Deriving 4th order approximation of a 2nd order derivative using Taylor Table method
Aim: Deriving 4th order approximation of a 2nd order derivative using Taylor Table method
Governing equations
Central difference scheme(Symmetric scheme)
Number of nodes in central schemes= p+q-1(p=order of derivative,q=order of approximation)
`=2+4-1=5`
`frac{d^2y}{dx^2} approx af(i-2)+bf(i-1)+cf(i)+df(i+1)+ef(i+2)`
`f(i-2)=f(i)-2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}-frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}-frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720}`
Multiplying the above equation by a
`a**(f(i-2))=a**(f(i)-2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}-frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}-frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720})`
`f(i-1)`can be calculated by Taylor theorem
`f(i-1)=f(i)-f'(i)Deltax+frac{f''(i)Deltax^2}{2}-frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}-frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720}`
Multiplying the above equation by b
`b**(f(i-1))=b**(f(i)-f'(i)Deltax+frac{f''(i)Deltax^2}{2}-frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}-frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720})`
`f(i+1)`can be calculated by Taylor theorem
`f(i+1)=f(i)+f'(i)Deltax+frac{f''(i)Deltax^2}{2}+frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}+frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720}`
Multiplying the above equation by d
`d**(f(i+1))=d**(f(i)+f'(i)Deltax+frac{f''(i)Deltax^2}{2}+frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}+frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720})`
`f(i+2)`can be calculated by Taylor theorem
`f(i+2)=f(i)+2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}+frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}+frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720}`
Multiplying the above equation by e
`e**(f(i+2))=e**(f(i)+2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}+frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}+frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720})`
Forming Taylor's table for keeping track of coefficient
| Numerical Stencil | Taylor's series terms | ||||||
| `f(i)` | `Deltaxf'(i)` | `Deltax^2**f''(i)` | `Deltax^3f'''(i)` | `Deltax^4f''''(i)` | `Deltax^5f'''''(i)` | `Deltax^6f''''''(i)` | |
| `a**f(i-2)` | `a` | `-2a` | `2a` | `frac{-8a}{6}` | `frac{16a}{24}` | `frac{-32a}{120}` | `frac{64a}{720}` |
| `b**f(i-1)` | `b` | `-b` | `frac{b}{2}` | `frac{-b}{6}` | `frac{b}{24}` | `frac{-b}{120}` | `frac{b}{720}` |
| `c**f(i)` | `c` | `0` | `0` | `0` | `0` | `0` | `0` |
| `d**f(i+1)` | `d` | `d` | `frac{d}{2}` | `frac{d}{6}` | `frac{d}{24}` | `frac{d}{120}` | `frac{d}{720}` |
| `e**f(i+2)` | `e` | `2e` | `2e` | `frac{8e}{6}` | `frac{16e}{24}` | `frac{32e}{120}` | `frac{64e}{720}` |
Summing each and every term in the table
`Sigma a**f(i-2)+b**f(i-1)+cf(i)+d**f(i+1)+e**f(i+2)=frac{d^2y}{dx^2}`
`Sigma a+b+c+d+e=0`(As such equation under consideration is second order)
`Sigma-2a-b+0+d+2e=0`(As such equation under consideration is second order)
`Sigma2a+frac{b}{2}+0+frac{d}{2}+2e=1`
`Sigmafrac{-8a}{6}+frac{-b}{6}+0+frac{d}{6}+frac{8e}{6}=0`
`Sigmafrac{16a}{24}+frac{b}{24}+0+frac{d}{24}+frac{16e}{24}=0`
The values of `a,b,c,d,e`can be calculated by Matlab
`a=frac{-1}{12}`
`b=frac{4}{3}`
`c=frac{-5}{2}`
`d=frac{4}{3}`
`e=frac{-1}{12}`
Thus `a=e and b=d`
Rearranging all the terms
