5-point discretisation problem

[mfarooq52003]

Hi
I will be grateful if someone knows about the following and explain it for me please.

What is finite difference scheme and how it leads to the discrete system? What is the effect of 5-point discretisation of the operator -d²/dx²-d²/dy²+$$\gamma$$d/dx on a rectangular region?

If anyone knows about it please explain it for me and then I'll post the problem whcih is arised by the mentioned differential operator on the rectangular region for more details.
Thanks
mfarooq52003

 

[gtoguy97]

Finite difference schemes are numerical methods used to approximate the solution of differential equations by replacing the continuous derivatives with discrete values. This is done by dividing the region where the equation is being solved into a finite number of grid points, and then approximating the derivatives at these points using polynomial interpolation. For example, the second order derivative -d2/dx2-d2/dy2+\gammad/dx can be approximated using a five-point discretization, which uses a five-point stencil to calculate the value of the derivative at a given point. This stencil consists of the current point, two points to the left and right, and one point above and below. This method is known as the five-point finite difference scheme. The effect of this discretization is to convert the continuous differential equation into a system of discrete equations, which can then be solved using numerical techniques. In the case of the example provided, the effect of the five-point discretization of the operator -d2/dx2-d2/dy2+\gammad/dx on a rectangular region is to create a system of equations for each grid point in the region, where the value of the derivative at that point is calculated using the values of the function at the surrounding points. This system of equations can then be solved using numerical techniques such as the Gauss-Seidel algorithm.