Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries.

*(English)*Zbl 1427.65172Summary: The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population. There are several numerical difficulties to efficiently handle such systems. Firstly extremely small time steps are usually demanded due to the stiffness of the system. Secondly it is always difficult to efficiently and accurately handle the moving boundaries. To overcome these difficulties, we first transform the one-dimensional problem with a moving boundary into a system with a fixed computational domain, and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems. Numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches, and it can be shown that Krylov IIF is superior to other three approaches in terms of stability and efficiency by direct comparison.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

92D25 | Population dynamics (general) |

35R37 | Moving boundary problems for PDEs |

35K57 | Reaction-diffusion equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L04 | Numerical methods for stiff equations |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65F50 | Computational methods for sparse matrices |

65F60 | Numerical computation of matrix exponential and similar matrix functions |

##### Keywords:

implicit integration factor methods; Krylov subspace; moving boundaries; stiffness; Stefan problem
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\textit{S. Liu} and \textit{X. Liu}, Discrete Contin. Dyn. Syst., Ser. B 25, No. 1, 141--159 (2020; Zbl 1427.65172)

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