Numerical Preservation of Velocity Induced Invariant Regions for Reaction-Diffusion Systems on Evolving Surfaces
Numerical Preservation of Velocity Induced Invariant Regions for Reaction-Diffusion Systems on Evolving Surfaces
复制标题
演化表面反应扩散系统速度诱导不变区域的数值保存
DOI：
10.1007/s10915-018-0741-7
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发表时间：
2018
影响因子：
2.5
通讯作者：
Frittelli M
中科院分区：
文献类型：
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作者：
Frittelli M
本文提出并分析了一种质量集总有限元法（LESFEM），用于求解在给定速度场下演化的曲面上的反应扩散方程组。一个完全离散的方法的基础上的隐式显式（IMEX）欧拉时间离散制定和膨胀率作为指标的表面演变的介绍。在假设网格保持Delaunay正则性的演化下，我们证明了一个充分条件，这取决于膨胀率，不变区域的存在（i）在空间离散水平上没有限制的网格大小和（ii）在完全离散水平下的时间步长的限制，取决于动力学，只有。在线性热方程的特定情况下，我们证明了半和全离散的最大值原理。对于著名的活化剂耗尽模型和托马斯反应扩散模型，我们证明了相空间中存在一族矩形，它们只在特定的增长律下是不变的。两个数值例子提供计算证明（i）的离散最大值原理和最佳收敛的热方程的线性增长的领域和（ii）的存在的不变区域的LESFEM-IMEX欧拉离散的RDS的逻辑增长的表面。
We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction–diffusion systems (RDSs) on surfaces inthat evolve under a given velocity field. A fully-discrete method based on the implicit–explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction–diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM–IMEX Euler discretisation of a RDS on a logistically growing surface.