论文标题
随机cahn- hilliard方程的明确方案的强收敛速率具有附加噪声
Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise
论文作者
论文摘要
在本文中，我们提出并分析了一种明确的时间步态方案，以便对随机cahn- hilliard方程进行添加噪声的空间离散。完全离散的近似结合了空间中的光谱盖金方法，并在时间上与驯服的欧拉法相结合。与文献中的隐式方案相反，这里的明确方案很容易实施，并在计算效率方面产生了显着提高。结果表明，完全离散的近似值强烈地收敛到精确溶液，并确定了强收敛速率。与随机allen方程的驯服时间稳定方案不同，由于非线性前面存在无限的线性算子，因此在分析中出现了必不可少的困难。为了克服它们，在本工作中开发了新的和非平凡的论点。据我们所知，这是关于随机Cahn--Hilliard方程的明确方案的第一个结果。最终进行数值实验以确认理论结果。
In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn--Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. Different from the tamed time-stepping schemes for stochastic Allen--Cahn equations, essential difficulties arise in the analysis due to the presence of the unbounded linear operator in front of the nonlinearity. To overcome them, new and non-trivial arguments are developed in the present work. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn--Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.
