2018-04-27 开胃学习数学系列 - 有限差分法 PDE

Finite Difference Methods for Solving PDEs

有限差分法求解偏微分方程
PDE具有封闭解是非常罕见，通常只能用数值解。
将重点放在有限差分法（FDM）上：其他数值方法存在可能更合适 - 具体在于实际问题。

Discretization

1. 使用FDM求解偏微分方程的第一步是将u（x，t）u（x，t）解作为离散的值集合，在空间和时间的网格点上，坐标上，的分布。 First step in solving PDEs using FDM is to represent the solution u(x,t)u(x,t) as a discrete collection of values at a well distributed grid points in space and time in the proper domain.

2. 然后是设计步长：

   
   

4. 任何其他所需点的值可以通过插值来近似。

5. A uniform mesh may not necessary be the most efficient form to work with, in fact, it rarely is. 一个统一的网格很少说是最有效的形式。

6. A greatly simplified rule-of-thumb: the mesh needs to be refined around the region where the function varies a lot 网格需要在功能变化很大的区域进行细化
   On the other hand, the mesh can be relatively coarse if the function is smooth and changing slowly 如果功能平滑并且变化缓慢，则网格可能相对较粗糙

7. In finance, the time grid is well advised to keep the important dates as grid points: such as cash flow dates, contractual schedule dates, etc. The spacing of these dates is most likely not uniform. 在金融方面，建议时间网格将重要日期保留为网格点，例如现金流量日期，contractual schedule dates等。这些日期的间隔很可能不统一。

Finite Difference Methods(FDM)

Toolkit for finite difference approximation

The Explicit Method

For convenience, reverse the time for Black-Scholes notation: t←T−t

比较原始：

套用一个