Calculus of Finite Differences - Simon Plouffe - Charles Jordan
Centered Finite Differences
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Centered finite difference coefficients on uniform grids Setup Establish uniform weights formula as given in Fornberg, B. "Calculation of Weights in Finite Difference Formulas." SIAM Review 40, no. 3 (1998): 685-691 and discussed within http : // reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html. In[273]:= UFDWeights@m_, n_, s_D := CoefficientList@Normal@Series@x s Log@xD m , 8x, 1, n<D h m D,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 h m DD<, 8lcm h m , weights h m lcm<DD First derivative In[275]:= CDCoefficients@1, 2D Out[275]= 82h, 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]= 9h 2 , 81, - 2, 1<= In[280]:= CDCoefficients@2, 4D Out[280]= 912h 2 , 8- 1, 16, - 30, 16, - 1<= In[281]:= CDCoefficients@2, 6D Out[281]= 9180 h 2 , 82, - 27, 270, - 490, 270, - 27, 2<= In[282]:= CDCoefficients@2, 8D Out[282]= 95040 h 2 , 8- 9, 128, - 1008, 8064, - 14350, 8064, - 1008, 128, - 9<=
description
How to generate centered finite difference coefficients on uniform grids with Mathematica, including first and second order finite difference derivatives with accuracy through eighth order.
Transcript of Centered Finite Differences
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<=
