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

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

    Research output: Contribution to journalArticle

    17 Citations (Scopus)

    Abstract

    This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.
    Original languageEnglish
    Pages (from-to)757-777
    Number of pages21
    JournalJournal of Computational Physics
    Volume215
    Issue number2
    DOIs
    Publication statusPublished - Jul 2006

    Fingerprint

    reaction-diffusion equations
    mesh
    Quenching
    quenching
    Derivatives
    viability
    infinity
    Program processors
    Differential equations
    differential equations
    arcs

    Keywords

    • Nonlinear reaction-diffusion equations
    • Quenching singularity
    • Moving mesh method
    • Peaceman–Rachford splitting
    • ADI monotonicity
    • Parallelization

    Cite this

    Liang, Kewei ; Lin, Ping ; Ong, Ming Tze ; Tan, Roger C. E. / A splitting moving mesh method for reaction-diffusion equations of quenching type. In: Journal of Computational Physics. 2006 ; Vol. 215, No. 2. pp. 757-777.
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    abstract = "This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.",
    keywords = "Nonlinear reaction-diffusion equations, Quenching singularity, Moving mesh method, Peaceman–Rachford splitting, ADI monotonicity, Parallelization",
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    A splitting moving mesh method for reaction-diffusion equations of quenching type. / Liang, Kewei; Lin, Ping; Ong, Ming Tze; Tan, Roger C. E.

    In: Journal of Computational Physics, Vol. 215, No. 2, 07.2006, p. 757-777.

    Research output: Contribution to journalArticle

    TY - JOUR

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    AU - Lin, Ping

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    AU - Tan, Roger C. E.

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    N2 - This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.

    AB - This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.

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