Numerical approach for bifurcation of nonlinear problem

关于不可压缩流体，我们得到了如下结果：第一个结果是关于二维Navier-Stokes方程的驻点解。在此基础上，奇尔德里斯等人（'89）研究了一个非定常的例子，即给出了大雷诺数下有限时间爆破的数值例子，并给出了临界雷诺数。然而，我们已经得到了不同的结果，使用修改的配方。我们用两种方法，即有限差分法和谱方法来检验它。（[1]）第二部分是关于常涡量重力波的分歧问题。我们的目标是看到全局分岔结构，结合我们的结果，我们已经得到的毛细重力波。主要研究结果如下：1.研究发现，随着涡量的变化，分岔结构基本保持不变.对于对称波，我们数值计算了存在多少种n型分岔解。我们通过模拟对n=1 <$6进行了检验。我们通过在流体区画流线给出了自由表面下流动的信息。可以看出，正涡度时出现涡动，涡动随着涡度的增大而扩大. Zufiria（'87）用数值方法给出了无限深重力波的非对称解。我们试图用我们算法的非对称版本来跟踪它们，但我们找不到。我们相信对于无限深的情况可能没有非对称解，但是对于有限深的情况。我们将进一步研究。

We have the following results on incompressible fluid.The first one is on the two-dimensional stagnation-point solution of the Navier-Stokes equations. On it Childress et al. ('89) investigated an unsteady example, namely they show some numerical examples of finite time blow-up for large Reynolds number, They also gave the critical Reynolds number numerically. However we have obtained different results using modified formulation. We had no blow-up and examined it by two methods, finite difference scheme and spectral mathod. ([1])The second one is on bifurcation problem of gravity waves with constant vorticity. Our object is to see the global bifurcation structure, combining the results with our results for capillary-gravity waves we have obtained before. The results on this work are as below :1. It is found that bifurcation structures are unchanged qualitatively as vorticity varies.2. As for symmetric waves, we conjectured numerically how many kinds of mode n bifurcation solutions exist. We checked it for n=1〜6 by simulations.3. We gave the information about the flow beneath the free surface by plotting stream-lines in the fluid region. It can be seen that eddy appears for positive vorticity and it expands as the vorticity becomes larger.4. Zufiria ('87) gave non-symmetric solutions for gravity waves of infinite depth numerically. We attempted to follow them by the nonsymmetric version of our algorithm, but we couldn't find any. We believe there might be no non-symmetric solutions for the case of infinite depth, but for the case of finite depth. We will further study it.