A linearized finite-difference method for the solution of some mixed concave and convex non-linear problems

doi:10.1016/j.amc.2007.07.051

Applied Mathematics and Computation

Volume 197, Issue 1, 15 March 2008, Pages 1-10

https://doi.org/10.1016/j.amc.2007.07.051 Get rights and content

Abstract

In the present work, a linearized finite-difference scheme is proposed in order to approximate the solution of some non-linear equation characterized by mixed concave and convex non-linearities. It is proved that the scheme is uniquely solvable and convergent. The resulting method is also analyzed for consistency and stability. Some numerical examples illustrating the method described in our work are given.

Introduction

The paper is devoted to the study of the Schrödinger equation with mixed concave and convex non-linearities such as $\{\begin{matrix} i \frac{\partial u}{\partial t} + \frac{\partial^{2} u}{\partial x^{2}} + | u |^{p - 1} u + ε | u |^{q - 1} u = 0, & L_{0} < x < L_{1}, t > t_{0} \\ u (t_{0}, x) = u_{0} (x), & L_{0} ⩽ x ⩽ L_{1} \\ \frac{\partial u}{\partial x} (t, L_{0}) = \frac{\partial u}{\partial x} (t, L_{1}) = 0, & t ⩾ t_{0} \end{matrix}$ where $i = \sqrt{- 1}$ , ε ⩾ 0, t₀ ⩾ 0, L₀ and L₁ are real parameters, u₀ is a complex continuous function of x and u(t, x) = v(t, x) + iw(t, x).

Non-linear Schrödinger equation plays an important role in the modeling of many phenomena such as optics of non-linear media, plasmas, condensed matter physics, etc. It then attracted the interest of researchers in physics and applied mathematics. However, it remains many open questions to do about this equation. We mention as examples the models of Bose–Einstein condensation and the stabilized solitons. In the latter case, the non-linear Schrödinger equation gives rise to soliton solutions in which the explicit expression can be well defined. For example, in the case of the non-linear cubic equation, u is given by (ε = 0 and p = 3) $u (x, t) = \sqrt{2 a} \exp (i (\frac{1}{2} cx - θ t + φ)) sech (\sqrt{a} (x - ct) + ϕ),$ where a, c and θ are some appropriate constants. It consists in fact of a soliton-type disturbance which travels with the speed c. For backgrounds on such a subject, the readers are referred to [5], [6], [10].

The idea behind this work is inspired from our works in [1], [4] and the works of Bratsos in [5] with some necessary and technical modifications. It consists in transforming the initial boundary-value problem (1) into a linear algebraic system. We develop a numerical method by replacing the time and the space partial derivatives by finite-difference replacements and the non-linear term by a linearized one. We then analyze the resulting method for local truncation error and stability and we prove that the scheme is uniquely solvable and convergent. Some numerical examples are studied at the end of the paper in order to validate our theoretical results. The paper is organized as follows: In Section 2, we introduce the finite difference scheme and then we transform problem (1) into a linear algebraic system. In Section 3, we prove that such an algebraic system is uniquely solvable. In Section 4, we study the convergence of the difference scheme. In Section 5, we analyze the consistency and the stability of the method. This is done by some preliminary computations based on Euler’s formula. In Section 6, a numerical example is exposed in order to validate the scheme.

We recall finally that the importance and the novelty of our work here is not in the method used of barycenter interpolation or extrapolation finite-difference scheme. This however is a well known procedure in numerical treatments of partial differential equations and is widely used. The novelty and the importance is firstly due to the presence of concave and convex non-linear terms and of non-lipschitzian term which may induce some bad error estimates in the rate of convergence. Recall that most of the studies that have been done about the non-linear Schrodinger equation have focused mainly on $i \frac{\partial u}{\partial t} + \frac{\partial^{2} u}{\partial x^{2}} \pm | u |^{p - 1} u = 0,$ where the non-linear term is a convex function depending in some cases on a Sobolev critical exponent $p_{c} = \frac{4}{N - 2}$ . Here N stands for the space dimension. We cite as the most recent studies in such a direction [5], [6], [7], [8], [9], [11], [12]. For more details, the readers can be referred to these cited references and those therein.

Throughout this paper, the constants used are generic, independent of step sizes h and l and not necessary the same at different occurrences.

