Centered finite difference coefficients on uniform grids

Setup

Establish uniform weights formula as given in Fornberg, B. "Calculation of Weights in Finite Difference Formulas." SIAMReview 40, no. 3 (1998): 685-691 and discussed withinhttp : // reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html.

In[273]:= UFDWeights@m_, n_, s_D := CoefficientList@Normal@Series@xs Log@xDm, 8x, 1, n<D � hmD, xD

Compute centered differences and rescale coefficients according to their least common multiple.

In[274]:= CDCoefficients@m_, n_D :=

With@8weights = UFDWeights@m, n, n � 2D<, With@8lcm = Apply@LCM, Denominator@weights hmDD<,8lcm hm, weights hm lcm<DD

First derivative

In[275]:= CDCoefficients@1, 2D

Out[275]= 82 h, 8-1, 0, 1<<

In[276]:= CDCoefficients@1, 4D

Out[276]= 812 h, 81, -8, 0, 8, -1<<

In[277]:= CDCoefficients@1, 6D

Out[277]= 860 h, 8-1, 9, -45, 0, 45, -9, 1<<

In[278]:= CDCoefficients@1, 8D

Out[278]= 8840 h, 83, -32, 168, -672, 0, 672, -168, 32, -3<<

Second derivative

In[279]:= CDCoefficients@2, 2D

Out[279]= 9h2, 81, -2, 1<=

In[280]:= CDCoefficients@2, 4D

Out[280]= 912 h2, 8-1, 16, -30, 16, -1<=

In[281]:= CDCoefficients@2, 6D

Out[281]= 9180 h2, 82, -27, 270, -490, 270, -27, 2<=

In[282]:= CDCoefficients@2, 8D

Out[282]= 95040 h2, 8-9, 128, -1008, 8064, -14 350, 8064, -1008, 128, -9<=