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In this paper, three energy preserving numerical methods are proposed, including the Crank–Nicolson Galerkin–Legendre spectral (CN–GLS) method, the SAV Galerkin–Legendre spectral (SAV–GLS) method, and the ESAV Galerkin–Legendre spectral (ESAV–GLS) method, for the space fractional nonlinear Schrodinger equation with wave operator. In theoretical analyses, we take the CN–GLS method as an example to analyze the boundness of numerical solution and the unconditional spectral–accuracy convergence in L 2 and L ∞ norms. The effective numerical implementations of the proposed spectral Galerkin methods are discussed in detail. Numerical comparisons are reported to illustrate that the theoretical results are reasonable and the proposed spectral Galerkin methods have high efficiency for energy preservation in long–time computations.

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar variable (ESAV) approach that can remove the bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products, which make it more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize the symplectic Runge-Kutta method for both solution variables and the auxiliary variable, where the values of internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms are of constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

In this paper, an efficient numerical scheme is presented for solving the space fractional nonlinear damped sine–Gordon equation with periodic boundary condition. To obtain the fully-discrete scheme, the modified Crank–Nicolson scheme is considered in temporal direction, and Fourier pseudo-spectral method is used to discretize the spatial variable. Then the dissipative properties and spectral-accuracy convergence of the proposed scheme in L ∞ norm in one-dimensional (1D) space are derived. In order to effectively solve the nonlinear system, a linearized iteration based on the fast Fourier transform algorithm is constructed. The resulting algorithm is computationally efficient in long-time computations due to the fact that it does not involve matrix inversion. Extensive numerical comparisons of one- and two-dimensional (2D) cases are reported to verify the effectiveness of the proposed algorithm and the correctness of the theoretical analysis.

In this paper we intend to construct a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrodinger (2D SFNS) equation with the integral fractional Laplacian. The temporal direction is discretized by the modified Crank-Nicolson method, and the spatial variable is approximated by a novel fractional central difference method. The mass and energy conservations and the convergence are rigorously proved for the proposed scheme. For 1D SFNS equation, the convergence relies heavily on the L ∞ -norm boundness of the numerical solution of the proposed scheme. However, we cannot obtain the L ∞ -norm boundness of the numerical solution by using the similar process for the 2D SFNS equation. One of the major significance of this paper is that we first obtain the L ∞ -norm boundness of the numerical solution and L 2 -norm error estimate via the popular “cut-off” function for the 2D SFNS equation. Further, we reveal that the spatial discretization generates a block-Toeplitz coefficient matrix, and it will be ill-conditioned as the spatial grid mesh number M and the fractional order α increase. Thus, we exploit an linearized iteration algorithm for the nonlinear system, so that it can be efficiently solved by the Krylov subspace solver with a suitable preconditioner, where the 2D fast Fourier transform (2D FFT) is applied in the solver to accelerate the matrix-vector product, and the standard orthogonal projection approach is used to eliminate the drift of mass and energy. Extensive numerical results are reported to confirm the theoretical analysis and high efficiency of the proposed algorithm.

In the paper, we aim to develop a class of high-order structure-preserving algorithms, which are built upon the idea of the newly introduced scalar auxiliary variable approach, for the multi-dimensional space fractional nonlinear Schrodinger equation. First, we reformulate the equation as an infinite-dimension canonical Hamiltonian system, and obtain an equivalent system with a modified energy conservation law by using the scalar auxiliary variable approach. Then, the new system is discretized by Gauss collocation methods to arrive at semi-discrete conservative systems. Subsequently, the Fourier pseudo-spectral method is applied for semi-discrete systems to obtain high-order fully-discrete schemes, which can both preserve the mass and the modified energy exactly in discrete scene. Finally, numerical experiments are provided to demonstrate the conservation and accuracy of the proposed schemes.

In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrodinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations.

This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we utilize the fractional centered difference formula to discrete the equivalent system derived by the invariant energy quadratization method in space direction, and obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully-discrete conservative scheme. The stability, solvability and convergence in the maximum norm of the numerical scheme are given. Furthermore, a fast algorithm based on the fast Fourier transformation technique is used to reduce the computational complexity in practical computation. Finally, numerical examples are provided to confirm our theoretical analysis results., Comment: 28pages, 5 figures

This paper aims to construct structure-preserving numerical schemes for the two-dimensional space fractional Klein-Gordon-Schrodinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as an infinite dimensional canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we use the Fourier pseudo-spectral method to discrete the equation in space direction and obtain a semi-discrete conservative system, which can be reformulated a finite dimensional canonical Hamiltonian system. Further applying the partitioned averaged vector field methods to the semi-discrete system gives a class of fully-discrete schemes that can preserve the mass and energy exactly. Numerical examples are provided to confirm our theoretical analysis results at last.

This paper aims to construct a numerical scheme for the damped nonlinear space fractional Schrodinger equation. First, the conservation laws of mass and energy for the continuous equation are derived. Then, based on the fractional centered difference formula, a semi-discrete scheme, which preserves the semi-discrete mass and energy conservation laws is proposed. Further applying the Crank–Nicolson method on the temporal direction gives a fully-discrete conservative scheme. Furthermore, the solvability, boundedness and convergence in the maximum norm of the numerical solutions are given. Some numerical examples are displayed to confirm the theoretical results.

We develop a paradigm for developing local energy dissipation rate preserving (LEDRP) approximations to general gradient flow models driven by source terms. In driven gradient flow models, the deduced energy density transport equation possesses an indefinite source. Local energy-dissipation-rate preserving algorithms are devised to respect the mathematical structure of both the driven gradient flow model and its deduced energy density transport equation. The LEDRP algorithms are also global energy-dissipation-rate preserving under proper boundary conditions such as periodic boundary conditions. However, the contrary may not be true. We then apply the paradigm to a phase field model for growth of a graphene sheet to produce a set of LEDRP algorithms. Numerical refinement tests are conducted to confirm the convergence property of the new algorithms and simulations of graphene growth are demonstrated to benchmark against existing results in the literature.

