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Asymptotic Behaviour of Time Stepping Methods for Phase Field Models.

Xinyu Cheng1, Dong Li2, Keith Promislow3

  • 1Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2 Canada.

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|January 28, 2021
PubMed
Summary
This summary is machine-generated.

Adaptive time stepping methods for metastable dynamics were analyzed. Some energy stable methods require more steps than others, contrary to expectations, but Backward Euler performs well for Allen-Cahn dynamics.

Keywords:
Allen–Cahn equationCahn–Hilliard equationEnergy stabilityPhase field modelTime stepping

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Materials science

Background:

  • Metastable dynamics in materials science are often modeled by the Allen-Cahn and Cahn-Hilliard equations.
  • Efficient numerical methods are crucial for simulating these complex systems, especially for capturing long-term behavior.
  • Adaptive time stepping is key to balancing accuracy and computational cost.

Purpose of the Study:

  • To investigate and compare the performance of various adaptive time stepping methods for metastable dynamics.
  • To formally predict the optimal time step sizes for different methods under specific stability conditions.
  • To analyze the trade-offs between accuracy, stability, and computational efficiency.

Main Methods:

  • Analysis of first and second-order time stepping methods in a semi-discrete setting.
  • Formal prediction of time step sizes based on local truncation error and small length scale parameters.
  • Introduction and application of the 'profile fidelity' concept to assess method stability.
  • Computational studies to validate analytical predictions.

Main Results:

  • Some energy stable and fully implicit methods require more time steps than anticipated.
  • Certain popular energy stable methods underperform compared to standard schemes.
  • Backward Euler method shows preserved energy decay and profile fidelity for larger time steps in Allen-Cahn dynamics than previously thought.
  • Eyre-type methods often perform worse due to loss of profile fidelity.

Conclusions:

  • The choice of time stepping method significantly impacts the efficiency and accuracy of simulating metastable dynamics.
  • Profile fidelity is a critical factor in determining the suitability of a method for these problems.
  • Standard methods like Backward Euler can be surprisingly effective under certain conditions, challenging existing assumptions.