This paper introduces three local structure-preserving algorithms for the one-dimensional nonlinear Schr & ouml;dinger equation with power law nonlinearity, comprising two local energy- conserving algorithms and o...
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This paper introduces three local structure-preserving algorithms for the one-dimensional nonlinear Schr & ouml;dinger equation with power law nonlinearity, comprising two local energy- conserving algorithms and one local momentum-conserving algorithm. Additionally, we extend these local conservation algorithms to achieve global conservation under periodic boundary conditions. Theoretical analyses confirm the conservation properties of these algorithms. In numerical experiments, we validate the advantages of these algorithms in maintaining longterm energy or momentum conservation by comparing them with a multi-symplectic Preissman algorithm.
This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d-dimensional ( d = 1 {d=1} or 2) nonlinear Schr & ouml;dinger equation. Based on the method of lin...
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This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d-dimensional ( d = 1 {d=1} or 2) nonlinear Schr & ouml;dinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.
The paper aims to construct a class of high-order explicit conservative schemes for the space fractional nonlinear Schrodinger equation by combing the invariant energy quadratization method and Runge-Kutta method. We ...
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The paper aims to construct a class of high-order explicit conservative schemes for the space fractional nonlinear Schrodinger equation by combing the invariant energy quadratization method and Runge-Kutta method. We first derive the Hamiltonian formulation of the equation, and obtain a new equivalent system via introducing a scalar variable. Then, we propose a semi-discrete conservative system by using the Fourier pseudo-spectral method to approximate the equivalent system in space. Further applying the fourth-order modified Runge-Kutta method to the semi-discrete system gives two classes of schemes for the equation. One scheme preserves the energy while the other scheme conserves the mass. Numerical experiments are provided to demonstrate the conservative properties, convergence orders and long time stability of the proposed schemes.
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 ...
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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. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplec...
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In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplectic algorithms, eight local energy-conserving algo- rithms and eight local momentum-conserving algorithms. Among these algorithms, some have been discussed and widely used while the most are new. The outstanding advantage of these proposed algorithms is that they conserve the local structures in any time-space re- gion exactly. Therefore, the local structure-preserving algorithms overcome the restriction of global structure-preserving algorithms on the boundary conditions. Numerical experiments are conducted to show the performance of the proposed methods. Moreover, the unified framework can be easily applied to many other equations.
For separable Hamiltonian systems, we construct first ever three symplectic explicit Runge-Kutta-Nystr & ouml;m methods of six orders with exact parameters. We numerically and theoretically compare these three new...
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For separable Hamiltonian systems, we construct first ever three symplectic explicit Runge-Kutta-Nystr & ouml;m methods of six orders with exact parameters. We numerically and theoretically compare these three new exact methods with the only existing exact 6-th order symplectic explicit partitioned Runge-Kutta method and two approximate 6-th order symplectic explicit Runge-Kutta-Nystr & ouml;m methods. These new methods are more accurate and stable.
Maxwell's equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper i...
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Maxwell's equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.
This paper considers the newly introduced generalized scalar auxiliary variable approaches to construct high-efficiency energy-preserving schemes for the sine-Gordon equation with Neumann boundary conditions. The equa...
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This paper considers the newly introduced generalized scalar auxiliary variable approaches to construct high-efficiency energy-preserving schemes for the sine-Gordon equation with Neumann boundary conditions. The equation is first reformulated into an equivalent system by defining a new auxiliary variable that is not limited to square root. Then, the cosine pseudo-spectral method is applied to the system and derive a semi-discrete conservative scheme. Subsequently, we combine the auxiliary variable with the nonlinear term and use an explicit technique discretization in time to derive a fully-discrete energy-preserving scheme. Furthermore, a fast algorithm based on the discrete cosine transform technique reduces the computational complexity in practical computation. Finally, various numerical experiments are displayed to verify the accuracy, efficiency and conservation of the proposed schemes.
In this paper, we focus on the numerical solutions of Maxwell's equations with Dirichlet boundary conditions in rectangular coordinate. A class of explicit methods is derived by using an effective solver for a sys...
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In this paper, we focus on the numerical solutions of Maxwell's equations with Dirichlet boundary conditions in rectangular coordinate. A class of explicit methods is derived by using an effective solver for a system of ordinary differential equations which is obtained by approximating on spatial fields. A significant advantage of this class of methods is their simplicity and their ease of implementation. The error estimates presented in this paper show that the numerical solutions obtained by this class of methods is of high-order. The main advantage of this class of methods is that it is divergence-free.
In this paper, we propose a class of stochastic exponential discrete gradient schemes for SDEs with linear and gradient components in the coefficients. The root mean-square errors of the schemes are analyzed, and the ...
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In this paper, we propose a class of stochastic exponential discrete gradient schemes for SDEs with linear and gradient components in the coefficients. The root mean-square errors of the schemes are analyzed, and the structure-preserving properties of the schemes for SDEs with special structures are investigated. Numerical tests are performed to verify the theoretical results and illustrate the numerical behavior of the proposed methods. (C) 2021 Elsevier B.V. All rights reserved.
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