Centered 7 point numerical second derivative approximation

Assume the following:

f 2 (x)= ∑ i=−3 i=3 c i f(x+ih) MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCa aaleqabaGaaGOmaaaakiaacIcacaWG4bGaaiykaiabg2da9maaqaha baGaam4yamaaBaaaleaacaWGPbaabeaakiaadAgacaGGOaGaamiEai abgUcaRiaadMgacaWGObGaaiykaaWcbaGaamyAaiabg2da9iabgkHi TiaaiodaaeaacaWGPbGaeyypa0JaaG4maaqdcqGHris5aaaa@4BC7@  

(1.1)

Where the 2 superscript after f denotes the second  derivative.

Define the Taylor series for each of 7 points

f(x+ih)= ∑ n=0 n=6 (ih) n n! f n (x) MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaey4kaSIaamyAaiaadIgacaGGPaGaeyypa0ZaaabCaeaa daWcaaqaaiaacIcacaWGPbGaamiAaiaacMcadaahaaWcbeqaaiaad6 gaaaaakeaacaWGUbGaaiyiaaaacaWGMbWaaWbaaSqabeaacaWGUbaa aOGaaiikaiaadIhacaGGPaaaleaacaWGUbGaeyypa0JaaGimaaqaai aad6gacqGH9aqpcaaI2aaaniabggHiLdaaaa@4F15@  

(1.2)

Using the coefficients of equation (1.1) in equations (1.2) we get:

                                                                                                 

For n=0 which is f0(x) term: ∑ −3 3 c i =0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGJbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaaWcbaGaeyOe I0IaaG4maaqaaiaaiodaa0GaeyyeIuoaaaa@3E67@  

(1.3) 

For n=1 which is f1(x) we have                     ∑ −3 3 c i (ih) 1 1! =0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGJbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacaGGOaGaamyAaiaa dIgacaGGPaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaGymaiaacgcaaa aaleaacqGHsislcaaIZaaabaGaaG4maaqdcqGHris5aOGaeyypa0Ja aGimaaaa@4407@    (1.4)

For n=2 which are the f2(x) (i.e. f’’(x)) coefficients we have ∑ −3 3 c i (ih) 2 2! =1 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGJbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacaGGOaGaamyAaiaa dIgacaGGPaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaacgcaaa aaleaacqGHsislcaaIZaaabaGaaG4maaqdcqGHris5aOGaeyypa0Ja aGymaaaa@440A@  

(1.5)

For n=3 which is f3(x) coefficients we have ∑ −3 3 c i (ih) 3 3! =0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGJbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacaGGOaGaamyAaiaa dIgacaGGPaWaaWbaaSqabeaacaaIZaaaaaGcbaGaaG4maiaacgcaaa aaleaacqGHsislcaaIZaaabaGaaG4maaqdcqGHris5aOGaeyypa0Ja aGimaaaa@440B@

For each other n we have: ∑ −3 3 c i (ih) n n! =0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGJbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacaGGOaGaamyAaiaa dIgacaGGPaWaaWbaaSqabeaacaWGUbaaaaGcbaGaamOBaiaacgcaaa aaleaacqGHsislcaaIZaaabaGaaG4maaqdcqGHris5aOGaeyypa0Ja aGimaaaa@4477@  

(1.6)

We therefore have the matrix equation:

