Abstract In this paper, we study high-order compact schemes for semilinear parabolic moving boundary problems. We first convert the original problem into an equivalent one defined on a rectangular region by introducing a linear transformation and the well-known exponential transformation. Next, we derive a compact scheme with fourth-order accuracy in the spatial dimension and second-order accuracy in the temporal dimension. Moreover, we prove that the numerical solutions are convergent strictly in the maximum norm by an energy argument. Extending to two-dimensional semilinear moving boundary problems is also provided. Finally, a series of numerical experiments including linear and semilinear examples are carried out to verify that our schemes have more advantages than the one proposed only for the linear moving boundary problem by Cao et al. (JCAM, 234 (2010) 2578–2586.).

中文翻译：

---

半线性抛物线移动边界问题的高阶紧致方案

摘要 在本文中，我们研究了半线性抛物线移动边界问题的高阶紧致方案。我们首先通过引入线性变换和众所周知的指数变换将原始问题转换为在矩形区域上定义的等效问题。接下来，我们推导出一个紧凑的方案，在空间维度上具有四阶精度，在时间维度上具有二阶精度。此外，我们通过能量论证证明数值解严格收敛于最大范数。还提供了对二维半线性移动边界问题的扩展。最后，进行了包括线性和半线性示例在内的一系列数值实验，以验证我们的方案比 Cao 等人仅针对线性移动边界问题提出的方案具有更多优势。(JCAM,