Search

In this paper, we develop a time-splitting method to investigate the evolution of the spin–orbit-coupled Bose–Einstein condensates. The fully discrete scheme is constructed by combining the second-order time-splitting method for temporal discretization and the framework of Galerkin finite element method for spatial discretization. Ample numerical simulations are performed to validate the features of the designed numerical method, including the optimal order of error norms, the conservation law and the center-of-mass. Moreover, the interaction of waves with typical trapping potentials in two and three dimensions is numerically simulated. [ABSTRACT FROM AUTHOR]

In this paper, a number of energy stable numerical schemes are proposed for the fractional Cahn–Hilliard equation. We prove mass conservation, unique solvability and energy stability for three time semi-discretized schemes based on the first-order semi-implicit scheme, the Crank–Nicolson scheme and the BDF2 scheme respectively. Then we present error analysis for these numerical schemes with the Fourier spectral approximation in space. Some numerical experiments are finally carried out to confirm accuracy and effectiveness of these proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose stable second-order numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations, which are based on the convex splitting in time and the Fourier spectral method in space. It is shown that the scheme for the fractional Cahn–Hilliard equation preserves mass. Meanwhile, the unique solvability and energy stability of the numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations are proved. Finally, we present some numerical experiments to confirm the accuracy and the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we study two-grid mixed finite element algorithms (MFEAs) for the stationary incompressible Stokes equations with damping and nonlinear slip boundary conditions. We set up the weak formulation and obtain the well-posedness. Then employing the mixed finite element method, some optimal error estimates between the exact solution and the approximation solution are derived. Two kinds of two-grid MFEAs based on the Stokes correction and Newton-type correction are established. Some error estimates for two-grid MFEAs are obtained. Several numerical results are presented to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, two multi-level stabilized algorithms for the Navier–Stokes equations with damping are studied. The algorithms combine the stabilized finite element technique with the multi-level method. Two kinds of multi-level stabilized algorithms based on the Oseen-type correction and Newton-type correction are presented. Some error estimates for the multi-level stabilized algorithms are derived. Moreover, the numerical tests agree with the theoretical results. [ABSTRACT FROM AUTHOR]

