This paper is concerned with the time-step condition of a linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in t...This paper is concerned with the time-step condition of a linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a 'lifting' technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schr¨odinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.展开更多
Wave equation model (WEM) first developed by Lynch and Gray [2] is one of accurate and effective numerical methods to resolve shallow water equations. This paper shows the numerical consistency of the second-order wav...Wave equation model (WEM) first developed by Lynch and Gray [2] is one of accurate and effective numerical methods to resolve shallow water equations. This paper shows the numerical consistency of the second-order wave equation and the first-order continuity equation, analyzes the error between them. This paper also shows that the numerical friction factor τ0 appearing in wave equation is of key importance to the numerical solutions and mass conservation of wave equation model. Numerical calculations of M2 tidal waves in rectangular harbor and a quarter annular harbor are made to demonstrate that it is possible to find a proper numerical friction factor To with which accurate solutions and satisfactory mass conservation can be achieved by wave equation model.展开更多
We develop an exponential spline interpolation method to solve the nonlinear Schrödinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be uncon...We develop an exponential spline interpolation method to solve the nonlinear Schrödinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be unconditionally stable. The conservation quantities are computed to determine the conservation properties of the problem. We will describe the method and present numerical tests by two problems. The numerical simulations results demonstrate the well performance of the proposed method.展开更多
A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is...A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space- time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles＇ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler-Lagrange （EL） equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.展开更多
In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass...In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11571181 and 11731014)Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project
文摘This paper is concerned with the time-step condition of a linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a 'lifting' technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schr¨odinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.
文摘Wave equation model (WEM) first developed by Lynch and Gray [2] is one of accurate and effective numerical methods to resolve shallow water equations. This paper shows the numerical consistency of the second-order wave equation and the first-order continuity equation, analyzes the error between them. This paper also shows that the numerical friction factor τ0 appearing in wave equation is of key importance to the numerical solutions and mass conservation of wave equation model. Numerical calculations of M2 tidal waves in rectangular harbor and a quarter annular harbor are made to demonstrate that it is possible to find a proper numerical friction factor To with which accurate solutions and satisfactory mass conservation can be achieved by wave equation model.
文摘We develop an exponential spline interpolation method to solve the nonlinear Schrödinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be unconditionally stable. The conservation quantities are computed to determine the conservation properties of the problem. We will describe the method and present numerical tests by two problems. The numerical simulations results demonstrate the well performance of the proposed method.
基金This research was supported by the Na- tional Magnetic Confinement Fusion Energy Research Project （Grant Nos. 2015GB111003 and 2014GB124005）, the National Natural Science Foundation of China （Grant Nos. NSFC- 11575185, 11575186, and 11305171）, JSPS-NRF-NSFC A3 Fore- sight Program （Grant No. 11261140328）, the Key Research Pro- gram of Frontier Sciences CAS （QYZDB-SSW-SYS004）, Geo- Algorithmic Plasma Simulator （GAPS） Project, and the National Magnetic Confinement Fusion Energy Research Project （Grant No. 2013GB111002B）.
文摘A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space- time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles＇ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler-Lagrange （EL） equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.
基金Project supported by the National Natural Science Foundation of China (Grant No 11171038)
文摘In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.
