Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman--Rachford Splitting

Cheng, Hanz Martin; Lin, Ping; Sheng, Quan Z.; Tan, Roger C. E.

doi:10.1137/s1064827501380691

Cited by 42 publications

(84 citation statements)

References 23 publications

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“…The example reflects the necessariness and urgency of new treatments, considerations, and analysis of traditional decomposition schemes based on (2.5) otherwise the reliability of the algorithms can be of concern. This has motivated a tremendous effort in the decomposition community in searching for a new generation of the difference methods [1,4,5,9,19,27,29,32,44,45].…”

Section: Definition 22mentioning

confidence: 99%

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Adaptive decomposition finite difference methods for solving singular problems—A review

Sheng

2009

Front. Math. China

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Decomposition, or splitting, finite difference methods have been playing an important role in the numerical solution of nonsingular differential equation problems due to their remarkable efficiency, simplicity, and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with various kinds of adaptations can be found in many recent publications. The novel approaches have been proven to be highly effective and reliable in operations. In this article, we will focus on some of the latest developments in the area. Key comments and discussion will be devoted to two particularly interesting issues in the research, that is, direct solutions of degenerate singular reactiondiffusion equations and nonlinear sine-Gordon wave equations. Numerical experiments with simulated demonstrations will be given.

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Section: Definition 22mentioning

confidence: 99%

“…One of the most exciting developments in the area is for the quenchingcombustion differential equations containing concentrated nonlinear source terms [1,6,8,9,30,45,46]. The preliminary study of the problems can be found in Ref.…”

Section: Decompositions For Singular Reaction-diffusion Problemsmentioning

confidence: 99%

Adaptive decomposition finite difference methods for solving singular problems—A review

Sheng

2009

Front. Math. China

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“…Our temporal coordinates and temporal increments will be determined in term of u t since blow-up takes place in u t . We adopt the following arc-length monitor function for u t [9,14,18,26],…”

Section: Discretizationsmentioning

confidence: 99%

“…Obviously, the irregular mesh obtained in section 2 make the system (8) uniquely solvable and an irregular difference scheme can be constructed based on approximate derivatives obtained in (9). We substitute approximate partial derivatives (9) into (4) and get an implicit difference scheme…”

Section: The Difference Schemementioning

confidence: 99%

“…Its accuracy may diminish when the length scale of the singularity is less than the step size. It has been demonstrated that significant improvements in accuracy and efficiency can be achieved by adaptive method in numerically solving problems with this type of singularities [4,9,5,10,22,26], since the mesh points can be concentrated locally in the regions with rapid variation of the solution.…”

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confidence: 99%

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Numerical solution of quenching problems using mesh-dependent variable temporal steps

Liang¹,

Lin²,

Tan³

2007

Applied Numerical Mathematics

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A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains

Beauregard

Sheng

2013

Numerical Methods Partial

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This article studies a fully adaptive finite difference method for solving quenching‐type nonlinear reaction‐diffusion equations over circular domains. Although an auxiliary condition at the origin and radial symmetry are imposed, adaptations are accomplished via arc‐length‐based monitoring functions in space and time, respectively. The monotonicity and positivity of the numerical solution are proved following a suitable grid constraint, and the numerical stability is ensured in the von Neumann sense. Theoretical bounds of the critical quenching radius are obtained and then refined through the computation. Computational examples are provided to illustrate the effectiveness and plausibility of the new adaptive computational procedure developed. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 472–489, 2014

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Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman--Rachford Splitting

Cited by 42 publications

References 23 publications

Adaptive decomposition finite difference methods for solving singular problems—A review

Adaptive decomposition finite difference methods for solving singular problems—A review

Numerical solution of quenching problems using mesh-dependent variable temporal steps

A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains

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