Optimal error estimate of a compact scheme for nonlinear Schrödinger equation

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

The use of lower order approximations in the neighborhood of boundaries coupled with higher order interior approximations is examined for the mixed initial boundary-value problem for hyperbolic partial differential equations.Uniform error can be maintained using smaller grid intervals with the lower order approximations near the boundaries.Stability results are presented for approximations to the initial boundary-value problem for the model equation uf + cux = 0 which are fourth order in space and second order in time in the interior and second order in both space and time near the boundaries.These results are generalized to a class of methods of this type for hyperbolic systems.Computational results are presented and comparisons are made with other methods.

Computation of steady and oscillatory convection in saturated porous media

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation ${u_t} = - c{u_x}$. A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.

Multidimensional difference schemes with fourth-order accuracy

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection–diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter-uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

A nonlinear discontinuity-preserving method for noise removal for band pass signals such as signals modulated with Binary Phase-Shift Keying (BPSK) modulation is proposed in this paper. This method is inspired by the anisotropic diffusion algorithm to remove noise and preserve discontinuities in band pass signals modulated with a single frequency. It is demonstrated here that nonlinear noise removal method for a real valued band pass signal requires a solution for a nonlinear partial differential equation which is of fourth order in space and second order in time. The results presented in this work show better performance in nonlinear noise removal for real valued band pass signals in comparison with the previous work in the literature.

In this paper, we construct a compact difference scheme for the time-fractional partial integro-differential equation. This model involves two nonlocal terms in time, i.e., a Caputo time-fractional derivative and an integral term with memory. We obtain the stability and the discrete $L_{2}$ convergence with second-order in time and fourth-order in space by the energy method. Two numerical examples are provided to confirm the theoretical results.

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrodinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h 4 +� 2 ) in the discrete L 1 -norm with time stepand mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme. Mathematics subject classification: 65M06, 65M12.

A second-order in time and fourth-order in space, alternating direction implicit finite difference time domain (ADI-FDTD) scheme is presented for radio-wave propagation in a isotropic collisionless plasma. At the same time, the unconditional stability of the higher order ADI-FDTD formulation for plasma is analytically proved. By analysis of the growth factor, the numerical dispersion relation is derived. In addition, the numerical dispersion errors are investigated. From the error analysis and phase velocity data, it is argued that the proposed approach will significantly minimize the dispersion errors compared to the ADI-FDTD method.

Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Based on the common equation of seismic wave fields and its Hamiltonian formulation, an implicit symplectic method (ISM) is proposed for seismic wave modelingin this paper. Combined with helix boundary conditions, the spectral factorization method is applied to give a fast way for the ISM modeling. The propagation of seismic wave can be described by a same scalar acoustic euqation for homogeneous, complex isotropic and anisotropic media. With the assumption of no dissipation, the scalar equation depicts the symplectic evolution of seismic wave fields which is an infinite dimensional Hamiltonian system. Therefore, the scalar equation can be rewritten into Hamiltonian formulation and consequently its symplectic schemes can be given. Compared with the explicit symplectic method (ESM), ISM is unconditionally stable and permits big time steps and long‐time calculations which may be impossible for ESM. But the inverse of large matrix must be calculated at each step of ISM, which is a time consuming process. Fortunately, the matrix is band‐shape with a helical boundary conditions. In addition, the nonzero units in each row of the matrix not only are located in the same relative positions but also have similar values. Therefore, spectral factorization method can be used to get the inversion of the matrix in a very quick way. Based on the spectral factorization method, an ISM scheme with second order in time is applied to model the wave fields in constant velocity, layer velocity and Marmousi velocity models. The results indicates that with the help of spectral factorization ISM is a quite good method in seismic wave modeling.

Good Boussinesq equations will be considered in this work. First we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then three fully discrete schemes are developed based on symplectic scheme in time direction, which are sympletic-structure preserving.Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical test indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.