Galerkin methods for the Davey–Stewartson equations

doi:10.1016/j.amc.2018.01.044

Applied Mathematics and Computation

Volume 328, 1 July 2018, Pages 144-161

https://doi.org/10.1016/j.amc.2018.01.044 Get rights and content

Abstract

In this paper, we propose two Galerkin methods to investigate the evolution of the Davey–Stewartson equations. The extrapolated Crank–Nicolson scheme and decoupled semi-implicit multistep scheme are employed to increase the order of the time discrete accuracy, which only requires the solutions of a linear system at each time step. Four numerical experiments are presented to illustrate the features of the proposed numerical methods, such as the optimal convergence order, the conservation variable and the application in rogue waves.

Introduction

A variety of phenomena across a range of disciplines, which in applied mathematics and physics from fluid dynamics, solid state physics, quantum machines, plasmas physics to nonlinear optics and so forth, and in chemistry and biology as well [1], [2], are described by nonlinear dispersive partial differential equations. For instance, the Schrödinger equation is the fundamental governing equation in quantum mechanics and quantum field theory [3], which is used to describe many-body theory and condensed matter physics like the Bose–Einstein condensate. It also has extensive applications in nonlinear optics [4] and water waves [5]. In view of the importance of these equations, many researchers tried their best to find analytical solutions or numerical solutions to the nonlinear dispersive equations by using various methods [6], [7], [8], [9], [10], [11], [12], [13], [14].

The Davey–Stewartson (DS) equations were originally introduced by Davey and Stewartson [15] to describe the evolution of a three-dimensional weakly nonlinear wave-packet on water in finite depth, in which the water waves have one direction of travel but the amplitude of waves is modulated in two spatial directions. The equations appear in studying the long-wave-short-wave resonances [16] and describing gravity-capillarity surface wave packets in the limit of shallow water [17]. The evolution of a plasma under the action of a magnetic field in plasma physics [18] and certain (2+1)-dimensional nonlinear localized excitations can be described by DS equations. Furthermore, the (2+1)-dimensional Schrödinger equation with cubic nonlinearity fails to be integrable. However, Davey–Stewartson equations, a pair of coupled nonlinear equations in two dependent variables, were shown to be integrable [19], [20], [21] as the extension of the (1+1)-dimensional nonlinear Schrödinger equation. In fact, when the DS system is considered with various parameters in it, there are only two integrable cases which are known as DS I and DS II [21]. These two integrable cases are of elliptic-hyperbolic and hyperbolic-elliptic type, depending on the strength of surface tension. The DS II equations are thus the hyperbolic-elliptic ones, which are in particular relevant for surface gravity waves without surface tension. In addition, for DS I and DS II equations, special localized solutions, called respectively dromions and lumps, can be explicitly found.

Many works for Davey–Stewartson equations have been done in recent years. A vast method finding analytical solutions has been proposed [6], [8], [21], [22], [23], [24], [25], [26], [27], [28]. Biswas [22] considered G’/G method and the solitary wave ansatz method to study traveling wave solution and 1-soliton solutions, respectively. El-Kalaawy and Ibrahim [23] obtained the solitary wave solutions by using extended mapping method technique. Ohta and Yang [27], [28] derived the general rogue waves by the bilinear method. In parallel with the analytical studies, a surge of efforts have been devoted to the numerics of this system [11], [13], [29], [30], [31], [32], [33], [34]. Stimming and Zhang [29] presented an efficient and accurate solver for the nonlocal potential in the Davey–Stewartson equations using nonuniform Fast Fourier Transform. Besse et al. [30] proposed a time splitting spectral method for numerical study of the DS equations, including exact soliton type solutions of the hyperbolic-elliptic equations and the blow up of focusing elliptic-elliptic equations. Klein and Roidot [13] investigated the semi-classical Davey–Stewartson equations by both fourth order time-stepping and spectral method. Gao et al. [31] proposed a Galerkin method with first-order time discretization. To the best of our knowledge, there are few results on numerics of the DS equations by finite element method, though there are numerous investigations for nonlinear dispersive equations [31], [35].

