本文的主要目的是實現文獻[11]中所提出的一種兩階段直接施力沉浸邊界投影方法模擬流體與移動固體交互作用的動力行為，其中每一沉浸固體都配有一個給定的速度。這個兩階段的方法結合了直接施力沉浸邊界投影方法和預測-修正策略，其中引入一個只分佈在固體上的離散虛擬力並將其附加到流體動量方程式來處理沉浸固體邊界上的無滑移邊界條件。具體來說，首先使用隱式尤拉公式去離散不可壓縮納維爾-史托克方程的時間變數並應用顯式一階的方法線性化其非線性對流項，然後採用預測-修正直接施力沉浸邊界投影法去求解時間離散後的方程式，在預測和修正階段中我們皆採用肖林時間一階的投影法。另外，對於投影法計算中的空間離散，我們採用交錯網格中央差分格式。我們執行兩個關於多個移動固體的數值實驗來說明此演算法的效率。數值結果顯示這個簡單的預測-修正沉浸邊界投影法對流固耦合問題可以求取合理的數值結果。

The aim of this thesis is to implement the two-stage direct-forcing immersed boundary
projection method proposed by Horng et al. [11] for simulating the dynamics
of fluid interacting with moving solid objects, where each immersed solid object is
equipped with a prescribed velocity. This two-stage approach combines a directforcing
immersed boundary projection method with a prediction-correction strategy,
in which a discrete virtual force distributed on the solid object is introduced and
appended to the fluid momentum equations to accommodate the no-slip boundary
condition at the immersed solid boundary. Specifically, we first use the implicit Euler
formula to discretize the temporal variable in the incompressible Navier-Stokes
equations and apply the explicit first-order approximation to linearize the nonlinear
convection term. We then employ a predicition-correction direct-forcing immersed
boundary projection method to solve the time-discretized equations, where we adopt
the first-order in time Chorin’s projection method in both prediction and correction
stages. For spatial discretization in the projection computations, we employ the central
difference scheme on the staggered grids. We give two numerical examples of
multiple moving solid objects to illustrate the performance of the algorithm. From the
numerical results, we find that this simple predicition-correction immersed boundary
approach can achieve reasonable results for fluid-solid interaction problems.

[1] O. Botella and R. Peyret, Benchmark spectral results on the lid-driven cavity flow,
Computers & Fluids, 27 (1998), pp. 421-433.
[2] D. L. Brown, R. Cortez, and M. L. Minion, Accurate projection methods for the incompressible
Naver-Stokes equatons, Journal of Computational Physics, 168 (2001),
pp. 464-499.
[3] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Mathematics of
Computation, 22 (1968), pp. 745-762.
[4] A. J. Chorin, On the convergence of discrete approximations to the Navier-Stokes
equations, Mathematics of Computation, 23 (1969), pp. 341-353.
[5] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 2nd
Edition, Springer-Verlag, New York, 1990.
[6] W. E and J.-G. Liu, Projection method I: convergence and numerical boundary
layers, SIAM Journal on Numerical Analysis, 32 (1995), pp. 1017-1057.
[7] W. E and J.-G. Liu, Projection method III: spatial discretization on the staggered
grid, Mathematics of Computation, 71 (2002), pp. 27-47.
[8] W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Communications
in Mathematical Sciences, 1 (2003), pp. 317-332.
[9] B. E. Griffith, An accurate and efficient method for the incompressible Navier-
Stokes equations using the projection method as a preconditioner, Journal of Computational
Physics, 228 (2009), pp. 7565-7595.
[10] J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for
incompressible flows, Computer Methods in Applied Mechanics and Engineering,195
(2006), pp. 6011-6045.
[11] T.-L. Horng, P.-W. Hsieh, S.-Y. Yang, and C.-S. You, A simple direct-forcing immersed
boundary projection method with prediction-correction for fluid-solid
interaction problems, preprint, 2016.
[12] P.-W Hsieh, M.-C Lai, S.-Y Yang, and C.-S You, An unconditionally energy stable
penalty immersed boundary method for simulating the dynamics of an inextensible
interface interacting with a solid particle, Journal of Scientific Computing, 64
(2015), pp. 289-316.
[13] C.-C. Liao, Y.-W. Chang, C.-A. Lin, and J. M. McDonough, Simulating flows with
moving rigid boundary using immersed-boundary method, Computers & Fluids,
39 (2010), pp. 152-167.
[14] J.-G. Liu, J. Liu, and R. L. Pego, Stable and accurate pressure approximation
for unsteady incompressible viscous flow, Journal of Computational Physics, 229
(2010), pp. 3428-3453.
[15] D. Z. Noor, M.-J. Chern, and T.-L. Horng, An immersed boundary method to
solve fluid-solid interaction problems, Computational Mechanics, 44 (2009), pp.
447-453.
[16] C. S. Peskin, Flow patterns around heart valves: a numerical method, Journal of
Computational Physics, 10 (1972), pp. 252-271.
[17] C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), pp. 479-
51
[18] A. Quarteroni, F. Saleri, and A. Veneziani, Factorization methods for the numerical
approximation of Navier-Stokes equations, Computer Methods in Applied Mechanics
and Engineering, 188 (2000), pp. 505-526.
[19] D. Russell and Z. J. Wang, A Cartesian grid method for modeling multiple moving
objects in 2D incompressible viscous flow, Journal of Computational Physics,
191 (2003), pp. 177-205.
[20] R. Temam, Sur l’approximation de la solution des ´equations de Navier-Stokes
par la m´ethode des pas fractionnaires II, Archive for Rational Mechanics and Analysis,
33 (1969), pp. 377-385.
[21] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Revised Ed.,
Elsevier Science Publishers B.V., Amsterdam, 1984.
[22] S. Xu and Z. J. Wang, An immersed interface method for simulating the interaction
of a fluidwith moving boundaries, Journal of Computational Physics, 10 (2006),
pp. 454-493.