Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices

Hoz, Francisco de la; Vadillo, Fernando

doi:10.1016/j.cpc.2011.12.012

Cited by 7 publications

(9 citation statements)

References 32 publications

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“…Therefore, we have tested different techniques: integrating factors in [8,11], an ETDRK4 scheme in [9], exact integration in time in [10], and, finally, an IMEX scheme in [12]. Remark that [10,12] involve the use of Sylvester-type equations.…”

Section: Discussionmentioning

confidence: 98%

“…Without trying to exhaust all the possible applications of these powerful methods, we have considered some extremely important (and frequent) particular cases, involving the use of trigonometric polynomials, as in [9,8,11]; the use of Chebyshev polynomials, as in [10]; and, finally, the use of Hermite functions, sinc functions and rational Chebyshev polynomials in unbounded domains, as in [12]. Moreover, while the use of FFT has been a basic tool in the implementation of these methods in [9,8], we have followed a primarily matrix approach in [11,12,10]. Indeed, as justified in Section 4.1, we think that the use of differentiation matrices should be labeled by no means a minor approach.…”

Section: Discussionmentioning

confidence: 99%

“…(28) In [11], we have combined the power of integrating factors, as in [8], with ideas taken from [10], such us the avoidance of tensorization. The developed scheme does not require the explicit calculation of matrix exponentials, and can be implemented without toil in an arbitrary number of spatial dimensions.…”

Section: Other Applications Of Differentiation Matricesmentioning

confidence: 99%

“…To finish this section, let us also mention [11], where we have used differentiation matrices to solve the sine-Gordon equation in N dimensions: 1 c 2 u tt = u − sin(u). (28) In [11], we have combined the power of integrating factors, as in [8], with ideas taken from [10], such us the avoidance of tensorization.…”

Section: Other Applications Of Differentiation Matricesmentioning

confidence: 99%

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Numerical simulations of time-dependent partial differential equations

Hoz

Vadillo

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tWhen a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness, with a rapid exponential decay of some modes (as in a dissipative PDE), or a rapid oscillation of some modes (as in a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation both in space and in time.Along our research, we have focused on a particular case of spectral methods, the socalled pseudo-spectral methods, to solve numerically time-dependent PDEs using different techniques: an integrating factor, in de la Hoz and Vadillo (2010); an exponential time differencing method, in de la Hoz and Vadillo (2008); and differentiation matrices in the theoretical frame of matrix differential equations, in de la Vadillo (2012, 2013a,b). This paper, which is a unified review of those contributions, aims at providing a better understanding of those methods, by illustrating their variety and, more importantly, their power. Furthermore, we also give emphasis to choosing adequate schemes to advance in time.

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Section: Discussionmentioning

confidence: 98%

Section: Discussionmentioning

confidence: 99%

Section: Other Applications Of Differentiation Matricesmentioning

confidence: 99%

Section: Other Applications Of Differentiation Matricesmentioning

confidence: 99%

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Numerical simulations of time-dependent partial differential equations

Hoz

Vadillo

2016

Journal of Computational and Applied Mathematics

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“…It is known to all that soliton solutions appear in the 1D and 2D SG equations, particularly solitons solutions to the 1D SG equation studied numerically and theoretically. In a very intuitive and easy way to express, Francisco et al 25 implemented operational matrices to SG equation. In a very intuitive and easy way to express, Francisco et al 25 implemented operational matrices to SG equation.…”

Section: History Of Solving Nonlinear Sg Equationsmentioning

confidence: 99%

Generalized Lagrange Jacobi‐Gauss‐Lobatto vs Jacobi‐Gauss‐Lobatto collocation approximations for solving (2 + 1)‐dimensional Sine‐Gordon equations

Latifi

Delkhosh

2019

Math Methods in App Sciences

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The Sine-Gordon (SG) equations are very important in that they can accurately model many essential physical phenomena. In this paper, the Jacobi-Gauss-Lobatto collocation (JGL-C) and Generalized Lagrange Jacobi-Gauss-Lobatto collocation (GLJGL-C) methods are adopted and compared to simulate the (2 + 1)-dimensional nonlinear SG equations. In order to discretize the time variable t, the Crank-Nicolson method is employed. For the space variables, two numerical methods based on the aforementioned collocation methods are applied. Furthermore, error estimation for both methods is provided. The present numerical method is truly effective, free of integration and derivative, and easy to implement. The given examples and the results assert that the GLJGL-C method outperforms the JGL-C method in terms of computation speed. Also, the presented methods are very valid, effective, and reliable.

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

Luo

Rabitz

2019

J Math Chem

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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

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Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices

Cited by 7 publications

References 32 publications

Numerical simulations of time-dependent partial differential equations

Numerical simulations of time-dependent partial differential equations

Generalized Lagrange Jacobi‐Gauss‐Lobatto vs Jacobi‐Gauss‐Lobatto collocation approximations for solving (2 + 1)‐dimensional Sine‐Gordon equations

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

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