I am doing the finite differencing for a pde and I am trying to expand the term f_m+1 with a superscript n+1 around say (f_m with a superscript n) to see whether or not the pde is consistent.
For forward in time, a partial derivative of time (df/dt)will be rewrite as [(f_m with a superscript n+1) - (f_m with a superscript n)]/(delta t)
Similarly, if i want to do forward differencing in space, df/dx can be rewrite as [(f_m+1 with a superscript n) - (f_m with a superscript n)]/(delta x)
I know how to do taylor expansion for (f_m+1 with a superscript n) around (f_m with a superscript n) and taylor expansion for (f_m with a superscript n+1) around (f_m with a superscript n). However, I do not know how to deal with f_m+1 with a superscript n+1. How should i do the taylor expansion. Thank you!