`frac{d^2y}{dx^2} approx af(i-2)+bf(i-1)+cf(i)+df(i+1)+ef(i+2)=(a+b+c+d+e)f(i)+(-2a-b+0+d+2e)Deltaxf'(i)+(2a+frac{b}{2}+0+frac{d}{2}+2e)Deltax^2f''(i)+(frac{-8a}{6}+frac{-b}{6}+0+frac{d}{6}+frac{8e}{6})Deltax^3f'''(i)+(frac{16a}{24}+frac{b}{24}+0+frac{d}{24}+frac{16e}{24})Deltax^4f''''(i)+(frac{-32a}{120}+frac{-b}{120}+0+frac{d}{120}+frac{32e}{120})Deltax^5f'''''(i)+(frac{64a}{720}+frac{b}{720}+0+frac{d}{720}+frac{64e}{720})Deltax^6f''''''(i)`
Equating LHS and RHS
`frac{d^2y}{dx^2}=frac{-1}{12}f(i-2)+frac{4}{3}f(i-1)+frac{-5}{2}f(i)+frac{4}{3}f(i+1)+frac{-1}{12}f(i+2)`=`0**f(i)+0**f'(i)+(-2a-b+0+d+2e)Deltax^2f''(i)+0**f'''(i)+0**f''''(i)+0**f'''''(i)+(some value)**Deltax^6f''''''(i)`
Dividing by `Deltax^2 throughout`
`frac{d^2y}{dx^2}=frac{-1}{12}f(i-2)+frac{4}{3}f(i-1)+frac{-5}{2}f(i)+frac{4}{3}f(i+1)+frac{-1}{12}f(i+2)``**frac{1}{Deltax^2}`=`(-2a-b+0+d+2e)**f''(i)`+order of (`Deltax^4`)
Thus equation for the skewed right-sided difference is
`frac{d^2y}{dx^2}=(frac{-1}{12}f(i-2)+frac{4}{3}f(i-1)+frac{-5}{2}f(i)+frac{4}{3}f(i+1)+frac{-1}{12}f(i+2))``**frac{1}{Deltax^2}`
Skewed right-sided difference
Solution
Number of nodes in right sided schemes= p+q (p=order of derivative,q=order of approximation)
=`2+4=6`
`frac{d^2y}{dx^2} approx af(i)+bf(i+1)+cf(i+2)+df(i+3)+ef(i+4)+gf(i+5)` `
`f(i+1)`can be calculated by Taylor theorem
`f(i+1)=f(i)+f'(i)Deltax+frac{f''(i)Deltax^2}{2}+frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}+frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720}`
Multiplying the above equation by b
`b**(f(i+1))=b**(f(i)+f'(i)Deltax+frac{f''(i)Deltax^2}{2}+frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}+frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720})`
`f(i+2)`can be calculated by Taylor theorem
`f(i+2)=f(i)+2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}+frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}+frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720}`
Multiplying the above equation by c
`c**(f(i+2))=c**(f(i)+2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}+frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}+frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720})`
`f(i+3)`can be calculated by Taylor theorem
`f(i+3)=f(i)+f'(i)**3Deltax+f''(i)frac{9}{2}Deltax^2+f'''(i)frac{27}[6}Deltax^3+f''''(i)frac{81}{24}Deltax^4+f'''''frac{243}{120}Deltax^5+f''''''(i)frac{729}{720}Deltax^6`
Multiplying the above equation by d
`d**f(i+3)=d**(f(i)+f'(i)**3Deltax+f''(i)frac{9}{2}Deltax^2+f'''(i)frac{27}[6}Deltax^3+f''''(i)frac{81}{24}Deltax^4+f'''''frac{243}{120}Deltax^5+f''''''(i)frac{729}{720}Deltax^6)`
`f(i+4)`can be calculated by Taylor theorem
`f(i+4)=f(i)+f'(i)4**Deltax+f''(i)frac{16}{2}Deltax^2+f'''(i)frac{64}{6}Deltax^3+f''''(i)frac{256}{24}Deltax^4+f'''''(i)frac{1024}{120}Deltax^5+f''''''(i)frac{4096}{720}Deltax^6`
Multiplying the above equation by e
`e**(f(i+4))=e**(f(i)+f'(i)4**Deltax+f''(i)frac{16}{2}Deltax^2+f'''(i)frac{64}{6}Deltax^3+f''''(i)frac{256}{24}Deltax^4+f'''''(i)frac{1024}{120}Deltax^5+f''''''(i)frac{4096}{720}Deltax^6)`
`f(i+5)`can be calculated by Taylor theorem
`f(i+5)=f(i)+f'(i)**5**Deltax+f''(i)frac{25}[2}Deltax^2+f'''(i)frac{125}{6}Deltax^3+f''''(i)frac{625}{24}Deltax^4+f'''''(i)frac{3125}{120}Deltax^5+f''''''(i)frac{15625}{720}Deltax^6`
Multiplying the above equation by g
`g**(f(i+5))=g**(f(i)+f'(i)**5**Deltax+f''(i)frac{25}[2}Deltax^2+f'''(i)frac{125}{6}Deltax^3+f''''(i)frac{625}{24}Deltax^4+f'''''(i)frac{3125}{120}Deltax^5+f''''''(i)frac{15625}{720}Deltax^6)`
Calculating Coefficients through Taylor's table.
