Difference Equation Representations for Second Derivatives

If a function f(x) is tabulated at a uniform interval h, then

f(x₀ + nh) = f(x_n) = f_n

where n is an integer and x₀ is any arbitrary value.

The values at the right in the table below are the leading error terms for each difference equation representation.

Centered Differences
f₁ - 2f₀ + f_-1 f''₀ = -------------- h² -f₂ + 16f₁ - 30f₀ + 16f_-1 - f_-2 f''₀ = ------------------------------ h² 2f₃ - 27f₂ + 270f₁ - 490f₀ + 270f_-1 + 2f_-2 f''₀ = ----------------------------------------- 180h²	h² f⁽⁴⁾₀ - ------- 12 h⁴ f⁽⁶⁾₀ + ------- 90 h⁶ f⁽⁸⁾₀ - ------- 560
Forward Differences
f₂ - 2f₁ + f₀ f''₀ = ------------- h² -f₃ + 4f₂ - 5f₁ + 2f₀ f''₀ = --------------------- h² 11f₄ - 56f₃ + 114f₂ - 104f₁ + 35f₀ f''₀ = ---------------------------------- 12h² -f₃ + 4f₂ + 6f₁ - 20f₀ + 11f_-1 f''₀ = ----------------------------- 12h² -10f₅ + 61f₄ - 156f₃ + 214f₂ - 154f₁ + 45f₀ f''₀ = ------------------------------------------ 12h² f₄ - 6f₃ + 14f₂ - 4f₁ - 15f₀ + 10f_-1 f''₀ = ----------------------------------- 12h² 137f₆ - 972f₅ + 2970f₄ - 5080f₃ + 5265f₂ - 3132f₁ + 812f₀ f''₀ = -------------------------------------------------------- 180h² -13f₅ + 93f₄ - 285f₃ + 470f₂ - 255f₁ - 147f₀ + 137f_-1 f''₀ = ---------------------------------------------------- 180h² 2f₄ - 12f₃ + 15f₂ + 200f₁ - 420f₀ + 228f_-1 - 13f_-2 f''₀ = ------------------------------------------------- 180h²	- h f⁽³⁾₀ 11h² f⁽⁴⁾₀ + --------- 12 5h³ f⁽⁵⁾₀ - -------- 6 h³ f⁽⁵⁾₀ + ------- 12 137h⁴ f⁽⁶⁾₀ + ---------- 180 13h⁴ f⁽⁶⁾₀ - --------- 180 7h⁵ f⁽⁷⁾₀ - -------- 10 11h⁵ f⁽⁷⁾₀ + --------- 180 h⁵ f⁽⁷⁾₀ - ------- 90
Backward Differences
f₀ - 2f_-1 + f_-2 f''₀ = --------------- h² 2f₀ - 5f_-1 + 4f_-2 - f_-3 f''₀ = ---------------------- h² 35f₀ - 104f_-1 + 114f_-2 - 56f_-3 + 11f_-4 f''₀ = ------------------------------------- 12h² 11f₁ - 20f₀ + 6f_-1 + 4f_-2 - f_-3 f''₀ = ------------------------------ 12h² 45f₀ - 154f_-1 + 214f_-2 - 156f_-3 + 61f_-4 - 10f_-5 f''₀ = --------------------------------------------- 12h² 10f₁ - 15f₀ - 4f_-1 + 14f_-2 - 6f_-3 + f_-4 f''₀ = ------------------------------------- 12h² 812f₀ - 3132f_-1 + 5265f_-2 - 5080f_-3 + 2970f_-4 - 972f_-5 + 137f_-6 f''₀ = ------------------------------------------------------------- 180h² 137f₁ - 147f₀ - 255f_-1 + 470f_-2 - 285f_-3 + 93f_-4 - 13f_-5 f''₀ = ------------------------------------------------------ 180h² -13f₂ + 228f₁ - 420f₀ + 200f_-1 + 15f_-2 - 12f_-3 + 2f_-4 f''₀ = --------------------------------------------------- 180h²	+ h f⁽³⁾₀ 11h² f⁽⁴⁾₀ + --------- 12 5h³ f⁽⁵⁾₀ + -------- 6 h³ f⁽⁵⁾₀ - ------- 12 137h⁴ f⁽⁶⁾₀ + ---------- 180 13h⁴ f⁽⁶⁾₀ - --------- 180 7h⁵ f⁽⁷⁾₀ + -------- 10 11h⁵ f⁽⁷⁾₀ - --------- 180 h⁵ f⁽⁷⁾₀ + ------- 90

Research Interests
First Derivatives