Stabilized Variational Formulations of Chorin-Type and Artificial Compressibility Methods for the Stochastic Stokes–Darcy Equations

IF 2.8 2区 数学 Q1 MATHEMATICS, APPLIED
Huangxin Chen, Can Huang, Shuyu Sun, Yahong Xiang
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Abstract

In this paper, we consider two different types of numerical schemes for the nonstationary stochastic Stokes–Darcy equations with multiplicative noise. Firstly, we consider the Chorin-type time-splitting scheme for the Stokes equation in the free fluid region. The Darcy equation and an elliptic equation for the intermediate velocity of free fluid coupled with the interface conditions are solved, and then the velocity and pressure in free fluid region are updated by an elliptic system. Secondly, we further consider the artificial compressibility method (ACM) which separates the fully coupled Stokes–Darcy model into two smaller subphysics problems. The ACM reduces the storage and the computational time at each time step, and allows parallel computing for the decoupled problems. The pressure in free fluid region only needs to be updated at each time step without solving an elliptic system. We utilize the RT\(_1\)-P\(_1\) pair finite element space and the interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM\(_1\)-P\(_0\) finite element space in the spatial discretizations. Under usual assumptions for the multiplicative noise, we prove that both of the Chorin-type scheme and the ACM are unconditionally stable. We present the error estimates for the time semi-discretization of the Chorin-type scheme. Numerical examples are provided to verify the stability estimates for both of schemes. Moreover, we test the convergence rate for the velocity in time for both of schemes, and the convergence rate for the pressure approximation in time average is also tested.

Abstract Image

用于随机斯托克斯-达西方程的乔林型和人工可压缩性方法的稳定变分公式
本文针对具有乘法噪声的非稳态随机斯托克斯-达西方程,考虑了两种不同类型的数值方案。首先,我们考虑了自由流体区域斯托克斯方程的 Chorin 型时间分割方案。首先,我们考虑了自由流体区域斯托克斯方程的 Chorin 型时间分割方案,求解了达西方程和自由流体中间速度的椭圆方程以及界面条件,然后用椭圆系统更新了自由流体区域的速度和压力。其次,我们进一步考虑了人工可压缩性方法(ACM),该方法将完全耦合的斯托克斯-达西模型分离成两个较小的子物理问题。人工可压缩性法减少了每个时间步的存储量和计算时间,并允许并行计算解耦问题。自由流体区域的压力只需在每个时间步更新,无需求解椭圆系统。我们利用 RT\(_1\)-P\(_1\) 对有限元空间和基于 BDM\(_1\)-P\(_0\) 有限元空间的内部惩罚非连续加勒金(IPDG)方案进行空间离散。在乘法噪声的通常假设下,我们证明 Chorin 型方案和 ACM 都是无条件稳定的。我们给出了 Chorin 型方案时间半离散化的误差估计。我们提供了数值示例来验证这两种方案的稳定性估计。此外,我们还测试了两种方案在时间上的速度收敛率,并测试了压力近似在时间平均上的收敛率。
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来源期刊
Journal of Scientific Computing
Journal of Scientific Computing 数学-应用数学
CiteScore
4.00
自引率
12.00%
发文量
302
审稿时长
4-8 weeks
期刊介绍: Journal of Scientific Computing is an international interdisciplinary forum for the publication of papers on state-of-the-art developments in scientific computing and its applications in science and engineering. The journal publishes high-quality, peer-reviewed original papers, review papers and short communications on scientific computing.
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