COMPARISON OF FINITE‐DIFFERENCE SCHEMES FOR THE GROSS‐PITAEVSKII EQUATION

2009 ◽  
Vol 14 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Vyacheslav A. Trofimov ◽  
Nikolai Peskov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Pitaevskii Equation ◽  
Conservative Scheme ◽  
Conservative Finite Difference Scheme

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L 2 norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

A NEW LINEAR AND CONSERVATIVE FINITE DIFFERENCE SCHEME FOR THE GROSS–PITAEVSKII EQUATION WITH ANGULAR MOMENTUM ROTATION

2019 ◽  
Vol 61 (02) ◽  
pp. 204-232
Author(s):  
JIN CUI ◽  
WENJUN CAI ◽  
CHAOLONG JIANG ◽  
YUSHUN WANG
Keyword(s):  
Angular Momentum ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Mesh Size ◽  
Pitaevskii Equation ◽  
Time Step ◽  
Lifting Technique ◽  
Conservative Finite Difference Scheme

A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal $H^{1}$ -error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order $O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step $\unicode[STIX]{x1D70F}$ and mesh size $h$ . Numerical experiments have been carried out to show the efficiency and accuracy of our new method.

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

scholarly journals On the Computation of the Mountain Torque from Gridded Global Datasets

2008 ◽  
Vol 136 (10) ◽  
pp. 4005-4009 ◽  
Author(s):  
Huei-Ping Huang ◽  
Klaus M. Weickmann
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Spectral Method ◽  
Surface Pressure ◽  
Finite Difference Scheme ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Axial Component ◽  
Practical Applications ◽  
Benchmark Calculation

This note evaluates the numerical schemes used for computing the axial component of the mountain torque from gridded global surface pressure and topography datasets. It is shown that the two formulas of the mountain torque based on (i) an integral of the product of the surface pressure and the gradient of topography, and (ii) an integral of the product of the topography and the surface pressure gradient, should produce identical results if a centered even-ordered finite-difference scheme or the spectral method is used to evaluate the integrand. Noncentered finite-difference schemes are not recommended not only because they produce extremely large errors but also because they produce different results for the two formulas. When compared with the benchmark calculation using the spectral method, it is found that the centered fourth-order finite-difference scheme is an efficient and generally accurate approximation for practical applications. Using the data from NCEP–NCAR reanalysis, the finite-difference schemes generally underestimate the global mountain torque compared to the benchmark. This negative error is interpreted as due to the asymmetry in the distribution of surface pressure and in the steepness of the topography between the western and eastern slopes of the mountains.

scholarly journals ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2014 ◽  
Vol 19 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Raimondas Čiegis ◽  
Natalija Tumanova
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
The Stability

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

scholarly journals Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Lei Zhang ◽  
Lisha Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Burgers Equation ◽  
Solitary Wave Solution ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Fisher Equation ◽  
Nonstandard Finite Difference Scheme ◽  
Nonstandard Finite Difference

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

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