An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

Ramos, Higinio; Vigo‐Aguiar, Jesús

doi:10.1002/num.20212

Cited by 13 publications

(11 citation statements)

References 16 publications

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“…Some of the options available for time integration when using a moving grid method of lines code is surveyed in [46]. A new technique is proposed in [47] for the numerical integration of the system of ordinary differential equations that arises in the method of lines solution of timedependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability.…”

Section: Methods Of Linesmentioning

confidence: 99%

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The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition

Shakeri

Dehghan

2008

Computers & Mathematics with Applications

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a b s t r a c tHyperbolic partial differential equations with an integral condition serve as models in many branches of physics and technology. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional nonlocal hyperbolic equation is presented by the method of lines. The method of lines (MOL) is a general way of viewing a partial differential equation as a system of ordinary differential equations. The partial derivatives with respect to the space variables are discretized to obtain a system of ODEs in the time variable and then a proper initial value software can be used to solve this ODE system. We propose two forms of MOL for solving the described problem. Several numerical examples and also some comparisons with finite difference methods will be investigated to confirm the efficiency of this procedure.

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Section: Methods Of Linesmentioning

confidence: 99%

“…It is applicable to a wide range of problems in many areas [47]. The reader can see the Appendix of [22] for some problems in physics, fluid dynamics, reactor models, automatic control, and more.…”

Section: Testmentioning

confidence: 99%

The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition

Shakeri

Dehghan

2008

Computers & Mathematics with Applications

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“…From Table 1, it is clear that the Block Backward Differentiation Formula (BBDF) yields a more accurate result than the result of the Crank-Nicholson computation and outperforms the results obtained by Cash [2] and Jator [7].…”

Section: Problemmentioning

confidence: 96%

A Collocation Based Block Multistep Scheme without Predictors for the Numerical Solution Parabolic Partial Differential Equations

Ehigie

2021

JRRS

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Introduction: Many life problems often result in differential equations models when formulated mathematically, particularly problems that depend on time and rates which give rise to Partial Differential Equations (PDE). Aims: In this paper, we advance the solution of some Parabolic Partial Dif-ferential Equations (PDE) using a block backward differentiation formula im-plemented in block matrix form without predictors. Materials and Methods: The block backward differentiation formula is devel-oped using the collocation method such that multiple time steps are evaluated simultaneously. Results: A five-point block backward differentiation formula is developed. The stability analysis of the methods reveals that the method is L 0 stable. Conclusion:The implementation of some parabolic PDEs shows that the method yields better accuracy than the celebrated Crank-Nicholson's method.

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“…It is noted that Vigo-Aguiar and Ramos [44] also constructed a special family of Runge-Kutta collocation algorithms based on Chebyshev-Gauss-Lobatto points, with A-stability and stiffly accurate characteristics. The interested reader may also refer to [34,35] for additional information.…”

Section: Introductionmentioning

confidence: 99%

A Chebyshev-Gauss Spectral Collocation Method for Ordinary Differential Equations

Zhongqing¹

2015

JCM

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In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.Mathematics subject classification: 65L05, 65L60, 41A10.

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An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

Cited by 13 publications

References 16 publications

The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition

The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition

A Collocation Based Block Multistep Scheme without Predictors for the Numerical Solution Parabolic Partial Differential Equations

A Chebyshev-Gauss Spectral Collocation Method for Ordinary Differential Equations

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