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

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

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“…Hence, the exact solution of Example 1 is uðx, y, tÞ = cos ðπxÞ cos ðπyÞ cos ðtÞ as in Kang et al [40].…”

Section: Example 1 Consider the Sine-gordon Equation [40]mentioning

confidence: 93%

“…In this paper, we investigate the solution of the twodimensional nonlinear sine-Gordon equation [40]:…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of Two-Dimensional Sine-Gordon Equation

Deresse

Mussa

Gizaw

2021

Advances in Mathematical Physics

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“…[8,25,36] for the one dimensional sine-Gordon equation and in Ref. [22] for the two dimensional case, respectively.…”

Section: Numerical Examplesmentioning

confidence: 97%

“…see Refs. [4,5,8,14,17,19,22,25,40] and references therein). However, most of existing energy-conserved schemes are fully implicit, which implies that one has to solve a system of nonlinear equations, at each time step.…”

Section: Introductionmentioning

confidence: 99%

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

2019

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In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energyconserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semidiscrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete H 1 h -norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

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“…More extensive applications of energy-stable or energy conservative method to a wide class of physical models also are available. See the related works for wave equations [7,8], the phase field crystal equation [9] and the Cahn-Hilliard equation [10][11][12], etc.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulations of the energy-stable scheme for Swift-Hohenberg equation

Jun

2019

THERM SCI

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A collocation Fourier scheme for Swift-Hohenberg equation based on the convex splitting idea is implemented. To ensure an efficient numerical computation, we propose a general framework with linear iteration algorithm to solve the non-linear coupled equations which arise with the semi-implicit scheme. Following the contraction mapping theorem, we present a detailed convergence analysis for the linear iteration algorithm. Various numerical simulations, including verification of accuracy, dissipative property of discrete energy and pattern formation, are presented to demonstrate the efficiency and the robustness of proposed method.

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

Cited by 9 publications

References 29 publications

Analytical Solution of Two-Dimensional Sine-Gordon Equation

Analytical Solution of Two-Dimensional Sine-Gordon Equation

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

Numerical simulations of the energy-stable scheme for Swift-Hohenberg equation

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