在研究流体和弹性体结构相互作用的问题时，浸入边界法(Immersed Boundary.Method)是一种非常重要而且常用的计算方法。在很多领域尤其是生物力学领域有着广泛的应用。但是浸没边界法本身也有一些局限，限制了其应用的范围，其中最突出的一个就是当使用显式格式时浸没边界法的数值刚性特别大，这就使得在计算时，时间步长必须取得非常小，那么如果想研究系统长时间的演化，计算量就会变得不可忍受。本项目致力于在浸入边界法的基础上，发展稳定高效的半隐式离散格式，克服时间步长的限制，以适应科学研究和工程应用的需要。我们将在现有研究的基础上，将时间滞后的离散方法与小尺度分解相结合，在保持隐式格式稳定性的基础上，降低其计算量，同时结合自适应网格等技术，提高空间精度，最终得到一种稳定性好，精度高，有广阔应用前景的数值方法。

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 have some limitations to limit its application. One of them is that it require small time steps to maintain stability when solved with an explicit method. If we want to study the long time evolution, the computational cost will be too expensive to afford. In order to overcome this.difficulty, based on our previous work, we propose to develop some stable and efficent semi-implicit algorithms for Immersed Boundary method to deal with the numerous problems emerge in the scientific research and applications. The main idea is to apply the small scale decomposition on some semi-implicit or implicit discretization, to reduce its computational cost, then combine the adaptive mesh to increase the spatial accuracy. Hopefully, we could get some stable, efficient algorithms with reasonable accuracy and capability to deal with the interface with complex geometric structure and nonlinear stress-strain relation. ??