−3h 0 0! −2h 0 0! −h 0 0! 0h 0 0! h 0 0! 2h 0 0! 3h 0 0! −3h 1 1! −2h 1 1! −h 1 1! 0h 1 1! h 1 1! 2h 1 1! 3h 1 1! −3h 2 2! −2h 2 2! −h 2 2! 0h 2 2! h 2 2! 2h 2 2! 3h 2 2! −3h 3 3! −2h 3 3! −h 3 3! 0h 3 3! h 3 3! 2h 3 3! 3h 3 3! −3h 4 4! −2h 4 4! −h 4 4! 0h 4 4! h 4 4! 2h 4 4! 3h 4 4! −3h 5 5! −2h 5 5! −h 5 5! 0h 5 5! h 5 5! 2h 5 5! 3h 5 5! −3h 6 6! −2h 6 6! −h 6 6! 0h 6 6! h 6 6! 2h 6 6! 3h 6 6! c −3 c −2 c −1 c 0 c 1 c 2 c 3 = 0 0 1 0 0 0 0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeWbhaaaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIZaGaamiA aaGaayjkaiaawMcaamaaCaaaleqabaGaaGimaaaaaOqaaiaaicdaca GGHaaaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIYaGaamiAaaGa ayjkaiaawMcaamaaCaaaleqabaGaaGimaaaaaOqaaiaaicdacaGGHa 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WaaSaaaeaadaqadaqaaiaaikdacaWGObaacaGLOaGaayzkaaWaaWba aSqabeaacaaIXaaaaaGcbaGaaGymaiaacgcaaaaabaWaaSaaaeaada qadaqaaiaaiodacaWGObaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI XaaaaaGcbaGaaGymaiaacgcaaaaabaWaaSaaaeaadaqadaqaaiabgk HiTiaaiodacaWGObaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa aaGcbaGaaGOmaiaacgcaaaaabaWaaSaaaeaadaqadaqaaiabgkHiTi aaikdacaWGObaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGc baGaaGOmaiaacgcaaaaabaWaaSaaaeaadaqadaqaaiabgkHiTiaadI gaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGa aiyiaaaaaeaadaWcaaqaamaabmaabaGaaGimaiaadIgaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaaiyiaaaaaeaa daWcaaqaamaabmaabaGaamiAaaGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaaaOqaaiaaikdacaGGHaaaaaqaamaalaaabaWaaeWaaeaa caaIYaGaamiAaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaO qaaiaaikdacaGGHaaaaaqaamaalaaabaWaaeWaaeaacaaIZaGaamiA aaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdaca GGHaaaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIZaGaamiAaaGa ayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaGGHa aaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIYaGaamiAaaGaayjk aiaawMcaamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaGGHaaaaa qaamaalaaabaWaaeWaaeaacqGHsislcaWGObaacaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaaGcbaGaaG4maiaacgcaaaaabaWaaSaaae aadaqadaqaaiaaicdacaWGObaacaGLOaGaayzkaaWaaWbaaSqabeaa caaIZaaaaaGcbaGaaG4maiaacgcaaaaabaWaaSaaaeaadaqadaqaai aadIgaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaakeaacaaI ZaGaaiyiaaaaaeaadaWcaaqaamaabmaabaGaaGOmaiaadIgaaiaawI cacaGLPaaadaahaaWcbeqaaiaaiodaaaaakeaacaaIZaGaaiyiaaaa aeaadaWcaaqaamaabmaabaGaaG4maiaadIgaaiaawIcacaGLPaaada ahaaWcbeqaaiaaiodaaaaakeaacaaIZaGaaiyiaaaaaeaadaWcaaqa amaabmaabaGaeyOeI0IaaG4maiaadIgaaiaawIcacaGLPaaadaahaa WcbeqaaiaaisdaaaaakeaacaaI0aGaaiyiaaaaaeaadaWcaaqaamaa bmaabaGaeyOeI0IaaGOmaiaadIgaaiaawIcacaGLPaaadaahaaWcbe qaaiaaisdaaaaakeaacaaI0aGaaiyiaaaaaeaadaWcaaqaamaabmaa baGaeyOeI0IaamiAaaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaa aaaOqaaiaaisdacaGGHaaaaaqaamaalaaabaWaaeWaaeaacaaIWaGa amiAaaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaaaaaOqaaiaais 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aaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIZaGaamiAaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOnaaaaaOqaaiaaiAdacaGGHaaaaa qaamaalaaabaWaaeWaaeaacqGHsislcaaIYaGaamiAaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGOnaaaaaOqaaiaaiAdacaGGHaaaaaqaam aalaaabaWaaeWaaeaacqGHsislcaWGObaacaGLOaGaayzkaaWaaWba aSqabeaacaaI2aaaaaGcbaGaaGOnaiaacgcaaaaabaWaaSaaaeaada qadaqaaiaaicdacaWGObaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI 2aaaaaGcbaGaaGOnaiaacgcaaaaabaWaaSaaaeaadaqadaqaaiaadI gaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiAdaaaaakeaacaaI2aGa aiyiaaaaaeaadaWcaaqaamaabmaabaGaaGOmaiaadIgaaiaawIcaca GLPaaadaahaaWcbeqaaiaaiAdaaaaakeaacaaI2aGaaiyiaaaaaeaa daWcaaqaamaabmaabaGaaG4maiaadIgaaiaawIcacaGLPaaadaahaa WcbeqaaiaaiAdaaaaakeaacaaI2aGaaiyiaaaaaaaacaGLOaGaayzk aaWaaeWaaeaafaqabeWbbaaaaeaacaWGJbWaaSbaaSqaaiabgkHiTi aaiodaaeqaaaGcbaGaam4yamaaBaaaleaacqGHsislcaaIYaaabeaa aOqaaiaadogadaWgaaWcbaGaeyOeI0IaaGymaaqabaaakeaacaWGJb WaaSbaaSqaaiaaicdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIXaaa beaaaOqaaiaadogadaWgaaWcbaGaaGOmaaqabaaakeaacaWGJbWaaS baaSqaaiaaiodaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9maabmaa baqbaeqabCqaaaaabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaG imaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaaaaiaawIcacaGLPaaa aaa@6C1B@  

