Computational Partial Differential Equations Using MATLAB

Li, Jichun; Chen, Yitung

doi:10.1201/9781420089059

Cited by 56 publications

(39 citation statements)

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“…and a 0 = 1 3 (−1 + 10α). For α = 0, we get 4th order accurate scheme while using α = 2/11 the scheme becomes sixth order accurate which leads to a 1 = 12 11 and a 0 = 3 11 [25], [30], [31]. Also near boundary points to construct sixth order compact scheme which sustain accuracy throughout the three dimensional domain [25], [30], [31].…”

Section: Higher Order Compact Difference Schemementioning

confidence: 97%

Numerical Study to Coupled Three Dimensional Reaction Diffusion System

et al. 2019

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The numerical solution of reaction diffusion systems may require more computational efforts if the change in concentrations occurs extremely rapid. This is because more time points are needed to resolve the reaction diffusion process accurately. In this paper, three finite difference implicit schemes are used which are unconditionally stable in order to enhance consistency. Novelty is reported by compact finite difference implicit scheme on a reaction diffusion system with higher accuracy measured by L 2 , L ∞ , and Relative error norms. Efficiency is observed by reducing grid space along small temporal steps. CPU performance, transmission capacity along comparison of three schemes shows excellent agreement with the analytical solution.INDEX TERMS Reaction diffusion systems, alternating direction implicit, Douglas scheme, higher order compact scheme, Thomas algorithm.

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Section: Higher Order Compact Difference Schemementioning

confidence: 97%

Numerical Study to Coupled Three Dimensional Reaction Diffusion System

et al. 2019

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“…The reader is referred to Refs. [5,7] for more details on how to generate compact finite-difference formulas. In this study, the spatial derivatives are approximated with the formulas introduced by Li in Ref.…”

Section: Sixth-order Compact Finite-difference Scheme Formulasmentioning

confidence: 99%

“…In this study, the spatial derivatives are approximated with the formulas introduced by Li in Ref. [5]. Below the formulas in Ref.…”

Section: Sixth-order Compact Finite-difference Scheme Formulasmentioning

confidence: 99%

“…When using CFDS to study complex problems, its high accuracy and stability have also attracted much attention from many scholars. Experience shows that the compact scheme is much more accurate than the corresponding explicit scheme of the same order [5]. Over the past three decades, the methods for developing high-order CFDS have made great progress.…”

Section: Introductionmentioning

confidence: 99%

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A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

et al. 2020

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This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4, 4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4, 4)-Padé approximation.

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“…In order to make those near-boundary points have the same order accuracy as interior nodes, they should be obtained by matching Taylor series expansions to the order of O(h 6 ) at boundary points 1, 2, N − 1 and N , hence we get the following formulae [21] φ 1 + 126…”

Section: High-order Numerical Schemementioning

confidence: 99%

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

Chen

Zhang

Liu

2020

Applied Mathematics and Computation

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This paper introduces a modification of n-dimensional Hopf-Cole transformation to the n-dimensional Burgers' system. We obtain the n-dimensional heat conduction equation through the modification of the Hopf-Cole transformation.Then the high-order exponential time differencing precise integration method (PIM) based on fourth-order Taylor approximation in combination with a spatially global sixth-order compact finite difference (CFD) scheme is presented to solve the equation with high accuracy. Moreover, coupling with the Strang splitting method, the scheme is extended to multi-dimensional (two,three-dimensional)Burgers' system, which also possesses high computational efficiency and accuracy. Several numerical examples verify the performance and capability of the proposed scheme. Numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method. In addition, the two-dimensional and three-dimensional examples demonstrate excellent adaptability, and the numerical simulation results also have very high accuracy in medium Reynolds numbers.

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Computational Partial Differential Equations Using MATLAB

Cited by 56 publications

References 0 publications

Numerical Study to Coupled Three Dimensional Reaction Diffusion System

Numerical Study to Coupled Three Dimensional Reaction Diffusion System

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

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