1st-Order Accurate, FD Approximation to [ xf ]0

From the Taylor series expansion, we can write f(x1) in terms of f(x0) and its derivatives as follows:

f(x1) = f(x0) + (x1 - x0)[ xf ]0 + (2!)-1(x1 - x0)2 [ x2 f ]0 + (3!)-1(x1 - x0)3 [ x3 f ]0 + ...

Now, solve for [ xf ]0 :

(x1 - x0)[ xf ]0 = [ f(x1) - f(x0) ] - { (2!)-1(x1 - x0)2 [ x2 f ]0 + (3!)-1(x1 - x0)3 [ x3 f ]0 + ... }
[ xf ]0 = [ f(x1) - f(x0) ] (x1 - x0)-1 - { (2!)-1(x1 - x0) [ x2 f ]0 + (3!)-1(x1 - x0)2 [ x3 f ]0 + ... }
[ xf ]0 [ f(x1) - f(x0) ] (x1 - x0)-1 + order[ (Dx) ]

With this expression, then, we have a FD expression for [ xf ]0 that is said to be accurate to 1st-order in Dx.


PHYS7412 Index | Finite Difference Schemes