(1.7)

We can simplify this if we multiply both sides by 1/h2 and drop all references to h in the matrix:

 

−3 0 0! −2 0 0! −1 0 0! 0 0 0! 1 0 0! 2 0 0! 3 0 0! −3 1 1! −2 1 1! −1 1 1! 0 1 1! 1 1 1! 2 1 1! 3 1 1! −3 2 2! −2 2 2! −1 2 2! 0 2 2! 1 2 2! 2 2 2! 3 2 2! −3 3 3! −2 3 3! −1 3 3! 0 3 3! 1 3 3! 2 3 3! 3 3 3! −3 4 4! −2 4 4! −1 4 4! 0 4 4! 1 4 4! 2 4 4! 3 4 4! −3 5 5! −2 5 5! −1 5 5! 0 5 5! 1 5 5! 2 5 5! 3 5 5! −3 6 6! −2 6 6! −1 6 6! 0 6 6! 1 6 6! 2 6 6! 3 6 6! c −3 c −2 c −1 c 0 c 1 c 2 c 3 = 0 0 1/ h 2 0 0 0 0 MathType@MTEF@5@5@+= feaahmart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeWbhaaaaaqaamaalaaabaWaaeWaaeaacqGHsislcaaIZaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIWaaaaaGcbaGaaGimaiaacgcaaa aabaWaaSaaaeaadaqadaqaaiabgkHiTiaaikdaaiaawIcacaGLPaaa daahaaWcbeqaaiaaicdaaaaakeaacaaIWaGaaiyiaaaaaeaadaWcaa qaamaabmaabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqa 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(1.8)

              

The results for c with 7 points are:

C7=1/h2{1/90,-3/20,3/2,-49/18,3/2,-3/20,1/90}

On the other hand, the results with 5 points are:

C5=1/h2{-1/12,4/3,-5/2,4/3,-1/12}

And the results for 3 points are

C3=1/h2{1,-2,1}

One sees that there are significant differences between 7 and 5 points approximations for the second derivative.  The computational cost of using 7 points Vs. using 5 points is just 7/5 larger for the 7 point result.  That is a very worthwhile tradeoff for an  oscillatory wave packet of the Schrodinger equation.

For your convenience here is a Mathematica Notebook that can compute the coefficients for any number of points:

A[j_,i_,n_]=If[i==0 && n-j==0,1,(j-n)^i/i!]

v[i_]=If[i==2,1,0]

p=8

mp=Table[A[j,i,p/2],{I,0,p},{j,0,p}]

bp=Table[v[i],{i,0,p}]

ci=LinearSolve[mp,bp]