generally , it depends on your geometry . for example if you solve the pde for a 2D case (for example 2D heat equation), in the higher part of your geometry you should use BW because there is no node above your boundary in which you cannot use Central or Forward .

another condition may be the behavior of your equation , for example in wave equation all nodes influenced from the past nodes so if you use the FW or Central its not stable and only the upwind discretization ( BW in space ) will be correct .

generally , it depends on your geometry . for example if you solve the pde for a 2D case (for example 2D heat equation), in the higher part of your geometry you should use BW because there is no node above your boundary in which you cannot use Central or Forward .

another condition may be the behavior of your equation , for example in wave equation all nodes influenced from the past nodes so if you use the FW or Central its not stable and only the upwind discretization ( BW in space ) will be correct .

The correct answer depends on some factors:

1) the mathematical character of the PDE. Elliptic equations have no specific direction of propagation and central FD should be used). Conversely, hyperbolic equations have specific characteristic directions and forward/backward derivatives are often used according to these directions.

2) Numerical stability properties. Sometimes forward/backward derivatives are used also to stabilize the numerical solution, preventing oscillations that can onset the numerical instability.

Often, a time-dependent PDE problem can be discretized using several stencils, that happens particularly close to the boundaries where non-symmetric stencils must be used.

Amirhossein Taran

Filippo Maria Denaro

Thanks a lot dear Amirhossein and Filippo, I found my answer

It's a good question and I am interested in it.

hello Dear Mr . Pooria Azarakhsh

Assuming you have a nice enough ff (lets say thrice continuously differentiable around xx), the central difference has error on the order of h2h2, which is smaller than the error from forward/backward differences (which is on the order of hh). You can see this by plugging in a Taylor polynomial of degree 3 into the definitions of the forward/backward/central difference, and see that the error is on the order stated prior. Or read about it on below site:

http://www.math.ohiou.edu/courses/math3600/lecture27.pdf

http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node9.html

Ahmad Keshavarzi Tnx Mr. Keshavarzi