A splitting moving mesh method for reaction–diffusion equations of quenching type

Liang, Kewei; Lin, Ping; Ong, Ming Tze; Tan, Roger C. E.

doi:10.1016/j.jcp.2005.11.019

Cited by 19 publications

(21 citation statements)

References 36 publications

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“…In order to increase accuracy with a moderate number of mesh grids, we apply the moving mesh strategy in our algorithm. Moving mesh method is an active study field in recent years and has many applications in computational physics [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

An adaptive algorithm for the Thomas–Fermi equation

Zhu

et al. 2011

Numer Algor

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A free boundary value problem is introduced to approximate the original Thomas-Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.Keywords Thomas-Fermi equation · Adaptive finite element method · Moving mesh method · Equidistribution · Free boundary value problem · Semi-infinite interval

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Section: Introductionmentioning

confidence: 99%

An adaptive algorithm for the Thomas–Fermi equation

Zhu

et al. 2011

Numer Algor

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“…Even so, reliable theoretical estimates have been established for rectangular domains. These estimates have been verified numerically in a number of resources [2,11,17].…”

Section: Theoretical Estimates Of Quenching Domainsmentioning

confidence: 71%

Numerical solutions to singular reaction–diffusion equation over elliptical domains

Beauregard

2015

Applied Mathematics and Computation

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Solid fuel ignition models, for which the dynamics of the temperature is independent of the single-species mass fraction, attempt to follow the dynamics of an explosive event. Such models may take the form of singular, degenerate, reaction-diffusion equations of the quenching type, that is, the temporal derivative blows up in finite time while the solution remains bounded. Theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies. Quenching domains are known to exist for piecewise smooth boundaries, but often lack theoretical estimates. Rectangular geometries have been primarily studied. Here, this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise, smooth, connected geometries. Elliptical domains are of primary interest and a Peaceman-Rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids. Rigorous numerical analysis ensures numerical solution monotonicity, positivity, and linear stability of the proposed algorithm. Simulation experiments are provided to illustrate the accomplishments.

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“…Following the idea in [23], we divide the analysis into two parts. One is the solution away from the quenching point.…”

Section: Accuracy and Stability Analysismentioning

confidence: 99%

“…It can also be shown from maximum principle (ref. [23]) that 0 ≤ u e < 1 (In our programme, we define a small positive constant δ as a tolerance, e.g., we could choose δ related to τ min . And we will say u quenches when u reaches 1 − δ).…”

Section: Stability Near Quenchingmentioning

confidence: 99%