| Numerical Stencil | Taylor's series terms | |||||
| `f(i)` | `Deltaxf'(i)` | `Deltax^2f''(i)` | `Deltax^3f'''(i)` | `Deltax^4f''''(i)` | `Deltax^5f'''''(i)` | |
| `af(i)` | a | 0 | 0 | 0 | 0 | 0 |
| `bf(i+1)` | b | b | `frac{b}{2}` | `frac{b}{6}` | `frac{b}{24}` | `frac{b}{120}` |
| `cf(i+2)` | c | 2c | `frac{4c}{2}` | `frac{8c}{6}` | `frac{16c}{24}` | `frac{32c}{120}` |
| `df(i+3)` | d | 3d | `frac{9d}{2}` | `frac{27d}{6}` | `frac{81d}{24}` | `frac{243d}{120}` |
| `ef(i+4)` | e | 4e | `frac{16e}{2}` | `frac{64e}{6}` | `frac{256e}{24}` | `frac{1024e}{120}` |
| `gf(i+5)` | g | 5g | `frac{25g}{2}` | `frac{125g}{6}` | `frac{625g}{24}` | `frac{3125g}{120}` |
Solving for `a,b,c,d,e,g`
Summing Taylor's series terms
`Sigmaa+b+c+d+e+g=0`
`Sigma0+b+2c+3d+4e+5g=0`
`Sigma0+frac{b}{2}+frac{4c}{2}+frac{9d}{2}+frac{16e}{2}+frac{25g}{2}=1`
`Sigma0+frac{b}{6}+frac{8c}{6}+frac{27d}{6}+frac{64e}{6}+frac{125g}{6}=0`
`Sigma0+frac{b}{24}+frac{16c}{24}+frac{81d}{24}+frac{256e}{24}+frac{625g}{24}=0`
`Sigma0+frac{b}{120}+frac{32c}{120}+frac{243d}{120}+frac{1024e}{120}+frac{3125g}{120}=0`
`a=frac{15}{4}`
`b=frac{-77}{6}`
`c=frac{107}{6}`
`d=-13`
`e=frac{61}{12}`
`g=frac{-5}{6}`
Rearranging all the terms
Thus equation for the skewed right-sided difference is
`frac{d^2y}{dx^2} = (frac{15}{4}f(i)+frac{-77}{6}f(i+1)+frac{107}{6}f(i+2)-13f(i+3)+frac{61}{12}f(i+4)+frac{-5}{6}f(i+5))**frac{1}{Deltax^2}`
Skewed left-sided difference
Number of nodes in left sided schemes= p+q (p=order of derivative,q=order of approximation)
=`2+4=6`
`frac{d^2y}{dx^2} approx af(i)+bf(i-1)+cf(i-2)+df(i-3)+ef(i-4)+gf(i-5)` `
`f(i-1)`can be calculated by Taylor theorem`
`f(i-1)=f(i)-f'(i)Deltax+frac{f''(i)Deltax^2}{2}-frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}-frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720}+..........`
Multiplying the above equation by b
`b**(f(i-1))=b**(f(i)-f'(i)Deltax+frac{f''(i)Deltax^2}{2}-frac{f'''(i)Deltax^3}{6}+frac{f''''(i)Deltax^4}{24}-frac{f'''''(i)Deltax^5}{120}+frac{f''''''(i)Deltax^6}{720}+)`
`f(i-2)`can be calculated by Taylor theorem
`f(i-2)=f(i)-2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}-frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}-frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720}+.....................`
Multiplying the above equation by c
`c**(f(i-2))=c**(f(i)-2**Deltaxf'(i)+frac{f''(i)4Deltax^2}{2}-frac{f'''(i)8Deltax^3}{6}+frac{f''''(i)16Deltax^4}{24}-frac{f'''''(i)32Deltax^5}{120}+frac{f''''''(i)64Deltax^6}{720}+.....................)`
`f(i-3)`can be calculated by Taylor theorem`
`f(i-3)=f(i)-f'(i)**3Deltax+f''(i)frac{9}{2}Deltax^2-f'''(i)frac{27}[6}Deltax^3+f''''(i)frac{81}{24}Deltax^4-f'''''(i)frac{243}{120}Deltax^5+f''''''(i)frac{729}{720}Deltax^6`
Multiplying the above equation by d
`d**(f(i-3))=d**(f(i)-f'(i)**3Deltax+f''(i)frac{9}{2}Deltax^2-f'''(i)frac{27}[6}Deltax^3+f''''(i)frac{81}{24}Deltax^4-f'''''(i)frac{243}{120}Deltax^5+f''''''(i)frac{729}{720}Deltax^6)`