In this paper, we study the numerical solutions of Davey–Stewartson equations by Galerkin finite element method with extrapolated Crank–Nicolson Scheme and decoupled multistep method. Due to the nonlinearity and coupling and complex structure system of DS equations, linearized semi-implicit schemes should be considered approximations in time direction since their implementation only requires the solution of a linear system. Among linearized schemes, the extrapolated Crank–Nicolson method is a popular one since it provides a second order accuracy in the time direction. However, linearized Crank–Nicolson method only solved the nonlinearity of the DS equations, we also need solve a coupled algebraic system. To overcome this shortcoming of coupled Crank–Nicolson method, we develop the decoupled scheme by using semi-implicit multistep method, which improves the efficiency of algorithm and remains the second order convergence rate in time as Crank–Nicolson time discretization.

The outline of this paper is as follows. In the next section, we introduce the governing equations for DS equations. In Section 3, numerical algorithms of the DS equations are given. In Section 3.1, we study the weak form and the proof of L² norm conservations. The full discretization based on linearized Crank–Nicolson method and decoupled multistep method are presented in Section 3.2 and Section 3.3, respectively. In Section 4, the L² error estimates for extrapolated Crank–Nicolson and decoupled implicit-explicit two-step schemes are presented and the proof are given. In Section 5, some details during the progress of implementation are presented. In Section 6, we propose four numerical experiments for DS equations, which illustrates the efficiency of the presented numerical algorithms by the L², L^∞, H¹ error accuracy and the relative error of L² norm conservation laws. Furthermore, we apply the proposed method to the rogue waves in DS equations. Finally, a brief conclusion is given in the last section.

Section snippets

Governing equations

Let $Ω = [L_{1}, L_{2}] \times [L_{3}, L_{4}]$ be a bounded subset of $R^{2}$ and T > 0 be a final time, $J = (0, T],$ $Ω_{T} = Ω \times (0, T)$ . We consider the following (2+1)-dimensional Davey–Stewartson equations with power law nonlinearity ${\begin{matrix} i \frac{\partial Ψ}{\partial t} + α \frac{\partial^{2} Ψ}{\partial x^{2}} + β \frac{\partial^{2} Ψ}{\partial y^{2}} + γ {| Ψ |}^{2} Ψ + δ Φ_{x} Ψ = 0, & in Ω_{T}, \\ \frac{\partial^{2} Φ}{\partial x^{2}} + \frac{\partial^{2} Φ}{\partial y^{2}} + θ \frac{{\partial | Ψ |}^{2}}{\partial x} = 0, & in Ω_{T}, \\ Ψ = 0, Φ = 0, & on \partial Ω \times [0, T], \\ Ψ = Ψ_{0}, Φ = Φ_{0}, & on Ω \times {t = 0}, \end{matrix}$ where Ψ(x, y, t) is a complex valued function which denotes the complex amplitude,while Φ(x, y, t) is a real valued function which denotes the velocity of a underlying mean flow. Using the

Numerical methods

We use the standard Sobolev space W^m,k(Ω), where m is a nonnegative integer with 1 ≤ k ≤ ∞, and the space involving time L^k(0, T; W^m,k) with norm ${∥ \cdot ∥}_{L^{k} (0, T; W^{m, k})}$ . Let $W^{m, 2} (Ω) = H^{m} (Ω)$ with the norm ‖ · ‖_m and ( · ,  · ) be the inner product on $L^{2} (Ω) = H^{0} (Ω)$ defined by $(u, v) = \int_{Ω} u (x) {(v (x))}^{*} d x d y, \forall u, v \in L^{2} (Ω),$ where v* denotes the conjugate of v. The norm ‖ · ‖_∞ denotes the essential supremum. Set $V = H_{0}^{1} (Ω) = {v \in H^{1} (Ω) : v |_{\partial Ω} = 0}$ .

