Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods

Asgari, Zohreh; Hosseini, Shahram

doi:10.1016/j.cpc.2012.10.009

Cited by 20 publications

(9 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2.4. In [2], the authors had simulated the discrete energy which looks to likely be conservative. However, it might be caused by the high accuracy of the proposed pseudospectral method.…”

Section: )mentioning

confidence: 99%

“…In this test, we consider the equation ( and the boundary condition is periodic. Similar to [2], in order to study the evolution of the ring solitons, we plot both the surfaces and the corresponding contours in terms of sin( u 2 ) with h = 0.1 and ∆t = 0.1. As seen from Fig.3, Fig.4, Fig.5 and Fig.6, ring soliton shrinks for initial stage (t = 0), but as time goes on, oscillations and radiations begin to form and continue to form up to t = 4.…”

Section: Circular Ring Solitonmentioning

confidence: 99%

“…Analytical solutions to the unperturbed sine-Gordon equation with zero damping have been obtained by Lambs methods [29] and Bäcklund transformations. Many efforts have been attempted to develop numerical methods, such as the finite difference method [3,4,10], the time-splitting pseudospectral and spectral method [2], the finite element method [1], the meshfree reproducing kernel particle Ritz method [8], the local weak meshless method [11], the boundary element method [12], the Differential quadrature method [15] and the radial basis functions method [13] for the 2D sine-Gordon equation. However, there exist few available error estimate results in the above-mentioned works for the 2D case.…”

Section: Introductionmentioning

confidence: 99%

“…As for the sine-Gordon equation, to our knowledge, only a few results considered this vital property. Although many verification results of the discrete energy for various numerical methods are reported [2,3,4], the analysis at a theoretical level was hardly shown.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An efficient finite difference scheme for the 2D sine-Gordon equation

Kang¹,

Feng²,

Cheng³

et al. 2017

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit the discrete energy conservation for the undamped model theoretically. Due to the semi-implicit treatment for the nonlinear term, it leads to a sequence of nonlinear coupled equations. We use a linear iteration algorithm, which can solve them efficiently, and the contraction mapping property is also proven. Based on truncation errors of the numerical scheme, the convergence analysis in the discrete l 2 -norm is investigated in detail. Moreover, we carry out various numerical simulations, such as verifications of the second order accuracy, tests of energy conservation and circular ring solitons, to demonstrate the efficiency and the robustness of the proposed scheme.

show abstract

Section: )mentioning

confidence: 99%

Section: Circular Ring Solitonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An efficient finite difference scheme for the 2D sine-Gordon equation

Kang¹,

Feng²,

Cheng³

et al. 2017

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

show abstract

“…and the Neumann boundary conditions are ∂u ∂l (x, t) = w(x, t), x ∈ Γ, t > 0 (4) where l denotes the (typically exterior) normal to the boundary of the domain, and Γ is the boundary of Ω, i.e Γ = ∂Ω. The real parameter β 0 weights the dissipative term.…”

mentioning

confidence: 99%

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

Luo

Rabitz

2019

J Math Chem

View full text Add to dashboard Cite

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.Keywords Sine-Gordon equations · Meshless methods · High dimensional model representation 1 IntroductionThe sine-Gordon equation (SGE) is a nonlinear hyperbolic partial differential equation (PDE) involving the d'Alembert operator and the sine of the unknown function, and the SGE plays an important role in many mathematical physics applications. It was originally introduced by Bour [6] and rediscovered

show abstract

A stable time–space Jacobi pseudospectral method for two-dimensional sine-Gordon equation

Mittal

2020

J. Appl. Math. Comput.

View full text Add to dashboard Cite

Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods

Cited by 20 publications

References 27 publications

An efficient finite difference scheme for the 2D sine-Gordon equation

An efficient finite difference scheme for the 2D sine-Gordon equation

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

A stable time–space Jacobi pseudospectral method for two-dimensional sine-Gordon equation

Contact Info

Product

Resources

About