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Here we compile the standard finite difference schemes of various orders on uniform grids.

Consider the derivative du/dxdu/dx of a function u(x)u(x), where xx can be any independent variable (e.g., time, space, etc.). Finite-difference approximations are used to represent the continuous function u(x)u(x) by a set of discrete points unu^n at evenly spaced intervals, where nn is the index of the point in the grid. The spacing between these points is denoted as Δx\Delta x.

Example of a one-dimensional grid with uniform spacing \Delta x.

Figure 1:Example of a one-dimensional grid with uniform spacing Δx\Delta x.

Forward Difference

First Derivative (∂u∂x)(\frac{\partial u}{\partial x})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un+1−unΔx\frac{u^{n+1} - u^n}{\Delta x}
(1)
−un+2+4un+1−3un2Δx\frac{-u^{n+2} + 4u^{n+1} - 3u^n}{2\Delta x}
(2)

Second Derivative (∂2u∂x2)(\frac{\partial^2 u}{\partial x^2})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un+2−2un+1+unΔx2\frac{u^{n+2} - 2u^{n+1} + u^n}{\Delta x^2}
(3)
−un+3+4un+2−5un+1+2unΔx2\frac{-u^{n+3} + 4u^{n+2} - 5u^{n+1} + 2u^n}{\Delta x^2}
(4)

Third Derivative (∂3u∂x3)(\frac{\partial^3 u}{\partial x^3})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un+3−3un+2+3un+1−unΔx3\frac{u^{n+3} - 3u^{n+2} + 3u^{n+1} - u^n}{\Delta x^3}
(5)
−3un+4+14un+3−24un+2+18un+1−5un2Δx3\frac{-3u^{n+4} + 14u^{n+3} - 24u^{n+2} + 18u^{n+1} - 5u^n}{2\Delta x^3}
(6)

Fourth Derivative (∂4u∂x4)(\frac{\partial^4 u}{\partial x^4})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un+4−4un+3+6un+2−4un+1+unΔx4\frac{u^{n+4} - 4u^{n+3} + 6u^{n+2} - 4u^{n+1} + u^n}{\Delta x^4}
(7)
−2un+5+11un+4−24un+3+26un+2−14un+1+3unΔx4\frac{-2u^{n+5} + 11u^{n+4} - 24u^{n+3} + 26u^{n+2} - 14u^{n+1} + 3u^n}{\Delta x^4}
(8)

Backward Difference

First Derivative (∂u∂x)(\frac{\partial u}{\partial x})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un−un−1Δx\frac{u^n - u^{n-1}}{\Delta x}
(9)
3un−4un−1+un−22Δx\frac{3u^n - 4u^{n-1} + u^{n-2}}{2\Delta x}
(10)

Second Derivative (∂2u∂x2)(\frac{\partial^2 u}{\partial x^2})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un−2un−1+un−2Δx2\frac{u^n - 2u^{n-1} + u^{n-2}}{\Delta x^2}
(11)
2un−5un−1+4un−2−un−3Δx2\frac{2u^n - 5u^{n-1} + 4u^{n-2} - u^{n-3}}{\Delta x^2}
(12)

Third Derivative (∂3u∂x3)(\frac{\partial^3 u}{\partial x^3})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un−3un−1+3un−2−un−3Δx3\frac{u^n - 3u^{n-1} + 3u^{n-2} - u^{n-3}}{\Delta x^3}
(13)
5un−18un−1+24un−2−14un−3+3un−42Δx3\frac{5u^n - 18u^{n-1} + 24u^{n-2} - 14u^{n-3} + 3u^{n-4}}{2\Delta x^3}
(14)

Fourth Derivative (∂4u∂x4)(\frac{\partial^4 u}{\partial x^4})

O(Δx)\mathcal{O}(\Delta x)O(Δx2)\mathcal{O}(\Delta x^2)
un−4un−1+6un−2−4un−3+un−4Δx4\frac{u^n - 4u^{n-1} + 6u^{n-2} - 4u^{n-3} + u^{n-4}}{\Delta x^4}
(15)
3un−14un−1+26un−2−24un−3+11un−4−2un−5Δx4\frac{3u^n - 14u^{n-1} + 26u^{n-2} - 24u^{n-3} + 11u^{n-4} - 2u^{n-5}}{\Delta x^4}
(16)

Central Difference

First Derivative (∂u∂x)(\frac{\partial u}{\partial x})

O(Δx2)\mathcal{O}(\Delta x^2)O(Δx4)\mathcal{O}(\Delta x^4)
un+1−un−12Δx\frac{u^{n+1} - u^{n-1}}{2\Delta x}
(17)
−un+2+8un+1−8un−1+un−212Δx\frac{-u^{n+2} + 8u^{n+1} - 8u^{n-1} + u^{n-2}}{12\Delta x}
(18)

Second Derivative (∂2u∂x2)(\frac{\partial^2 u}{\partial x^2})

O(Δx2)\mathcal{O}(\Delta x^2)O(Δx4)\mathcal{O}(\Delta x^4)
un+1−2un+un−1Δx2\frac{u^{n+1} - 2u^n + u^{n-1}}{\Delta x^2}
(19)
−un+2+16un+1−30un+16un−1−un−212Δx2\frac{-u^{n+2} + 16u^{n+1} - 30u^n + 16u^{n-1} - u^{n-2}}{12\Delta x^2}
(20)

Third Derivative (∂3u∂x3)(\frac{\partial^3 u}{\partial x^3})

O(Δx2)\mathcal{O}(\Delta x^2)O(Δx4)\mathcal{O}(\Delta x^4)
un+2+2un+1−2un−1−un−22Δx3\frac{u^{n+2} + 2u^{n+1} - 2u^{n-1} - u^{n-2}}{2\Delta x^3}
(21)
−un+3+8un+2−13un+1+13un−1−8un−2+un−38Δx3\frac{-u^{n+3} + 8u^{n+2} - 13u^{n+1} + 13u^{n-1} - 8u^{n-2} + u^{n-3}}{8\Delta x^3}
(22)

Fourth Derivative (∂4u∂x4)(\frac{\partial^4 u}{\partial x^4})

O(Δx2)\mathcal{O}(\Delta x^2)O(Δx4)\mathcal{O}(\Delta x^4)
un+2−4un+1+6un−4un−1+un−2Δx4\frac{u^{n+2} - 4u^{n+1} + 6u^n - 4u^{n-1} + u^{n-2}}{\Delta x^4}
(23)
−un+3+12un+2−39un+1+56un−39un−1+12un−2−un−36Δx4\frac{-u^{n+3} + 12u^{n+2} - 39u^{n+1} + 56u^n - 39u^{n-1} + 12u^{n-2} - u^{n-3}}{6\Delta x^4}
(24)
Introduction
Summary
Appendix
Time Stepping