• We propose L 2 − 1 σ /Hermite–Galerkin spectral method to solve nonlinear fractional differential equation in R d. • Arbitrary time-step size of our scheme can be used without resulting in the nonsense numerical results. • Our method can arrive at the spectral (resp. second-order) accuracy for spatial (resp. temporal) discretization. • Effects of the fractional orders on the pattern formations of fractional Allen–Cahn and Gray–Scott models are studied. • Comparisons between the fractional models and the corresponding integer-order ones are investigated. The aim of this paper is to develop an efficient finite difference/Hermite–Galerkin spectral method for the time-fractional nonlinear reaction–diffusion equation in unbounded domains with one, two, and three spatial dimensions. For this purpose, we employ the L 2 − 1 σ formula to discretize the temporal Caputo derivative. Additionally, we apply the Hermite–Galerkin spectral method with scaling factor for the approximation in space. The stability of the fully discrete scheme is established to show that our method is unconditionally stable. Numerical experiments including one-, two-, and three-dimensional cases of the problem are carried out to verify the accuracy of our scheme. The scheme is show-cased by solving two problems of practical interest, including the fractional Allen–Cahn and Gray–Scott models, together with an analysis of the properties of the fractional orders. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a highly accurate and conservative split-step spectral Galerkin (SSSG) scheme, which combines the standard second-order Strang split-step method for handling the nonlinear and the potential terms with the Legendre spectral Galerkin method for approximating the Riesz space-fractional derivatives, for the two-dimensional nonlinear space-fractional Schrödinger equation. The mass conservation property of the numerical solution is proved and the optimal error estimate with respect to spatial discretization is established by introducing an orthogonal projection operator. In addition, a matrix diagonalization technique is introduced to resolve the multi-dimensional difficulty and reduce the computational complexity in the implementation. Numerical experiments are presented to illustrate the accuracy and robustness of the SSSG method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this letter, we consider the numerical approximation of a two-dimensional distributed-order time–space fractional reaction–diffusion equation. The time- and space-fractional derivatives are considered in the senses of Caputo and Riesz, respectively. By using the composite mid-point quadrature, the original fractional problem is approximated by a multi-term time–space fractional differential equation. Then the multi-term Caputo fractional derivatives are discretized by the L 2 - 1 σ formula. We apply the Legendre–Galerkin spectral method for the spatial approximation. Two numerical experiments with smooth and non-smooth initial conditions, respectively, are performed to illustrate the robustness of the proposed method. The results show that: our scheme can arrive at the spectral accuracy (resp. algebraic accuracy) in space for the problem with smooth (resp. non-smooth) initial condition. For both of these two cases, our scheme can lead to the second-order accuracies in time. Additionally, the convergence rates in both spatial and temporal distributed-order variables are two. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we propose Galerkin finite element methods to investigate the evolution of the generalized Klein–Gordon–Zakharov equations. The spatial discretization is based on Galerkin finite element method. The combination of time-splitting method and finite difference method is used for temporal discretization. The accuracy and efficiency of our numerical schemes are verified by the error norms, conservation laws and the application in nonrelativistic limit regime and in three spatial dimension of different numerical experiments. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop an efficient numerical treatment for the two-dimensional fractional nonlinear reaction–diffusion-wave equation with the time-fractional derivative of order α ( 1 < α < 2 ). For this purpose, we employ the alternating direction implicit (ADI) method based on the Crank–Nicolson scheme for the time stepping, while we apply the Legendre–Galerkin spectral method for the space discretization. The stability and convergence analysis are rigorously set up. In addition, the proposed method is extended to solve the time-fractional Klein–Gordon and sine-Gordon models. Numerical experiments are included, which verifies the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, implicit–explicit multistep Galerkin methods are studied for two-dimensional nonlinear Schrödinger equations and coupled nonlinear Schrödinger equations. The spatial discretization is based on Galerkin method using linear and quadratic basis functions on triangular and rectangular finite elements. And the implicit–explicit multistep method is used for temporal discretization. Linear and nonlinear numerical tests are presented to verify the validity and efficiency of the numerical methods. The numerical results record that the optimal order of the error in L 2 and L ∞ norm can be reached. [ABSTRACT FROM AUTHOR]

In this paper, two second-order stable schemes based on the scalar auxiliary variable (SAV) approach are constructed for the fractional-in-space Allen–Cahn equation. We use the Crank–Nicolson and BDF2 schemes for time discretization, and prove the unique solvability and energy stability. The fully discrete schemes are carried out with the Fourier spectral approximation in space. Finally, we provide numerical examples to verify the accuracy of the proposed methods. In particular, we develop an adaptive time-stepping approach related to the accuracy requirement to speed up the calculation. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical scheme, prove stability, existence of solutions, uniqueness and convergence of the incompressible Navier–Stokes equations. It has the advantage of being defined from strictly algebraic considerations. A significant feature of the present method is that the structure of the stabilization term based on the multiscale enrichment and derived from the Navier–Stokes problem itself. Ample numerical experiments are carried out to confirm the theory and illustrate the effectiveness of the scheme on incompressible fluid. [ABSTRACT FROM AUTHOR]

In this paper, mixed Galerkin finite methods for modified regularized long wave (MRLW) equation are studied. We discrete in space by finite elements and for the time discretization by extrapolated Crank–Nicolson scheme and explicit linear multistep scheme. Linear stability analysis based on Von Neumann theory are performed. Numerical experiments verify the efficiency and accuracy of the proposed numerical schemes. [ABSTRACT FROM AUTHOR]

