An explicit subparametric spectral element method of lines applied to a tumour angiogenesis system of partial differential equations

J. Valenciano, M. A. J. Chaplain

    Research output: Contribution to journalArticle

    13 Citations (Scopus)

    Abstract

    In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.
    Original languageEnglish
    Pages (from-to)165-187
    Number of pages23
    JournalMathematical Models and Methods in Applied Sciences
    Volume14
    Issue number2
    DOIs
    Publication statusPublished - 2004

    Fingerprint

    Spectral Element Method
    Angiogenesis
    Method of Lines
    Systems of Partial Differential Equations
    Partial differential equations
    Tumors
    Tumor
    Numerical Solution
    Ordinary differential equations
    Stiff Ordinary Differential Equations
    Augmented System
    Semidiscretization
    Polynomials
    Complex Geometry
    Chebyshev Polynomials
    Legendre
    Fourth Order
    Geometry
    Analytical Solution
    Computer simulation

    Keywords

    • Tumour angiogenesis
    • Spectral elements

    Cite this

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    title = "An explicit subparametric spectral element method of lines applied to a tumour angiogenesis system of partial differential equations",
    abstract = "In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.",
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    An explicit subparametric spectral element method of lines applied to a tumour angiogenesis system of partial differential equations. / Valenciano, J.; Chaplain, M. A. J.

    In: Mathematical Models and Methods in Applied Sciences, Vol. 14, No. 2, 2004, p. 165-187.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - An explicit subparametric spectral element method of lines applied to a tumour angiogenesis system of partial differential equations

    AU - Valenciano, J.

    AU - Chaplain, M. A. J.

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    PY - 2004

    Y1 - 2004

    N2 - In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.

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    KW - Tumour angiogenesis

    KW - Spectral elements

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