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An efficient semi-implicit immersed boundary method for the Navier-Stokes equations

机译：Navier-Stokes方程的有效半隐式沉浸边界方法

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摘要

The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier-Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semiimplicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method. (c) 2008 Elsevier Inc. All rights reserved.
机译：浸入边界法是研究流体结构相互作用的最有用的计算方法之一。另一方面，当使用显式方法求解时，浸入边界方法也需要少量时间来保持稳定性。在文献中已经提出了许多隐式或近似隐式的方法来消除这种严重的时间步长稳定性约束，但是没有一个方法能提供令人满意的性能。在本文中，我们提出了一种有效的半隐式方案，以从Navier-Stokes方程的浸入边界方法中消除这种刚度。我们的半隐式方案的构建包括两个步骤。首先，我们得到了半隐式离散化，证明它是无条件稳定的。在实践中，这种无条件稳定的半隐式方案仍然非常昂贵。接下来，我们将小规模分解应用于无条件稳定的半隐式方案，以构建有效的半隐式方案。与文献中提出的其他隐式或半隐式方案不同，我们的半隐式方案可以在频谱空间中明确求解。因此，我们的半隐式方案的计算成本与显式方案的计算成本相当。我们广泛的数值实验表明，我们的半隐式方案具有比显式方案更好的稳定性。使用沉浸边界方法可以节省大量计算时间。 （c）2008 Elsevier Inc.保留所有权利。

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