From 7-Point Scheme to World Gravitational Model

The Newton/Laplace World Gravitational Model for the gravitational field u(x,y,z) created by a mass distribution f(x,y,z) takes the form

• Δu = f  in 3-dimensional space with coordinates (x,y,z),

where Δ is the Laplace operator. In discrete form the equation takes the form of the 7-point scheme:

•  u(x+h,y,z)+u(x-h,y,z)+u(x,y+h,z)+u(x,y-h,z)+u(x,y,z+h)+u(x,y,z-h) –               6*u(x,y,z) =h*h* f(x,y,z)

on a grid with mesh size h. The discrete equation can be solved by time stepping according to a one-line code: See 2d example.

This code can be written by a child of early age and the output can be studied. It is thus possible to get a grasp of a World Gravitational Model at an early age. This is a revolution in mathematics teaching.

Observe that if f = 0, then the 7-point scheme expresses that

• u(x,y,z) is the mean value of the value of u at surrounding points (x+h,y,z), (x-h,y,z), (x,y+h,z), (x,y-h,z), (x,y,z+h) and (x,y,z-h).

Recall that a function u satisfying Δu = 0 is said to be a harmonic function and a basic theorem of harmonic analysis about harmonic functions is that the value at one point is the mean value over spheres centered at the point, referred to as the mean value property.

From the practice of the one-line code of the 7-point scheme, we are thus directly led to the mean value property as a basic theoretical result of harmonic analysis. From practise to theory.

On the other hand, the equation Δu = 0 in theoretical symbolic form does not invite to understanding of the mean value property.