Laguerre-Gauss collocation method for initial value problems of second order ODEs

“…Due to their high accuracy, spectral methods (including spectral collocation methods) have been applied to the numerical integration of ODEs in recent years. For example, Guo et al developed several Laguerre spectral collocation methods [20,23,51,52] and Legendre spectral collocation methods [21,22,24,47] for nonlinear first and second-order IVPs of ODEs. In [1,2], several spectral Galerkin and collocation methods were introduced for the numerical solutions of nonlinear Hamiltonian (ODE) systems.…”

Section: Introductionmentioning

confidence: 99%

Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications To Wave Equations

Wang¹,

Tong²,

Zhang³

2024

JCM

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We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.

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“…Hence, they are not suitable for long-term computation since we can not choose the polynomial degree N very large in actual computation (cf. [24]). The other point to be emphasized is that no error analysis has been developed for these methods.…”

Section: Introductionmentioning

confidence: 99%

A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations

Guo¹,

Huang²

2017

Commun. Comput. Phys.

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A domain decomposition based spectral collocation method is proposed for numerically solving Lane-Emden equations, which are frequently encountered in mathematical physics and astrophysics. Compared with the existing methods, this method requires less computational cost and is particularly suitable for long-term computation. The related error estimates are also established, indicating the spectral accuracy of the method. The numerical performance and efficiency of the method are illustrated by several numerical experiments.

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Laguerre-Gauss collocation method for initial value problems of second order ODEs

Abstract: This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.

Cited by 6 publications

References 31 publications

Spectral and pseudospectral methods for unbounded domains

Spectral and pseudospectral methods for unbounded domains

Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications To Wave Equations

A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations

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