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

Book 9 Computational Mathematics Chapter 10 Partial Differential Equations

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 points (the point A in the figure above), and the remaining ones as boundary points ( points B and C).

Let uik denote the values of the unknown function u = u(x,y) at the mesh points (x0 + ih, y0 + 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.