We consider a special type of fast reaction-diffusion systems in which the coefficients of the reaction terms of the two substances are much larger than those of the diffusion terms while the diffusive motion to the substrate is negligible. Specifically speaking, the rate constants of the reaction terms are $O(1/\epsilon)$ while the diffusion coefficients are $O(1)$ where the parameter $\epsilon$ is small. When the rate constants of the reaction terms become highly large, i.e. $\epsilon$ tends to 0, the singular limit behavior of such a fast reaction-diffusion system is inscribed by the Stefan problem with latent heat, which brings great challenges in numerical simulations. In this paper, we adopt a semi-implicit scheme, which is first-order accurate in time and can accurately approximate the interface propagation even when the reaction becomes extremely fast, that is to say, the parameter $\epsilon$ is sufficiently small. The scheme satisfies the positivity, bound preserving properties and has $L^2$ stability and the linearized stability results of the system. For better performance on numerical simulations, we then construct a semi-implicit Runge-Kutta scheme which is second-order accurate in time. Numerous numerical tests are carried out to demonstrate the properties, such as the order of accuracy, positivity and bound preserving, the capturing of the sharp interface with various $\epsilon$ and to simulate the dynamics of the substances and the substrate, and to explore the heat transfer process, such as solid melting or liquid solidification in two dimensions.

翻译：我们考虑一类特殊的快速反应扩散系统，其中两种物质的反应项系数远大于扩散项系数，而底物的扩散运动可忽略不计。具体而言，反应项的速率常数为$O(1/\epsilon)$，扩散系数为$O(1)$，其中参数$\epsilon$很小。当反应项速率常数变得非常大（即$\epsilon$趋近于0）时，此类快速反应扩散系统的奇异极限行为由带有潜热的Stefan问题描述，这给数值模拟带来了巨大挑战。本文采用一种半隐格式，该格式在时间上一阶精确，即使反应变得极快（即参数$\epsilon$足够小）也能精确逼近界面传播。该格式满足正性、保界性质，并具有$L^2$稳定性和系统的线性化稳定性结果。为提升数值模拟性能，我们进一步构造了时间上二阶精确的半隐Runge-Kutta格式。通过大量数值实验验证了精度阶、正性和保界性、不同$\epsilon$下尖锐界面的捕捉能力，模拟了物质与底物的动力学行为，并探索了二维空间中固体熔化或液体凝固等热传导过程。