Finite Difference Methods
The FDM approximations are been started with the help of Taylor Series for the function of
temperature ‘T’ in the environs [next point] of the point (i, j).
𝑇𝑖+1,𝑗
(∆𝑥)2
(∆𝑥)3
𝜕𝑇
𝜕2𝑇
𝜕3𝑇
= 𝑇𝑖,𝑗 + ( ) ∆𝑥 + ( 2 )
+ ( 3)
+ ……
𝜕𝑥 𝑖,𝑗
𝜕𝑥 𝑖,𝑗 2
𝜕𝑥 𝑖,𝑗 6
(1)
Modify the equation 1 as per the differential term requirement,
(∆𝑥)2
𝑇𝑖+1,𝑗 − 𝑇𝑖,𝑗
𝜕𝑇
𝜕2𝑇
∆𝑥
𝜕3𝑇
− ( 2)
− ( 3)
− …………
( ) =
𝜕𝑥 𝑖,𝑗
∆𝑥
𝜕𝑥 𝑖,𝑗 2
𝜕𝑥 𝑖,𝑗 6
(
𝑇𝑖+1,𝑗 − 𝑇𝑖,𝑗
𝜕𝑇
+ 𝑂(∆𝑥)
) =
𝜕𝑥 𝑖,𝑗
∆𝑥
(2)
(3)
Equation (3) is commonly known as Forward Difference Approximation [FDA]. In the
equation (3), the higher terms are neglected, which may chance o induce the Truncation Error
[𝑂(∆𝑥)] in the FDM approach.
The Taylor Series extended for the other approximation of FDM, which is Backward
Difference so the optimized equation is,
𝑇𝑖−1,𝑗
(−∆𝑥)2
(−∆𝑥)3
𝜕𝑇
𝜕2𝑇
𝜕3𝑇
= 𝑇𝑖,𝑗 + ( ) (−∆𝑥) + ( 2 )
+ ( 3)
+ ⋯ (4)
𝜕𝑥 𝑖,𝑗
𝜕𝑥 𝑖,𝑗
2
𝜕𝑥 𝑖,𝑗
6
Modify the equation 4 as per the differential term requirement,
(∆𝑥)2
𝑇𝑖,𝑗 − 𝑇𝑖−1,𝑗
𝜕𝑇
𝜕2𝑇
∆𝑥
𝜕3𝑇
+ ( 2)
− ( 3)
+ …………
( ) =
𝜕𝑥 𝑖,𝑗
∆𝑥
𝜕𝑥 𝑖,𝑗 2
𝜕𝑥 𝑖,𝑗 6
(5)
(
𝑇𝑖,𝑗 − 𝑇𝑖−1,𝑗
𝜕𝑇
+ 𝑂(∆𝑥)
) =
𝜕𝑥 𝑖,𝑗
∆𝑥
(6)
Equation (6) is commonly known as Backward Difference Approximation [BDA]. In the
equation (6), the higher terms are neglected, which may chance to induce the Truncation
Error [𝑂(∆𝑥)] in the FDM approach.
To get Central Difference approximation, subtract the Equation (4) from (1)
𝑇𝑖+1,𝑗 − 𝑇𝑖−1,𝑗 = 2. ∆𝑥 (
(
𝜕𝑇
) + 𝐻𝑖𝑔ℎ𝑒𝑟 𝑂𝑟𝑑𝑒𝑟 𝑇𝑒𝑟𝑚𝑠
𝜕𝑥 𝑖,𝑗
𝑇𝑖+1,𝑗 − 𝑇𝑖−1,𝑗
𝜕𝑇
+ 𝑂(∆𝑥 2 )
) =
𝜕𝑥 𝑖,𝑗
2. ∆𝑥
(7)
(8)
Equation (8) is commonly known as Central Difference Approximation [CDA]. In the
equation (8), the higher terms are neglected, which may chance to induce the Truncation
Error [𝑂(∆𝑥 2 )] in the FDM approach. But the Truncation Error generation in CDA is lesser
than FDA and BDA Truncation errors. This decrement of Truncation Error is because of the
subtraction process. Owing to this subtraction most of the higher order terms are been
removed from the equations (4) and (1).
To get Second Order Derivative, Add the Equation (4) and (1),
𝜕2 𝑇
𝑇𝑖+1,𝑗 + 𝑇𝑖−1,𝑗 = 2. 𝑇𝑖,𝑗 + (
(
)
𝜕𝑥2 𝑖,𝑗
(∆𝑥)2 + 𝑂(∆𝑥)2
𝑇𝑖+1,𝑗 − 2. 𝑇𝑖,𝑗 + 𝑇𝑖−1,𝑗
𝜕2𝑇
)
=
+ 𝑂(∆𝑥)2
2
2
(∆𝑥)
𝜕𝑥 𝑖,𝑗
Equation (10) is commonly known as Second Order Derivative Approximation.
(9)
(10)