In this paper, we aim to propose and analyze a linearized three-level Galerkin finite element method (FEM) for the nonlinear Schrödinger equation with a general nonlinearity and an external potential. Compared with the existing results in literature, under a weaker assumption on both the exact solution and the nonlinear term,  we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.  Besides the standard energy method, the key tools used in our analysis are an induction argument and several Sobolev inequalities. Numerical results are reported to verify our theoretical analysis.

In this paper, we present an efficient energy stable finite element method for the two phase incompressible Magnetohydrodynamic (MHD) flow which is governed by the incompressible MHD equations and the Cahn-Hilliard equation. The strong nonlinear system governs the dynamics and the coupling of multiple physical fields which are, respectively, the velocity $\mathbf{u}$, the pressure $p$, the magnetic induction $\mathbf{B}$, the concentration $\phi$, and the chemical potential $\mu$. To solve the problem efficiently, we propose a linearized finite element scheme which is absolutely stable in time. Several numerical experiments are shown for demonstrating the competitive behavior of the method.

The traditional convergent analysis of two-level method (TLM) will fail when Nédélec finite element is employed to approximate Maxwell's system. In this paper, based on the superclose theory, we develop a new analysis framework for the nonlinear conductivity problem in Maxwell's system, which remedies the weakness of Nédélec finite element for two-level method. This method can save computational cost and improve the efficiency. We obtain the optimal convergent rate $\mathcal{O}(\Delta t+h^2)$ in spatial space. A numerical example verifies our theoretical analysis.

In this paper, we present a numerical scheme and an alternating direction implicit (ADI) scheme for the one-dimensional and two-dimensional time-space fractional vibration equations (FVEs), respectively. Firstly, the considered time-space FVEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Riemann-Liouville derivative. This transformation can weaken the smoothness requirement in time when discretizing the partial integro-differential problems. Secondly, we use the Crank-Nicolson technique combined with the midpoint formula, the weighted and shifted Grünwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fractional central difference formula are applied to approximate the second order derivative and the Riesz derivative in spatial direction, respectively. Further, an ADI scheme is constructed for the two-dimensional case. Then, the convergence and unconditional stability of the proposed schemes are proved rigorously. Both of the schemes are convergent with the second order accuracy in time and space. Finally, two numerical examples are given to support the theoretical results.

This article presents well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes to solve the shallow water equations (SWEs). Well-balanced schemes are characterized by preservation of the steady state  exactly at the discrete level. The well-balanced property is of paramount importance in practical applications where many studied phenomena are regarded as small perturbations to equilibrium states. To achieve the well-balanced property, numerical fluxes presented here are constructed by means of a suitable conservative variables decomposition and the hydrostatic reconstruction idea. This decomposition strategy allows us to realize a novel simple source term approximation. Both rigorous theoretical analysis and extensive numerical examples all verify that the resulting schemes maintain the well-balanced property exactly. Furthermore, numerical results strongly imply that the proposed schemes can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions.

In this paper, the vibration characteristics of beams with arbitrarily and continuously varying thickness and resting on Pasternak elastic foundations were analytically studied based on the elasticity theory directly. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, was derived. The frequency equation governing the free vibration of varying-thickness beams resting on a Pasternak elastic foundation can be obtained by using the Fourier series expansion of the boundary conditions on the upper and lower surfaces of the beam. Convergence and comparison studies were conducted to demonstrate the high accuracy and efficiency of the present method. Application of the proposed analytical method to some typical beams with different geometry, Poisson's ratio, elastic coefficients of foundation were conducted further, and some new results are reported which may be used as an alternative of benchmark or standard solutions for numerical or other approximate results.

In this paper, a viscous Cahn-Hilliard equation with logarithmic Flory-Huggins energy potential is solved numerically by using a convex splitting scheme. This numerical scheme is based on the Backward Differentiation Formula (BDF) method in time and mixed finite element method in space. A regularization procedure is applied to logarithmic potential, which makes the domain of the regularized function $F(u)$ to be extended from $(-1,1)$ to $(-\infty,\infty)$. The unconditional energy stability is obtained in the sense that a modified energy is non-increasing. By a carefully theoretical analysis and numerical calculations, we derive discrete error estimates. Subsequently, some numerical examples are carried out to demonstrate the validity of the proposed method.

The aim of the paper is to propose a second-order accurate Crank-Nicolson scheme for solving semilinear parabolic equations. This scheme combing two-grid finite volume element method involves solving a small nonlinear system on a coarse grid space and a linear system on a fine grid space, which can improve computing efficiency while keeping the accuracy. It means that we can use large time steps in the actual calculation. We further prove the optimal error estimates of the scheme strictly and present numerous simulations to demonstrate the theoretical results.

In this paper, the nonlinear vibrations analysis of functionally graded (FG) rectangular plate in contact with fluid are investigated analytically using first order shear deformation theory (FSDT) for the first time. The pressure exerted on the free surface of the plate by the fluid is calculated using the velocity potential function and the Bernoulli equation. With the aid of von Karman nonlinearity strain-displacement relations and Galerkin procedure the partial differential equations of motion are developed. The nonlinear equation of motion is then solved by modified Lindstedt-Poincare method (MLPM). The effects of some system parameters such as vibration amplitude, fluid density, fluid depth ratio, volume fraction index and aspect ratio on the nonlinear natural frequency of the plate are discussed in detail.

Direct numerical simulations of self-sustained oscillations in two-dimensional rectangular cavity were performed in both subsonic and supersonic flow. A fifth order weighted essentially non-oscillatory (WENO) scheme was used for the nonlinear term of the two-dimensional unsteady compressible Navier-Stokes equations. A third order Runge-Kutta method was used to discretize the time derivative. The pressure signals on the cavity wall were analyzed by the short time Fourier transform (STFT) method. The formation mechanism of the component with lower frequency for the subsonic cases was analyzed. The detailed oscillating structures in and above the cavity were identified and the interactions between different waves were discussed for both the subsonic and supersonic cases. The results show the oscillating system would finally reach to the self-sustained state. The switching phenomenon of two waves in the cavity was identified for the supersonic case. An improved model of the shock wave patterns for supersonic case was given.

The purpose of this work is to shed some light on the dynamics of the equilateral restricted four-body problem with equally massed radiating bodies. The positions along with the linear stability of the coplanar equilibria are determined by using numerical methods. Specifically, we conduct a rigorous and systematic analysis for elucidating the influence of the mass parameter $m_3$ and the radiation pressure factor $q$ on the dynamics of the system. Our results indicate that these two parameters are very influential in the equilibria of the system.

The generating function method plays an important role in the construction of symplectic methods and closely depends on different generating functions. The three typical generating functions are widely applied in practical computations. This paper follows the general framework of the generating function method proposed by Feng Kang to produce a simple generating function with parameterized coordinates. This new generating function is more practical and covers the three typical ones by fixing the parameter to specific values. The relationship between symplectic transformation and new generating function and the Hamilton-Jacobi equation are discussed. A new family of arbitrary high-order symplectic methods with free parameter is obtained. Through the composition of the obtained low-order symplectic method, we derive another new class of any high-order symmetric symplectic methods with free parameter. These parametric symplectic methods will have more freedom of adjustment to design integrators which preserve energy or non-quadratic invariants. Computational examples illustrate the effectiveness of the proposed methods.