Numerical studies of a class of reaction–diffusion equations with Stefan conditions
Numerical studies of a class of reaction–diffusion equations with Stefan conditions
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DOI：
10.1080/00207160.2019.1599868
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发表时间：
2019-04
影响因子：
1.8
通讯作者：
Shuang Liu;Yihong Du;Xinfeng Liu
中科院分区：
文献类型：
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作者：
Shuang Liu;Yihong Du;Xinfeng Liu
摘要有效而准确地求解具有运动自由边界的微分方程组一直是一个非常困难的问题，而这种微分方程组已经被广泛地应用于描述许多物理/生物现象，如种群扩散的动力学。本文的主要目的是在一般框架内引入有效的数值方法来求解这类具有移动自由边界的方程组。主要的数值挑战是跟踪移动的自由边界，特别是对于高空间维度。为了克服这一问题，首先针对径向对称的二维模型引入了一种结合隐式求解器的前沿跟踪框架。对于一般的2D模型，采用水平集方法来更有效地处理复杂的拓扑变化。讨论了所提方法的精度和收敛阶，数值模拟结果与理论结果吻合较好。
ABSTRACT It is always very difficult to efficiently and accurately solve a system of differential equations coupled with moving free boundaries, while such a system has been widely applied to describe many physical/biological phenomena such as the dynamics of spreading population. The main purpose of this paper is to introduce efficient numerical methods within a general framework for solving such systems with moving free boundaries. The major numerical challenge is to track the moving free boundaries, especially for high spatial dimensions. To overcome this, a front tracking framework coupled with implicit solver is first introduced for the 2D model with radial symmetry. For the general 2D model, a level set approach is employed to more efficiently treat complicated topological changes. The accuracy and order of convergence for the proposed methods are discussed, and the numerical simulations agree well with theoretical results.