To seek the exact double non-traveling wave solutions of nonlinear partial differential equations, the compound ( G ′ G ) -expansion method is firstly proposed in this paper. With the aid of symbolic computation, this new method is applied to construct double non-traveling wave solutions of (2+1)-dimensional Painlevé integrable Burgers equation and (2+1)-dimensional breaking soliton equation. As a result, abundant double non-traveling wave solutions including double hyperbolic function solutions, double trigonometric function solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via the proposed method. These exact solutions contain arbitrary functions, which may be helpful to explain some complex phenomena. When the parameters are taken as special values, the double solitary-like wave solutions can be derived from double hyperbolic function solutions. Furthermore, the time evolutions of double solitary-like wave solutions are discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we consider a defect-correction stabilized finite element method for incompressible Navier–Stokes equations with friction boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. In the defect step, an artificial viscosity parameter σ is added to the Reynolds number as a stability factor, and the Oseen iterative scheme is applied in the correction step. H 1 × L 2 error estimations are derived for the one-step defect-correction stabilized finite element method. In the end, some numerical results are presented to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents one-level and two-level iterative penalty finite element methods to approximate the solutions of steady Navier–Stokes equations. First, one-level iterative penalty finite element method is applied to solve the steady Navier–Stokes equations numerically, and its and error estimates are derived. Then, two-level iterative penalty scheme is given and its error estimates are obtained for velocity and pressure. Finally, the numerical results are displayed to verify the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we use the fractional variational homotopy perturbation iteration method (FVHPIM) with modified Riemann–Liouville derivative to solve a time-fractional diffusion equation. Using this method, a rapid convergent sequence tending to the exact solution of the equation can be obtained. To show the efficiency of the considered method, some numerical examples are presented. [Copyright &y& Elsevier]

Abstract: A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 7/4 away from the characteristic layers. Numerical experiments support these theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we present explicit multistep method for regularized long wave (RLW) equation. The discretization in space is based on the Galerkin method using linear space finite elements, and for the time discretization the explicit multistep method is used. Propagation of solitary waves and interaction of two solitary waves are simulated using the proposed method to confirm the theoretical results. [Copyright &y& Elsevier]

Abstract: This study explores the influence of epidemics by numerical simulations and analytical techniques. Pulse vaccination is an effective strategy for the treatment of epidemics. Usually, an infectious disease is discovered after the latent period, H1N1 for instance. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. So we put forward a SVEIRS epidemic model with two time delays and nonlinear incidence rate, and analyze the dynamical behavior of the model under pulse vaccination. The global attractivity of ‘infection-free’ periodic solution and the existence, uniqueness, permanence of the endemic periodic solution are investigated. We obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. The main feature of this study is to introduce two discrete time delays and impulse into SVEIRS epidemic model and to give pulse vaccination strategies. [Copyright &y& Elsevier]

Abstract: In this paper, the extended Riccati equation mapping method is proposed to seek exact solutions of variable-coefficient nonlinear evolution equations. Being concise and straightforward, this method is applied to certain type of variable-coefficient diffusion–reaction equation and variable-coefficient mKdV equation. By means of this method, hyperbolic function solutions and trigonometric function solutions are obtained with the aid of symbolic computation. It is shown that the proposed method is effective, direct and can be used for many other variable-coefficient nonlinear evolution equations. [Copyright &y& Elsevier]

Abstract: A new algorithm is presented for solving Troesch’s problem. The numerical scheme based on the modified homotopy perturbation technique is deduced. Some numerical experiments are made. Compared with the variational iteration method and the Adomian decomposition method, the scheme is shown to be highly accurate, and only a few terms are required to obtain accurate computable solutions. Finally, the algorithm is applied to other problems. [Copyright &y& Elsevier]

Abstract: In this article, stabilized finite element methods are considered for the non-stationary Stokes equations, based on some lowest equal-order finite elements space pair (X h , M h ) which do not satisfy the discrete inf–sup condition. The stability of two kinds of methods is derived under some regularity assumptions. Then, the convergence of the penalty method and the pressure-Poisson stabilized method is compared. The result shows that the former error limits the order of approximation to , and the latter yields the optimal error estimate O(h). [Copyright &y& Elsevier]

Abstract: This paper describes multigrid algorithms for solving multigroup neutron transport equations. Local mode analysis of multilevel is performed for transport problems, and numerical Fourier analysis is carried out to study the smoothing and convergence properties of the multigrid algorithms for neutron diffusion and general P N approximate case respectively. Numerical test calculations validate the theoretical predictions. [Copyright &y& Elsevier]

In this paper, we develop a new space-fractional modified phase field crystal equation which has some similar properties including the mass conservation and energy dissipation. Then, we propose a second-order scheme based on a new Lagrange multiplier method that conserves the mass and dissipates the energy. For the new method, there are only two decoupled linear equations with constant coefficients and one nonlinear algebraic system to be solved at each time step which makes it efficient. Finally, we give some numerical experiments to verify the accuracy and stability of the proposed methods. [ABSTRACT FROM AUTHOR]