Objective
Derive the following 4th order approximations of the second order derivative
1. Central difference
2. Skewed right sided difference
3. Skewed left sided difference
4. plot the graph for diffrence of error for above scheme
Derivation of central diffrence
For derive 4th order approximation of second order derivative.folllowing node are consider for central diffrence method. nodes are i,(i2),(i1),(i+2),(i+1). the value of af(i2),b(i1)..etc find out by taylor series expansion. like this there is use of taylor series expansion for skewed left & right 4th order approx.
`af(i2)= a[f_(i)(2Δxf'(i))/(1!)+f_(i)''((2Δx)^2)/(2!) + f_(i)'''((2Δx)^2)/(2!) ] +......`
Taylor table for find out 4th order approximation
stensil 
f(i)

dxf'(i)

dx^2f''(i)

dx^3f'''(i)

dx^4f^iv(i)

dx^5f^v(i)

dx^6f^vi(i)

af(i2)

a

2a

4a/2

8a/2

16a/24

32a/120

64a/720

bf(i1)

b

b

b/2

b/6

b/24

b/120

b/240

cf(i)

c

0

0

0

0

0

0

df(i+1)

d

d

d/2

d/6

d/6

d/120

d/720

ef(i+2)

e

2e

4e/2

8e/6

8e/6

32e/120

64e/720

sum

0

0

1

0

0

?

?

after that made 5 equation for finding the value a,b,c,d,e
equation1: a+b+c+d+e=0
equation2: (2a)b+0+d+2e=0
equation3: 2a+b/2+0+d/2+4e/2=1
equaion4: (4a)+(b/6)+0+d/6+8e/6=0
equation5: (16a/24)+b/24+0+d/6+8e/6=0
the value of a,b,c,d,e is find out by use of matrix in 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];
B=[0;0;1;0;0];
x=A\B;
disp(x)
command window shows the value for coefficent
a=0.0833, b=1.3333, c=2.5000, d=1.3333, e=0.0833
after putting the value in in main equation we got final equation of 4th order approx.
`(d^2f)/dx^2= (0.8333f_(i2)+1.333f_(i1)2.5f_(i)+1.33f_(i+1)0.0833f_(i+2))/dx^2`
Derivation for Skewed right side approximation
For derive 4th order approximation of second order derivative.folllowing node are consider for skewed right side approx. nodes are i,(i+1),(i+2),(i+3),(i+4).

f(i)

dxf'(i)

dx^2f''(i)

dx^3f'''(i)

dx^4f^iv(i)

dx^5f^v(i)

dx^6f^vi(i)

af(i)

a

0

0

0

0

0

0

bf(i+1)

b

b

b/2

b/6

b/24

b/120

b/720

cf(i+2)

c

2c

4c/2

8c/2

16c/24

32c/120

64c/720

df(i+3)

d

3d

9d/2

27d/6

81d/24

243d/120

729d/720

ef(i+4)

e

4e

16e/2

64e/6

256e/24

1024e/120

4096e/720

sum

0

0

1

0

0

?

?

after that made 5 equation for finding the value a,b,c,d,e
equation1: a+b+c+d+e=0
equation2: 0+b+2c+3d+4e=0
equation3: 0+b/2+4c/2+9d/2+16e/2=1
equaion4: 0+(b/6)+(8c/2)+27d/6+64e/6=0
equation5: 0+(b/24)+16c/24+81d/24+256e/24=0
a,b,c,d,e value find out by use of matrix in matlab
A=[1 1 1 1 1;
0 1 2 3 4;
0 1/2 2 9/2 8;
0 1/6 8/6 27/6 64/6;
0 1/24 16/24 81/24 256/24];
B=[0;0;1;0;0];
x=A\B;
disp(x)
command window shows the value for coefficent
a=2.9167 b=8.6667 , c=9.5000 , d=4.6667 , e=0.9167
after putting the value in in main equation we got final equation of 4th order approx.
`(d^2f)/dx^2= (2.9167f_(i)8.6667f_(i+1)+9.5f_(i+2)4.6667f_(i+3)+0.9167f_(i+4))/dx^2`
Derivation for Skewed left side approximation
For derive 4th order approximation of second order derivative.folllowing node are consider for skewed left side approx. nodes are i,(i1),(i2),(i3),(i4).
stensil

f(i)

dxf'(i)

dx^2f''(i)

dx^3f'''(i)

dx^4f^iv(i)

dx^5f^v(i)

dx^6f^vi(i)

af(i4)

a

4a

16a/2

64a/6

256a/24

1024a/120

4096a/720

bf(i3)

b

3b

9b/2

27b/6

81b/24

243b/120

729b/720

cf(i2)

c

2c

4c/2

8c/6

16c/24

32c/120

64c/720

df(i1)

d

d

d/2

d/6

d/24

d/120

d/720

ef(i)

e

0

0

0

0

0

0

sum

0

0

1

0

0

?