Applying Greens formula to problem (2.1) and using the boundary condition in the

Error estimates

In this section, we shall analyze the optimal error estimates for the full discrete scheme presented in Algorithm 3.1 with Crank–Nicolson method and Algorithm 3.2 with two-step backward differentiation formula. We introduce the projection $Π_{h}^{Ψ} : V \to V_{h},$ $Π_{h}^{Φ} : V \to V_{h}$ defined by, $α (\frac{\partial}{\partial x} (Π_{h}^{Ψ} v - v), \frac{\partial χ}{\partial x}) + β (\frac{\partial}{\partial y} (Π_{h}^{Ψ} v - v), \frac{\partial χ}{\partial y}) = 0, \forall χ \in V_{h}, v \in V .$ $(\nabla (Π_{h}^{Φ} v - v), \nabla χ) = 0, \forall χ \in V_{h}, v \in V .$ Then, it may be shown that $∥ v - Π_{h}^{Ψ} v ∥ + h ∥ ▿ (v - Π_{h}^{Ψ} v) ∥ \leq C h^{r + 1} {∥ v ∥}_{r + 1}, \forall v \in H^{r + 1} (Ω) \cap H_{0}^{1} (Ω),$ $∥ v - Π_{h}^{Φ} v ∥ + h ∥ ▿ (v - Π_{h}^{Φ} v) ∥ \leq C h^{r + 1} {∥ v ∥}_{r + 1}, \forall v \in H^{r + 1} (Ω) \cap H_{0}^{1} (Ω) .$

Let denote $\begin{matrix} Ψ_{h}^{n} - Ψ (t_{n}) \end{matrix}$

Basis functions and implementation issues

Let ${φ_{j}}_{j = 0}^{2}$ be the linear basis functions on a triangle element Ω_e, namely, φ_j ∈ P₁(Ω_e), $j = 0, 1, 2,$ $V_{h} |_{e} = span {φ_{0}, φ_{1}, φ_{2}}$ . Assume (Uⁿ, Zⁿ) be the approximation of the exact solutions $(Ψ^{n}, Φ^{n}) = (Ψ (t^{n}), Φ (t^{n}))$ .

To overcome the deficiency of solving complex algebraical system (3.8), we decompose the complex value function Ψ into real and imaginary parts, namely, $Ψ (t_{n}) = u_{1} (t_{n}) + i u_{2} (t_{n}),$ and approximate $u_{j} (t_{n}) = u_{j}^{n}$ by $U_{j}^{n} \in V_{h},$ namely, $Ψ_{h}^{n} = U_{1}^{n} + i U_{2}^{n}$ . We denote U_j and Z on an element by $U_{j}^{e} = \sum_{m = 0}^{2} U_{j m}^{e} (t) φ_{m} (x, y), Z^{e} =$

Numerical experiments

In this section, four experiences: the propagation of elliptic-elliptic and hyperbolic-elliptic DS equations are presented to illustrate the accuracy and L² norm conservation laws of the numerical schemes based on CCN method and DBDF2 method. Moreover, we extend the proposed numerical methods to study the dynamic of rogue waves including line rogue wave for elliptic-elliptic and hyperbolic–elliptic DS equations, two-rogue waves for elliptic–elliptic DS equations.

To illustrate the accuracy of

Conclusion

In this paper, numerical algorithms for the numerical solutions of the Davey–Stewartson equations are proposed using Galerkin finite element with second order time discretization, which employs linearized Crank–Nicolson scheme and decoupled multistep scheme. The high efficiency and accuracy of our numerical methods are demonstrated by the numerical experiments: the propagation of elliptic-elliptic and hyperbolic-elliptic DS equations. The numerical results recorded in Tables 1–4 and Figs. 1 and

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^☆: The project is supported by China Scholarship Council (201506280175) and NSF of China (11371289, 11501441).

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Galerkin methods for the Davey–Stewartson equations☆