一个愚蠢的问题

sddz77

在微分方程的离散化中，有限区域法和有限差分法有什么区别，请用最简单的语言和最简单的例子来说明。谢谢！

boling2000

我个人感觉是这样的：有限区域法——我猜想你指的是有限体积方法(finite volume method)，是把质量守恒、动量守恒和能量守恒直接应用于每个控制体（往往是每个网格单元），从而得到一系列非线性方程组进行求解——我感觉是从物理上出发得到；而有限差分法是在每个网格点上利用差分近似代替微分导数，得到一系列非线性方程组进行求解——感觉更像从数学上出发。

wangxq

我的理解是这样的：当仅仅从数学的角度来求解偏微分方程，可以忽略方程的求解域，认为求解域无限大或足够大。这样适用于有限差分法。但是当求解具体的问题时，求解域可能比较复杂，形状不规则，这样就需要结合具体的求解域来解偏微分方程，这时用有限区域法就便于把具体的求解域形状考虑进去

in-hk

On fluid dynamics, the conservation is the key issue to be considered. Both you guys had provided a very simple yet correct description. I would like to supplyment with some details. On some problem, expecially those involve discontinuities, conservation is the primary concern, i.e. the conservation law. For those working on the compressible flow problems would clearly understand it. Also, pay more attentation on the physics, it is also important.
Personally, I highly recommend the book Computational fluid dynamics, theory and applications, by J.D. Anderson. It has a very good descriptions on the need of conservation form of equations & very important, the classifications of PDE, ellptic, parabolic & hyperblic. The photocopy version is very cheap in China, only around RMB34. My hardcover verison is simply too expensive.
In fact some new solution techiques had been developed using the concept of space-time conservation. It can capture discontinuity accurately the FDM or FVM would need much more afford.
FDM is also needed for understand of many issue on CFD, such as CFL condiitons, stability, solution techniques.

周华

我的理解是，有限差分法是直接在网格点间进行插值等离散化计算，而有限体积法则是先求出单元边界面上的通量，在用通量计算控制点上的物理量变化。有限体积法从物理上和数学上看都更直观容易理解，与守恒形式的控制方程相结合，更容易保证格式的守恒性，推荐参考书：《An introduction to Computational Fluid Dynamics: The Finite Volume Method》H K Versteeg & W Malalasekera编著。