Fourth-order compact difference and alternating direction implicit schemes for telegraph equations

Xie, Shusen; Yi, Su-Cheol; Kwon, Tae In

doi:10.1016/j.cpc.2011.11.023

Cited by 38 publications

(13 citation statements)

References 29 publications

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“…Yousefi developed a new Legendre multiwavelet Galerkin method for the same purpose. In , other methods based on difference schemes were introduced.…”

Section: Introductionmentioning

confidence: 99%

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

Abd-Elhameed

Doha

Youssri

et al. 2016

Numerical Methods Partial

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The telegraph equation describes various phenomena in many applied sciences. We propose two new efficient spectral algorithms for handling this equation. The principal idea behind these algorithms is to convert the linear/nonlinear telegraph problems (with their initial and boundary conditions) into a system of linear/nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of our algorithm in the linear case is that the resulting linear systems have special structures that reduce the computational effort required for solving them. The numerical algorithms are supported by a careful convergence analysis for the suggested Chebyshev expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithms. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1553–1571, 2016

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“…Yousefi developed a new Legendre multiwavelet Galerkin method for the same purpose. In , other methods based on difference schemes were introduced.…”

Section: Introductionmentioning

confidence: 99%

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

Abd-Elhameed

Doha

Youssri

et al. 2016

Numerical Methods Partial

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“…In this comparison, we consider the domain ( x , y ) ∈ [ − 5,5] × [ − 5,5] in time t = 4. The numerical results show the high accuracy of the present method as compared with the method of .…”

Section: Numerical Experimentsmentioning

confidence: 79%

“…In this subsection, the MDPS method is tested for the two‐dimensional SG equation in large domain. We also present the comparison of the MDPS method with the SDPS method and the compact finite difference scheme of . The numerical experiments are conducted by taking N x = N y = N points.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…with initial conditions

u_{normalexact} (x MathClass-punc, y MathClass-punc, t) MathClass-rel= 4 {normaltan}^{MathClass-bin- 1} (normalexp (x MathClass-bin+ y MathClass-bin- t)) MathClass-punc,

Table shows the comparison of the maximum error between MDPS and SDPS methods and the alternating direction implicit compact finite difference method presented in . In this comparison, we consider the domain ( x , y ) ∈ [ − 5,5] × [ − 5,5] in time t = 4.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Also, a local weak meshless technique based on the radial point interpolation method is proposed in . Cui employed a fourth‐order compact finite difference scheme for the sine‐Gordon equation and Xie et al presented the alternating direction implicit (ADI) compact difference schemes for the hyperbolic equations such as the sine‐Gordon equation. In , authors proposed the meshless local Petrov–Galerkin method for the sine‐Gordon equation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

Taleei

Dehghan

2013

Math Methods in App Sciences

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In this article, we study an explicit scheme for the solution of sine-Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge-Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix-vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sineGordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions.

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New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

Zhang

Miao

2017

Numerical Methods Partial

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This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017

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Fourth-order compact difference and alternating direction implicit schemes for telegraph equations

Cited by 38 publications

References 29 publications

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

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