We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps.

翻译：我们研究欧拉坐标系下移动域上热方程的Crank-Nicolson型时间步进格式。由于空间域在连续时间步之间会发生变化，需要从前一时间步对解进行扩展。遵循Lehrenfeld与Olskanskii [ESAIM: M2AN, 53(2):585-614, 2019]的方法，我们应用基于所谓鬼影罚项的隐式扩展。空间离散采用切割有限元法。我们推导出完整的时空先验误差分析，特别在抛物型CFL条件下证明了时间方向的二阶收敛性。最后，我们展示二维和三维空间的数值结果，证实了理论分析的有效性，即使对于更大的时间步长也成立。