In this paper, two novel linear-implicit and momentum-preserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank-Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete $L^{\infty}$ norm. Then by using the standard energy method, both the linear-implicit Crank-Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of $\mathcal{O}(\tau^2+N^{-r})$ in the discrete $L^{\infty}$ norm without any restrictions on the grid ratio, where $N$ is the number of nodes and $\tau$ is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes.

In this paper, we propose and analyze a conservative high-order compact finite difference scheme for the Klein–Gordon–Schrodinger (KGS) equation with Dirichlet boundary condition. By introducing the time-derivative of one solution as a separate dependent variable, we rewrite the KGS equation as a system of partial differential equations (PDEs) and obtain a two-level compact finite difference scheme. Based on the energy method and matrix knowledge on the vector form, we convert the point-wise form of the proposed compact scheme into an equivalent vector form and analyze its conservative and convergence properties. We prove that the proposed scheme preserves the total energy and charge in the discrete forms and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of O ( h 4 + τ 2 ) in L 2 -norm, where h and τ are spatial and temporal steps, respectively. Numerical experiments are carried out to verify the performance of the scheme.

A class of high order compact methods combined with local one-dimensional method have been studied to numerically solve multidimensional convection–diffusion equations. The methods are widely accepted due to their compactness, high accuracy. In this kind of methods the spatial derivatives are approximated implicitly rather than explicitly with smaller stencil but with higher accuracy. The local one-dimensional strategy is adopted in time to reduce the scale of algebraic equations resulting from numerical methods. This makes the multidimensional problems be easily coded. Based on analyzing the splitting error of the local one-dimensional method, a more accurate scheme is obtained through minor modification on the original scheme. By Von Neumann approach, we can find that the proposed schemes are unconditionally stable. Some numerical results are reported to illustrate that the schemes are robust, efficient and accurate.

Many PDEs can be recast into the general multi-symplectic formulation possessing three local conservation laws. We devote the present paper to some systematic methods, which hold the discrete versions of the local conservation laws respectively, for the general multi-symplectic PDEs. For the original problem subjected to appropriate boundary conditions, the proposed methods are globally conservative. The proposed methods are successfully applied to many one-dimensional and multi-dimensional Hamiltonian PDEs, such as KdV equation, G–P equation, Maxwell’s equations and so on. Numerical experiments are carried out to verify the theoretical analysis.

In this paper, we show that the Cahn–Hilliard equation possesses a local energy dissipation law, which is independent of boundary conditions and produces much more information of the original problem. To inherit the intrinsic property, we derive three novel local structure-preserving algorithms for the 2D Cahn–Hilliard equation by the concatenating method. In particular, when the nonlinear bulk potential f ( ϕ ) in the equation is chosen as the Ginzburg–Landau double-well potential, the method discussed by Zhang and Qiao (2012) [50] is a special case of our scheme II. Thanks to the Leibnitz rules and properties of operators, the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time–space region. Under periodic boundary conditions, the schemes are proven to possess the discrete mass conservation and total energy dissipation laws. Numerical experiments are conducted to show the performance of the proposed schemes.

A Crank–Nicolson Galerkin spectral element method for solving the nonlinear Schrodinger (NLS) equation in two dimensions is proposed in this paper. Our key idea is twofolds. First, the 2D NLS equation is rewritten as an infinite-dimensional Hamiltonian PDE and the Hamiltonian PDE is discreted by using the Galerkin spectral element (GSE) method in space. Second, we cast the resulted ODEs into a finite-dimensional canonical Hamiltonian system and discrete the system by using the Crank–Nicolson (CN) method. The relay leads to a fully discretized and energy-preserved scheme. Without grid ratio restrictions, the order of convergence of our new method is O ( τ 2 + h 2 ) if the discrete L 2 -norm is employed. The Fast Fourier Transform and the matrix diagonalization method are applied to the new method to increase computing efficiency. Numerical examples are given to further illustrate the conservation properties and convergence of the energy-preserving scheme constructed.

Effect of V2O5 additive in silicate-containing electrolyte on AZ91D magnesium alloys treated by micro-arc oxidation (MAO) technology under different loading voltages was investigated. The results showed that vanadium was well up-taken into the coating chemically. Moreover, a new phase of MgV2O4 with spinel structure was obtained in MAO coatings due to V2O5 added into the electrolyte. The MgV2O4 phase was responsible for the coatings exhibiting brown color and also was beneficial to improving the anti-corrosion property. In spotting tests, the corrosion resistances of coatings prepared under the high voltage are about 6&ndash, 9 times higher than those of the low voltage because of the thicker coatings of the former. In potentiodynamic polarization tests, the coatings&rsquo, corrosion resistances were improved with the addition of V2O5, which was more significant under the low voltage than that under the high voltage. When the concentration of V2O5 was 0.2 g/L, the corrosion current density of the coating was the lowest, which means that the coating&rsquo, s corrosion resistance under the low voltage is the best. Hence, it is necessary to carry out targeted design of the coating&rsquo, s microstructure according to the different applications.

In this paper, we present a linearly implicit energy-preserving scheme for the Camassa-Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradient systems. The Camassa-Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank-Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme., Comment: 21 pages, 13 figures

In this paper, we propose a linearly implicit Fourier pseudo-spectral scheme, which preserves the total mass and energy conservation laws for the damped nonlinear Schrodinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal L2-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.