`f(i-4)`can be calculated by Taylor theorem `
`f(i-4)=f(i)-f'(i)4**Deltax+f''(i)frac{16}{2}Deltax^2-f'''(i)frac{64}{6}Deltax^3+f''''(i)frac{256}{24}Deltax^4-f'''''(i)frac{1024}{120}Deltax^5+f''''''(i)frac{4096}{720}Deltax^6`
Multiplying the above equation by e
`e**(f(i-4))=e**(f(i)-f'(i)4**Deltax+f''(i)frac{16}{2}Deltax^2-f'''(i)frac{64}{6}Deltax^3+f''''(i)frac{256}{24}Deltax^4-f'''''(i)frac{1024}{120}Deltax^5+f''''''(i)frac{4096}{720}Deltax^6)`
`f(i-5)` can be calculated by Taylor theorem`
`f(i-5)=f(i)-f'(i)**5**Deltax+f''(i)frac{25}[2}Deltax^2-f'''(i)frac{125}{6}Deltax^3+f''''(i)frac{625}{24}Deltax^4-f'''''(i)frac{3125}{120}Deltax^5+f''''''(i)frac{15625}{720}Deltax^6`
Multiplying the above equation by g
`g**(f(i-5))=g**(f(i)-f'(i)**5**Deltax+f''(i)frac{25}[2}Deltax^2-f'''(i)frac{125}{6}Deltax^3+f''''(i)frac{625}{24}Deltax^4-f'''''(i)frac{3125}{120}Deltax^5+f''''''(i)frac{15625}{720}Deltax^6)`
Calculating Coefficients through Taylor's table.
| Numerical Stencil | Taylor series table | |||||
| `f(i)` | `Deltaxf'(i)` | `Deltax^2f''(i)` | `Deltax^3f'''(i)` | `Deltax^4f''''(i)` | `Deltax^5f'''''(i)` | |
| `af(i)` | a | 0 | 0 | 0 | 0 | 0 |
| `bf(i-1)` | b | `-b` | `frac{b}{2}` | `frac{-b}{6}` | `frac{b}{24}` | `frac{-b}{120}` |
| `cf(i-2)` | c | `-2c` | `frac{4c}{2}` | `frac{-8c}{6}` | `frac{16c}{24}` | `frac{-32c}{120}` |
| `df(i-3)` | d | `-3d` | `frac{9d}{2}` | `frac{-27d}{6}` | `frac{81d}{24}` | `frac{-243d}{120}` |
| `ef(i-4)` | e | `-4e` | `frac{16e}{2}` | `frac{-64e}{6}` | `frac{256e}{24}` | `frac{-1024e}{120}` |
| `gf(i-5)` | g | `-5g` | `frac{25g}{2}` | `frac{-125g}{6}` | `frac{625g}{24}` | `frac{-3125g}{120}` |
Summation of all the terms will yield
`Sigmaa+b+c+d+e+g=0`
`Sigma0-b-2c-3d-4e-5g=0`
`Sigma0+frac{b}{2}+frac{4c}{2}+frac{9d}{2}+frac{16e}{2}+frac{25g}{2}=1`
`Sigma0+frac{-b}{6}+frac{-8c}{6}+frac{27d}{6}+frac{-64e}{6}+frac{-125g}{6}=0`
`Sigma0+frac{b}{24}+frac{16c}{24}+frac{81d}{24}+frac{256e}{24}+frac{625g}{24}=0`
`Sigma0+frac{-b}{120}+frac{-32c}{120}+frac{-243d}{120}+frac{-1024e}{120}+frac{-3125g}{120}=0`
`a=frac{15}{4}`
`b=frac{-77}{6}`
`c=frac{107}{6}`
`d=-13`
`e=frac{61}{12}`
`g=frac{-5}{6}`
Thus equation for the skewed left-sided difference is
`frac{d^2y}{dx^2} = (frac{15}{4}f(i)+frac{-77}{6}f(i-1)+frac{107}{6}f(i-2)-13f(i-3)+frac{61}{12}f(i-4)+frac{-5}{6}f(i-5))**frac{1}{Deltax^2}`
Matlab coding
Calculating central difference scheme coefficient through Matlab
>> A=[1,1,1,1,1;-2,-1,0,1,2;2,1/2,0,1/2,2;-8/6,-1/6,0,1/6,8/6;16/24,1/24,0,1/24,16/24]
A =
1.0000 1.0000 1.0000 1.0000 1.0000
-2.0000 -1.0000 0 1.0000 2.0000
2.0000 0.5000 0 0.5000 2.0000
-1.3333 -0.1667 0 0.1667 1.3333
0.6667 0.0417 0 0.0417 0.6667
>> B=[0;0;1;0;0]
B =
0
0
1
0
0
>> X=AB
X =
-0.0833
1.3333
-2.5000
1.3333
-0.0833Calculating skewed right-sided scheme coefficient through Matlab
