Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finite-difference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data.

在驱动力的作用下，通过界面运动可能会发生多晶材料形态的变化。本文考虑的一个重要的特殊情况是，当晶界与刚凝固的金属板的平面相交时，就会产生热切槽，从而形成了对称的细槽。如果瞬态表面扩散是主要的形成机制，则这将产生一个四阶时间相关的偏微分方程，而该方程具有未知的时间相关的表面扩散率。为了确定它，及时记录了固定位置的自由切槽表面的轮廓。开槽边界由自伴边界条件支持。我们在输入数据上提供了充分的条件，对于这些条件，事实证明所引起的系数识别问题是恰当的。此外，我们开发了一种预测器-校正器有限差分样条方法，以获取非线性系数识别问题的准确且稳定的数值解。数值结果说明了精确数据和噪声数据的反演性能。