发表机构
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对三类带自由边界的反应-扩散方程，采用固定边界法转化问题，用有限差分法构造系数矩阵为M-矩阵的数值格式，并证明其正性、单调性和稳定性，仅需温和的时间步长限制，数值算例验证了理论。
AI 中文摘要
本文考虑数值求解三类带自由边界的反应-扩散方程。首先，采用流行的固定边界法将所考虑的自由边界问题转化为固定边界问题。然后，通过使用有限差分方法，为这些转化后的固定边界问题开发了以其系数矩阵为$\text{M}$-矩阵的数值格式。接下来，对于所开发的数值格式，我们建立了涉及正性保持、单调性保持和稳定性的数值理论。值得注意的是，与一些现有工作对时间步长施加严格限制以讨论数值理论不同，所提出的数值格式仅需要温和的限制。最后，提供了数值算例来测试所开发的数值格式并验证理论结果。
英文摘要
This work considers to numerically solve three kinds of reaction-diffusion equations with free boundaries. First, the popular front-fixing method is used to transform the considered free boundary problems into fixed boundary problems. Then, by employing the finite difference method, numerical schemes with $\mathrm{M}$-matrices as their coefficient matrices are developed for these transformed fixed-boundary problems. Next, for the developed numerical schemes, we establish numerical theory involving positivity preservation, monotonicity preservation, and stability. It is noteworthy that, unlike some existing works that impose tight restrictions on the time step-size to discuss the numerical theory, the proposed numerical schemes require only a mild restriction. Finally, numerical examples are provided to test the developed numerical schemes and to confirm the theoretical results.