Section snippets

Finite-difference scheme

Consider a time step l = Δt and denote for $k \in N$ , t_k = t₀ + kl. Consider also a space step $h = Δ x = \frac{L_{1} - L_{0}}{N + 1}$ with some integer N. We subdivide the interval [L₀, L₁] into intervals [x_m, x_m+1] where x_m = L₀ + mh for m = 0, … , N + 1. Denote $u_{m}^{k}$ the approximation of u(t_k, x_m) and $U_{m}^{k}$ the numerical solution. We introduce the following notations $\begin{matrix} \partial_{m}^{k} U = \frac{U_{m}^{k + 1} - U_{m}^{k - 1}}{2 l} and Δ_{k}^{m} U = \frac{U_{m + 1}^{k} - 2 U_{m}^{k} + U_{m - 1}^{k}}{h^{2}}, \\ \frac{\partial U_{m}^{k}}{\partial t} = λ \partial_{m - 1}^{k} U + (1 - 2 λ) \partial_{m}^{k} U + λ \partial_{m + 1}^{k} U, \\ \frac{\partial U_{m}^{k}}{\partial x} = \frac{U_{m + 1}^{k} - U_{m - 1}^{k}}{2 h}, \\ \frac{\partial^{2} U_{m}^{k}}{\partial x^{2}} = μ Δ_{k + 1}^{m} U + (1 - 2 μ) Δ_{k}^{m} U + μ Δ_{k - 1}^{m} U . \end{matrix}$ Denote g(u) = ∣u∣^p−1u + ε∣u∣^q−1u. We then

Solvability of the difference scheme

From Eqs. (6), (7), (8) we obtain the following matrix form: ${AU}^{k + 1} = {BU}^{k} + {CU}^{k - 1},$ where A, B and C are the matrices defined below: $\begin{matrix} A = (\begin{matrix} a_{1} + a_{2} & a_{2} & 0 & \dots & \dots & 0 \\ a_{2} & a_{1} & a_{2} & ⋱ & ⋱ & ⋮ \\ 0 & ⋱ & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & a_{2} & a_{1} & a_{2} \\ 0 & \dots & \dots & 0 & a_{2} & a_{1} + a_{2} \end{matrix}), \\ B = (\begin{matrix} b_{1} + b_{2} & b_{2} & 0 & \dots & \dots & 0 \\ b_{2} & b_{1} & b_{2} & ⋱ & ⋱ & ⋮ \\ 0 & ⋱ & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & b_{2} & b_{1} & b_{2} \\ 0 & \dots & \dots & 0 & b_{2} & b_{1} + b_{2} \end{matrix}), \end{matrix}$ and finally $C = (\begin{matrix} c_{1} + c_{2} & c_{2} & 0 & \dots & \dots & 0 \\ c_{2} & c_{1} & c_{2} & ⋱ & ⋱ & ⋮ \\ 0 & ⋱ & ⋱ & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & c_{2} & c_{1} & c_{2} \\ 0 & \dots & \dots . & 0 & c_{2} & c_{1} + c_{2} \end{matrix}),$ where the coefficients a_j’s, b_j’s and c_j’s are those defined above.

The following result is immediate.

Theorem 3.1

The problem (9) is uniquely solvable.

Convergence of the difference scheme

The main result of this section is to prove the convergence of the difference scheme. We will prove precisely that the method is unconditionally convergent. To do this, assume that $U_{m}^{k} = e^{i k ψ} e^{i m θ}$ , where ψ is a complex parameter and θ is real. Denote X = e^iψ and Y = e^iθ. By replacing in Eq. (6) and canceling e^i(k−1)ψe^i(m−1)θ we obtain $A_{0} (Y) X^{2} + B_{0} (Y) X + C_{0} (Y) = 0,$ where $A_{0} (Y) = (2 μ σ + i λ) - (4 μ σ - i (1 - 2 λ)) Y + (2 μ σ + i λ) Y^{2},$ $B_{0} (Y) = 2 (1 - 2 μ) σ - (4 (1 - 2 μ) σ - 2 \tilde{g} ν l) Y + 2 (1 - 2 μ) σ Y^{2}$ and $C_{0} (Y) = (2 μ σ - i λ) - (4 μ σ - 2 \tilde{g} (1 - ν) l + i (1 - 2 λ) Y + (2 μ σ - i λ) Y^{2} .$ By

Consistency and stability of the method

The principal part of the local truncation error of the method arising from the scheme (2) is $L (t, x) = \frac{l}{2} \frac{\partial^{2} u}{\partial t^{2}} + h^{2} (- \frac{i}{12} \frac{\partial^{4} u}{\partial x^{4}} + λ \frac{\partial^{3} u}{\partial t \partial x^{2}}) + o ({lh}^{2}) .$ So, it tends to 0 as l and h tend to 0. This means that the method is consistent.

