带有奇异噪声的表面生长模型的数值研究
Numerical Study of a Surface Growth Model with Singular Noise
AI总结：
该研究针对奇异噪声驱动的外延薄膜生长随机模型，采用谱Galerkin投影离散非线性项、指数Euler格式积分时间，推导强误差估计并量化截断随机卷积增长与非线性消失速率，通过数值实验揭示粗糙度参数对生长动力学的影响。
AI中文摘要：
我们研究了一个用于外延薄膜生长的随机模型，该模型由空间粗糙的加性噪声驱动，其中噪声是奇异的，并且通过傅里叶空间中的截断进行正则化以赋予解以意义，导致极限下非线性消失。为了数值研究这一现象，非线性项通过谱Galerkin投影进行离散化，而时间积分则采用指数Euler格式执行。对于比时空白噪声更强的粗糙度，我们在$L^p(\Omega;C([0,T];\mathcal H^1))$中推导了强误差估计，该估计明确显示了空间截断、时间步长与非线性电流衰减之间的相互作用。我们还量化了截断随机卷积的增长以及相应的非线性消失速率。数值实验说明了随着粗糙度参数增大，从持续的山丘形成到噪声主导动力学的转变。
英文摘要：
We study a stochastic model for epitaxial thin-film growth driven by spatially rough additive noise in a regime where the noise is singular and regularization via truncation in Fourier space is used to give a meaning to the solution, leading to a vanishing nonlinearity in the limit. In order to study this phenomenon numerically, the nonlinearity is discretized by a spectral Galerkin projection, while time integration is performed with an exponential Euler scheme. For roughness stronger than space-time white noise, we derive strong error estimates in $L^p(Ω;C([0,T];\mathcal H^1))$ that display explicitly the interaction between the spatial cut-off, the time step, and the decay of the nonlinear current. We also quantify the growth of the truncated stochastic convolution and the corresponding vanishing rate of the nonlinearity. Numerical experiments illustrate the transition from persistent hill formation to noise-dominated dynamics as the roughness parameter increases.
