Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models

Authors

  • X. Wu,

    Corresponding author
    1. Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
    • Correspondence to: K.G. van der Zee and X. Wu, Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

      E-mail: k.g.v.d.zee@tue.nl and x.wu@tue.nl

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  • G. J. van Zwieten,

    1. Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
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  • K. G. van der Zee

    Corresponding author
    1. Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
    • Correspondence to: K.G. van der Zee and X. Wu, Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

      E-mail: k.g.v.d.zee@tue.nl and x.wu@tue.nl

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SUMMARY

We present unconditionally energy-stable second-order time-accurate schemes for diffuse-interface (phase-field) models; in particular, we consider the Cahn–Hilliard equation and a diffuse-interface tumor-growth system consisting of a reactive Cahn–Hilliard equation and a reaction–diffusion equation. The schemes are of the Crank–Nicolson type with a new convex–concave splitting of the free energy and an artificial-diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor-growth system, a semi-implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second-order accuracy, unconditional energy-stability, and superiority compared with their first-order accurate variants. Copyright © 2013 John Wiley & Sons, Ltd.

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