein).
The idea in this work is inspired from the works [2, 3, 10, 11]. In the articles [2, 3] the existence and the uniqueness of the solution of the nonlocal boundary value problem (1.1) and its general form under some conditions are studied. In the article , to find an approximate solution of the problem (1.1), first-order of accuracy Rothe difference scheme and second-order of accuracy Crank-Nicolson difference scheme are presented. The stability estimates for the solution of this problem and the stability of these difference schemes are established.
The main aim of this paper is to study modified Crank-Nicolson difference schemes for the approximate solution of problem (1.1). The paper is organized as follows. In Section 2, we establish estimates for the stability of higher order derivatives of the solution of problem (1.1). In Section 3, the second-order of accuracy modified Crank-Nicolson difference schemes for the approximate solution of problem (1.1) are presented. The stabilities of these difference schemes are established. In Section 4, we study the convergence of these difference schemes. In Section 5, a numerical example is exposed in order to validate the schemes. A procedure involving the modified Gauss elimination method is used for solving these difference schemes.
Throughout this paper, the constants used are not necessarily the same at different occurrences.
2. Nonlocal Boundary Value Problem
Theorem 2.1. Assume that and Then there exists a unique solution of problem (1.1) and the following inequalities are satisfied:
Proof. The proof of the estimate (2.2) is given in . Now we will obtain the estimate (2.3).
It is known that for smooth data of the problem there exists a unique solution of the problem (1.1), and the following formula holds: Therefore we have So that we get the estimate Using the condition and the formula (2.6) we get where By using estimates and the assumption we get By using the estimates (2.7) and (2.11) we obtain an estimate for Then by using the estimate for the relation and the triangle inequality we can obtain estimate (2.3). This completes the proof of Theorem 2.1.
3. Difference Schemes, Stability
In this section, we present -modified Crank-Nicolson difference schemes for the approximate solutions of problem (1.1) and establish the stabilities of these difference schemes. It is assumed that for Let us associate the nonlocal boundary value problem (1.1) with the corresponding second-order of accuracy -modified Crank-Nicolson difference schemes: for the approximate solutions of this nonlocal boundary value problem. denotes here the set and , , where stands for the greatest integer part of the real number .
By , is the solution of the -modified Crank-Nicolson difference schemes for the approximate solutions of Cauchy problem Here For using the formula (3.2) and the condition we obtain where Note that, here we considered for So, for the solution of problem (3.2), we have the following formula:
Theorem 3.1. Assume that and Then the solutions of the difference schemes (3.1) satisfy the stability inequalities
Proof. Using the estimates
and the formula (3.2), we can obtain
Using the spectral representation of the self-adjoint operators one can establish
Estimate for should also be examined. By using formula (3.7), the triangle inequality, and estimates (3.11), (3.13) the following estimate is obtained:
The proof of the estimate (3.9) for the difference schemes (3.1) is based on the last estimate and estimate (3.12).
Now, estimate (3.10) will be obtained. Using (3.2), we get So that For the estimate (3.10) the two cases should be examined separately: (i) (ii) Let Then, using (3.16) we get Therefore, Estimate for should also be obtained. Using the formula (3.5) and the formula (3.16) we get So that Therefore, using the estimates (3.18) and (3.20) we obtain Then using the estimate for the relation , and the triangle inequality we get the estimate Now, let Then using the formula (3.16) and the identity we get So that Therefore, using the estimates (3.20) and (3.24), the estimate is obtained. Then, by using the estimate (3.25), the relation , and the triangle inequality we get the estimate The result (3.10) follows from the estimates (3.22) and (3.26). So the proof is complete.
Theorem 4.1. Assume that Assume also that and are continuous, then the solution of the difference scheme (3.1) satisfies the convergence estimate where does not depend on but depends on
Proof. If we subtract (1.1) from (3.1) we obtain
where and is defined by the formula
Then the difference problem (4.2) has a solution in the form (3.7), but instead of , , we take , , , respectively. Using the estimates
and the formula obtained for the solution of (4.2), we can obtain
By the estimate (3.14) we have
Therefore, in order to obtain the inequality (4.1) we need estimates for for
For , by the use of the triangle inequality, Taylor's formula, continuity of and , the estimates are obtained. From the last estimates the result follows.
5. Numerical Results
In this section, the numerical experiments of the nonlocal boundary value problem by using modified Crank-Nicolson difference scheme (3.1) are investigated. The exact solution of this problem is For the approximate solution of problem (5.1), the set of a family of grid points depending on the small parameters and is defined.
Applying the second-order of accuracy modified Crank-Nicolson difference schemes (3.1) we present following second-order of accuracy difference schemes for the approximate solutions of problem (5.1) So for each , we have system of linear equations which can be written in the matrix form as where In the above matrices entries are given as Thus, we have the second-order difference equation (5.5) with respect to with matrix coefficients. To solve this difference equation we have applied the same modified Gauss elimination method for the difference equation with respect to with matrix coefficients. Hence, we seek a solution of the matrix in the following form: where are square matrices and are column matrices defined by Note that for obtaining first we need to find As in , we take is an identity matrix, is the zero column vector.
For their comparison, first the errors computed by of the numerical solutions of problem (5.1) are recorded for different values of and where represents the exact solution and represents the numerical solution at . The results are shown in Table 1 for , and , respectively.
Second, for their comparison, the relative errors are computed by and Table 2 is constructed for , and , respectively.
In the article  it can also be found, an example that Crank-Nicolson difference scheme is divergent but modified Crank-Nicolson is convergent.
The authors are grateful to Mr. Tarkan Aydın (Bahcesehir University, Turkey) for his comments and suggestions on implementation.
M. E. Mayfield, Non-reflective boundary conditions for Schrödinger's equation, Ph.D. thesis, University of Rhode Island, Kingston, RI, USA, 1989.
X. Antoine, C. Besse, and V. Mouysset, “Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions,” Mathematics of Computation, vol. 73, no. 248, pp. 1779–1799, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Ashyralyev and A. Sirma, “A note on the modified Crank-Nicolson difference schemes for Schrödinger equation,” in Complex Analysis and Potential Theory (Proceedings of the Conference Satellite to ICM 2006), pp. 256–271, World Scientific Press, River Edge, NJ, USA, 2007.View at: Google Scholar | Zentralblatt MATH
P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izland. Voronezh. Gosud. Univ., Voronezh, Russia, 1975.