论文标题
随机Stokes方程的高阶时间离散方法具有乘法噪声
Higher order time discretization method for the stochastic Stokes equations with multiplicative noise
论文作者
论文摘要
在本文中，我们提出了一种新的方法，用于具有乘法噪声的不可压缩随机stokes方程的时间。我们的新策略是基于随机微分方程的经典米尔斯坦方法。我们使用能量方法进行误差分析，并显示速度和压力近似值最多的强大收敛顺序。该证明是基于速度解决方案的新Hölder连续性估计。虽然在标准$ l^2 $ - 和$ h^1 $ norms中估算了速度近似的误差，但由于压力解决方案的规律性较低，因此在特殊规范中仔细分析了压力误差。另外，还引入了对计算中非常有用的压力解决方案的新解释。还提供了数值实验来验证误差估计及其清晰度。
In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of at most $1$ for both velocity and pressure approximations. The proof is based on a new Hölder continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard $L^2$- and $H^1$-norms, the pressure errors are carefully analyzed in a special norm because of the low regularity of the pressure solution. In addition, a new interpretation of the pressure solution, which is very useful in computation, is also introduced. Numerical experiments are also provided to validate the error estimates and their sharpness.