>> A=[1,1,1,1,1,1;0,1,2,3,4,5;0,1/2,4/2,9/2,16/2,25/2;0,1/6,8/6,27/6,64/6,125/6;0,1/24,16/24,81/24,256/24,625/24;0,1/120,32/120,243/120,1024/120,3125/120]
A =
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0 1.0000 2.0000 3.0000 4.0000 5.0000
0 0.5000 2.0000 4.5000 8.0000 12.5000
0 0.1667 1.3333 4.5000 10.6667 20.8333
0 0.0417 0.6667 3.3750 10.6667 26.0417
0 0.0083 0.2667 2.0250 8.5333 26.0417
>> B=[0;0;1;0;0;0]
B =
0
0
1
0
0
0
>> X=AB
X =
3.7500
-12.8333
17.8333
-13.0000
5.0833
-0.8333
Note: All the coefficients have been turned in fractions for ease in the calculation.
Calculating second-order derivative of exp(x)*cos(x) and comparing it with the three schemes.
Matlab code
function error_cds=central_difference_scheme(x,dx)
Exact_derivative=-2*exp(x)*sin(x);
Numerical_derivative=((-1/12)*exp(x-2*dx)*cos(x-2*dx)+(4/3)*exp(x-dx)*cos(x-dx)+(-5/2)*exp(x)*cos(x)+(4/3)*exp(x+dx)*cos(x+dx)+(-1/12)*exp(x+2*dx)*cos(x+2*dx))/dx^2;
error_cds=abs(Numerical_derivative-Exact_derivative);
end
function error_srs=skewed_right_sided_scheme(x,dx)
Exact_derivative=-2*exp(x)*sin(x);
Numerical_derivative=((15/4)*exp(x)*cos(x)+(-77/6)*exp(x+dx)*cos(x+dx)+(107/6)*exp(x+2*dx)*cos(x+2*dx)-13*exp(x+3*dx)*cos(x+3*dx)+(61/12)*exp(x+4*dx)*cos(x+4*dx)+(-5/6)*exp(x+5*dx)*cos(x+5*dx))/dx^2;
error_srs=abs(Numerical_derivative-Exact_derivative);
end
function error_sls=skewed_left_sided_scheme(x,dx)
Exact_derivative=-2*exp(x)*sin(x);
Numerical_derivative=((15/4)*exp(x)*cos(x)+(-77/6)*exp(x-dx)*cos(x-dx)+(107/6)*exp(x-2*dx)*cos(x-2*dx)-13*exp(x-3*dx)*cos(x-3*dx)+(61/12)*exp(x-4*dx)*cos(x-4*dx)+(-5/6)*exp(x-5*dx)*cos(x-5*dx))/dx^2;
error_sls=abs(Numerical_derivative-Exact_derivative);
end
% evaluating second order derivative of the function using three schemes
clear
close all
clc
x=pi/3;
dx=linspace(pi/4000,pi/40,80);
for i=1:length(dx)
error_cds(i)=central_difference_scheme(x,dx(i));
error_srs(i)=skewed_right_sided_scheme(x,dx(i));
error_sls(i)=skewed_left_sided_scheme(x,dx(i));
end
% plotting
figure(1)
loglog(dx,error_cds,'r');
hold on
loglog(dx,error_srs,'g');
hold on
loglog(dx,error_sls,'b');
xlabel('Range of dx values')
ylabel('Error');
legend('central difference scheme','skewed right sided scheme','skewed left sided scheme');
title('Comparison of errors from different schemes');
Graph
Conclusion
The central difference scheme will give the lowest error for increasing values of dx compared to the right-sided and left-sided scheme.
The error pattern follows the trend of Central difference scheme`lt` Skewed left-sided scheme `lt` Skewed right-sided scheme
Although the central difference scheme gives less error compared to skewed but in some cases skewed (right and left) is more useful. For eg if an error at the boundary to be calculated then skewed schemes are applied. The central difference scheme is not applicable in this regard.