It will usually come down to your best guess. When s = 1, you have forward euler (first order), at s = 0 you obtain backward euler (first order), and when s = 1/2 you get the second order accurate central difference. You could try looking into weno/eno methods. These methods will utilize multiple stencils to estimate the local derivative.

Weighted finite differences are used perversely, see for example:

http://eprints.usm.my/29569/1/A_study_of_several_explicit_finite_difference_schemes_for_the_linear_convection_equation.pdf

Dear Muhammad Ali

There are different numerical method for solving equations based on weighted residual method such as point collocation, minimum squares method, sub-domain, Galerkin method, Rayleigh-Ritz method. I think the suitable method for your equation is the Galerkin method. This method can be found in the following book:

http://s2.picofile.com/file/7897233545/BOUNDARY_ELEMENT_Brebbia_and_Dominguez.pdf.htm

best regards

Panji