For the stability of the method, remark that Eq. (6) can be written in the following form $(2 μ σ + i λ) (U_{m + 1}^{k + 1} - 2 U_{m}^{k + 1} + U_{m - 1}^{k + 1}) + {iU}_{m}^{k + 1} = - 2 (1 - 2 μ) σ (U_{m + 1}^{k} - 2 U_{m}^{k} + U_{m - 1}^{k}) + (- 2 μ σ + i λ) (U_{m + 1}^{k - 1} - 2 U_{m}^{k - 1} + U_{m - 1}^{k - 1}) + {iU}_{m}^{k - 1} - 2 \tilde{g} l (ν U_{m}^{k} + (1 - ν) U_{m}^{k - 1}) .$ Let for m and k $S_{m}^{k} = U_{m}^{k} - {\tilde{U}}_{m}^{k} .$ Eq. (16) implies that $(2 μ σ + i λ) (S_{m}$

Numerical implementations

We present in this section some illustrative examples in order to validate the methods and the results just described above. We consider the propagation of a soliton and the interaction of two colliding solitons. But before going on doing this, we want to recall firstly the fact that, to compute a solution, we need first to make some additional hypothesis based on the fact that one can not compute the solution in the whole real line. We therefore must suppose that it is compactly supported. We

Some additional results

We expose in this section some additional results that we have noticed in our work. The first one deals with the rate of convergence of the scheme described in our paper. Its proof is not complicated and follows quite similar techniques as in [1], [4]. For this reason, it will not be exposed here.

Theorem 7.1

Let u and Uⁿ be the solutions of (1) and ((2), (3), (4), (5)), respectively. Assume further that u is sufficiently regular and denote uⁿ its value at the time t_n. Then, for l small enough, we have $| | U^{n} - u$

Conclusion

In this paper, we analyzed a linearized three-level difference scheme for the Schrödinger equation with concave and convex non-linearities. The idea in the numerical method is based on replacing the time and the space partial derivatives by parametric finite-difference replacements and the non-linear term by a linearized one. We proved that the scheme is uniquely solvable and convergent. The resulting method is also analyzed for local truncation error and stability. The numerical

Acknowledgements

The authors thank Prof. Khaled Omrani from the Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Tunisie for the helpful suggestions and discussions with him. The first author thanks Professor Mohamed Amara, Director of Laboratoire de Mathématiques Appliquées, Université de Pau, France, where this work has been started.

References (12)

S. Keraani
On the blow-up phenomenon of the critical Schrödinger equation
J. Funct. Anal.
(2006)
Y. Martel et al.
Multi solitary waves for non-linear Schrödinger equations
Ann. I. H. poincaré.
(2006)
A. Ben Mabrouk, M.L. Ben Mohamed, Theoretical and numerical studies of some mixed sublinear-superlinear elliptic and...
A. Ben Mabrouk et al.
On some critical and slightly super-critical sub-superlinear equations
Far East J. Appl. Math.
(2006)
A. Ben Mabrouk et al.
Nodal solutions for some non-linear elliptic equations
Appl. Math. Comput.
(2007)
A. Ben Mabrouk et al.
Finite-difference approximate solutions for a mixed sub-superlinear equation
Appl. Math. Comput.
(2007)

There are more references available in the full text version of this article.

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A linearized finite-difference method for the solution of some mixed concave and convex non-linear problems

Abstract

Introduction

Section snippets

Finite-difference scheme

Solvability of the difference scheme

Convergence of the difference scheme

Consistency and stability of the method

Numerical implementations

Some additional results

Conclusion

Acknowledgements

J. Funct. Anal.

Ann. I. H. poincaré.

On some critical and slightly super-critical sub-superlinear equations

Far East J. Appl. Math.

Nodal solutions for some non-linear elliptic equations

Appl. Math. Comput.

Finite-difference approximate solutions for a mixed sub-superlinear equation

Appl. Math. Comput.