2.1.11.9.9
Chapter |
2 Background .1
Cosmology .11 Math |
Section 10.2 |

**Section** 1 **Finite differences**

The net method is the method of finite differences, at present it is on of the most widely spread methods of numerical solution of equations with partial derivitives. It is based on the idea of replacing the derivatives with finite difference equations. Consideration will be confined to the case of two independent variables.

Figure 4 |

Let there be given some domain *G* with boundary *G*
in the *xOy* plane. Let us construct in this plane two families of parallel straight lines:

The points of intersection of these straight lines will be called **mesh
points**. Two mesh points are said to be **neighbours**
if the distance between them in the direction of the axis *Ox* or *Oy* is equal to *h.* Consider mesh points
which do belong to the domain *G+**G* together with such
points outside this domain which lie distances less than *h* from the boundary *G*.
All points, the neighbours of which belong to this set, will be referred to as** ****interior
point****s** (the point A in the figure
above), and the remaining ones as **boundary points **(
points *B* and* C*).

Let *u*_{ik }denote the values of the unknown function *u* = *u*(*x,y*)
at the **mesh points** *(x*_{0} + *ih*,
*y*_{0} + *kl*). At **interior points**
of *G*, replace the partial derivatives by the finite differences

(1.1)

at boundary points by the less accurate expressions

(1.2)

**Partial derivatives of the second
order** are replaced in an analogous manner:

(1.3)

**These replacements of derivatives by finite differences allow to reduce the **solution
of partial differential equations to systems of difference equations.