Taylor Expansion for fm+1n+1 - Tips and Tricks

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SUMMARY

The discussion focuses on performing Taylor expansion for the term fm+1n+1 in the context of finite differencing for partial differential equations (PDEs). The user is familiar with expanding fm+1 and fn+1 separately but seeks guidance on combining these expansions for fm+1n+1. The key takeaway is that the user needs to apply Taylor series expansion techniques to both the spatial and temporal derivatives to achieve consistency in the PDE formulation.

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Homework Statement


how to do taylor expansion for fm+1n+1; f(t,x) with sub script m+1 and a super script n+1


Homework Equations


I know how to do taylor expansion for fm+1 and fn+1, but not fm+1n+1


The Attempt at a Solution

 
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Can you explain what you mean by fm+1n+1
 
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!
 

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