Fourier Spectral Methods for Degasperis–Procesi Equation with Discontinuous Solutions

In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The \(L^2\) stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method. By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear DP equation are provided to illustrate the accuracy and capability of the methods.     

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted $L^2$- and $L^{\infty }$-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order $min\{2?\alpha ,1+\alpha \}$, where $\alpha \in (0,1)$ is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.     

Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation

In this paper, four alternating direction implicit (ADI) schemes are presented for solving two-dimensional cubic nonlinear Schrödinger equations. Firstly, we give a Crank–Nicolson ADI scheme and a linearized ADI scheme both with accuracy $O(\mathrm{\Delta }{t}^2+h^2)$, with the same method, use fourth-order Padé compact difference approximation for the spatial discretization; two HOC-ADI schemes with accuracy $O(\mathrm{\Delta }{t}^2+h^4)$ are given. The two linearized ADI schemes apply extrapolation technique to the real coefficient of the nonlinear term to avoid iterating to solve. Unconditionally stable character is verified by linear Fourier analysis. The solution procedure consists of a number of tridiagonal matrix equations which make the computation cost effective. Numerical experiments are conducted to demonstrate the efficiency and accuracy, and linearized ADI schemes show less computational cost. All schemes given in this paper also can be used for two-dimensional linear Schrödinger equations.     

A space–time spectral collocation method for the two-dimensional variable-order fractional percolation equations

In this article, we introduce a space–time spectral collocation method for solving the two-dimensional variable-order fractional percolation equations. The method is based on a Legendre–Gauss–Lobatto (LGL) spectral collocation method for discretizing spatial and the spectral collocation method for the time integration of the resulting linear first-order system of ordinary differential equation. Optimal priori error estimates in $L^2$ norms for the semi-discrete and full-discrete formulation are derived. The method has spectral accuracy in both space and time. Numerical results confirm the exponential convergence of the proposed method in both space and time.     

Two-grid method for two-dimensional nonlinear Schrödinger equation by mixed finite element method

A conservative two-grid mixed finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse-grid solution as the initial guess. Moreover, error estimates are conducted for the two-grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy $H=O\left(h^{\frac{1}{2}}\right)$ in the two-grid method. The numerical results show that this method is very effective.     

A fast algorithm for solving the space–time fractional diffusion equation

In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ${(?\Delta )}_s^{\alpha /2}$ results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.     

Modeling the quantum tunneling effect for a particle with intrinsic structure in presence of external magnetic field in the Lobachevsky space

Generalized Schrödinger equation for Cox spin zero particle is studied in presence of magnetic field on the background of Lobachevsky space. Separation of the variables is performed. An equation describing motion along the axis $z$ turns out to be much more complicated than when describing the Cox particle in Minkowski space.The form of the effective potential curve indicates that we have a quantum-mechanical problem of the tunneling type. The derived equation has 6 regular singular points. To physical domains $z=\pm \infty$ there correspond the singular points 0 and 1 of the derived equation. Frobenius solutions of the equation are constructed, convergence of the relevant series is examined by Poincaré–Perron method. These series are convergent in the whole physical domain $z\in (?\infty ,+\infty )$. Visualization of constructed solutions and numerical study of the tunneling effect are performed.     

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both $\left(n\right)$th and $(n+1)$th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.     

Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term

In this article, we study and analyze a Galerkin mixed finite element (MFE) method combined with time second-order discrete scheme for solving nonlinear time fractional diffusion equation with fourth-order derivative term. We firstly introduce an auxiliary variable $\sigma =△u$, reduce the fourth-order problem into a coupled system with two equations, discretize the obtained coupled system at time $t_{k?\frac{\alpha }{2}}$ by a second-order difference scheme with second-order approximation for fractional derivative, then formulate mixed weak formulation and fully discrete MFE scheme. Further, we give the detailed proof for stability of scheme, the existence and uniqueness of MFE solution, and a priori error estimates. Finally, by some numerical computations, we test the theoretical results, which illustrate that we can obtain the numerical results for two variables, moreover, we arrive at second-order time convergence orders, which are higher than the ones yielded by the $L1$-approximation.     

The Laplacian spectral radius of bicyclic graphs with a given girth

Let $?(n,g)$ be the class of bicyclic graphs on $n$ vertices with girth $g$. Let ${?}_1(n,g)$ be the subclass of $?(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and ${?}_2(n,g)=?(n,g)?{?}_1(n,g)$. This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ${?}_1(n,g)$ and ${?}_2(n,g)$, respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for $?(n,g)$ are also obtained.     

Some generalized Ostrowski–Grüss type integral inequalities

In this paper, we establish some new Ostrowski–Grüss type integral inequalities involving $(k?1)$ interior points in 1D case, which are generalizations of some known results in the literature, and one of which is sharp. Then we deduce an Ostrowski–Grüss type integral inequality in 2D case involving ${(k?1)}^2$ interior points for the first time. We also present one application on the estimate of error bound for numerical integration formula, in which a sharp error bound for a new numerical integration formula is provided by the results established.