?

after that made 5 equation for finding the value a,b,c,d,e
equation1: a+b+c+d+e=0
equation2: 4a3b2cd+0=0
equation3: 16a/2+9b/2+4c/2+d/2+0=1
equaion4: (64a/6)+(27b/6)+(8c/6)+(d/6)+0=0
equation5: (256a/24)+81b/24+16c/24+d/24+0=0
A=[1 1 1 1 1;
4 3 2 1 0;
8 9/2 4/2 1/2 0;
64/6 27/6 8/6 1/6 0;
256/24 81/24 16/24 1/24 0];
B=[0;0;1;0;0];
x=A\B;
disp(x)
command window shows the value for coefficent.
a=2.9167 , b=8.6667 , c=9.5000 , d=4.6667 ,e=0.9167
after put the value in main equation we got final equation of 4th order approx.
`(d^2f)/dx^2= (2.9167f_(i)8.6667f_(i1)+9.5f_(i2)4.6667f_(i3)+0.9167f_(i4))/dx^2_`
Function out code for central diffrence
function out= central_fourth_order_approximation(x,dx)
% of analyatical function=exp(x)*cos(x)
% of analyatical derivative
% f''(x)=2*e(x)*sin(x)
second_order_analytical_derivative= 2*exp(x)*sin(x);
%numerical derivative
% use of cenrtal diffrencing
% fourth order approximation
central_diffrence=(((0.08333)*((exp(x(2*dx)))*(cos(x(2*dx)))))+((1.3333)*((exp(xdx))*(cos(xdx))))((2.5)*((exp(x))*(cos(x))))+((1.3333)*((exp(x+dx))*(cos(x+dx))))+((0.0833)*((exp(x+(2*dx)))*(cos(x+(2*dx))))))/(dx^2);
out=abs(central_diffrence  second_order_analytical_derivative);
end
Function out code for skewed right side
function out= skewed_right_side_approximation(x,dx)
% of analyatical function=exp(x)*cos(x)
% of analyatical derivative
% f''(x)=2*e(x)*sin(x)
second_order_analytical_derivative= 2*exp(x)*sin(x);
%numerical derivative
% use of skewed right side
% fourth order approximation
skewed_right_side =(((2.9167)*((exp(x))*(cos(x))))((8.6667)*((exp(x+dx))*(cos(x+dx))))+((9.5)*((exp(x+(2*dx))*(cos(x+(2*dx))))))((4.6667)*((exp(x+(3*dx)))*(cos(x+(3*dx)))))+((0.9167)*((exp(x+(4*dx)))*(cos(x+(4*dx))))))/(dx^2);
out=abs(skewed_right_side second_order_analytical_derivative);
end
Function out code for skewed left side
function out= skewed_left_side_approximation(x,dx)
% of analyatical function=exp(x)*cos(x)
% of analyatical derivative
% f''(x)=2*e(x)*sin(x)
second_order_analytical_derivative= 2*exp(x)*sin(x);
%numerical derivative
% use of skewed left side
% fourth order approximation
skewed_left_side =(((0.9167)*((exp(x(4*dx))*(cos(x(4*dx))))))((4.6667)*((exp(x(3*dx))*(cos(x(3*dx))))))+((9.5)*((exp(x(2*dx))*(cos(x(2*dx))))))((8.6667)*((exp(xdx))*(cos(xdx))))+((2.9167)*((exp(x))*(cos(x)))))/(dx^2);
out=abs(skewed_left_side second_order_analytical_derivative);
end
Main code
clear all
close all
clc
x=pi/2;
dx=linspace(pi/4,pi/40,10);
second_order_analytical_derivative=2*sin(x)*exp(x);
for i=1:length(dx)
central_diffrence_error(i)= central_fourth_order_approximation(x,dx(i));
skewd_right_side_error(i)=skewed_right_side_approximation(x,dx(i));
skewd_left_side_error(i)=skewed_left_side_approximation(x,dx(i));
end
figure(1)
loglog(dx,central_diffrence_error,'b')
hold on
loglog(dx,skewd_right_side_error,'r')
hold on
loglog(dx,skewd_left_side_error,'g')
legend('central_diffrence_error','skewd_right_side_error','skewd_left_side_error')
xlabel('dx value')
ylabel('error value')
Plot for comparison of absolute error between CDS, SRS,SLS
result is found at x=pi/2 & dx=linspace(pi/4,pi/40,10). so from graph the it is found that if we use central diffrence method for 4th order approximation there is error is less one compare with the other two method of skewed left & right side method. the central diffrence error gives less error because it take the data from symeterical node point from left & side right side.
why skewed system is useful?
in case of skewed for right side & left side system. it take inforamtion from current point & neighbouring point on left & right side so we can find out any order approximation. but in case of Cental diffrence system if we want to find out higher order approximation at boundry node, in that case there is no node is available on any side of node. , so it is not useful for Cental diffrence system.