Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method

Aim:- To derive the fourth order approximations of a second order derivative using central differencing scheme, skewed right side difference and skewed left side difference with the help of taylor table method and to compare analytical derivative obtained.

SCHEMES:

CENTRAL DIFFERENCING

NO OF NODES:

The no of nodes = p+q-1;
Where p= order derivative & q= order of approximation

So nodes = 2 +4 -1 =5

DERIVATION

the linear equation obtained from above can be used in matrix inversion method

A.X=B
So, X= inv(A)*B
We then get coefficients data

a= -.0833, b=1.3333 , c= -2.5000, d= 1.3333 & e = -0.0833

substituting coefficients in equation we get,

`(del^2u)/(delx^2)=(-0.0833) f(i-2) +(1.3333)f(i-1) +(-2.5000)f(i) +(1.3333)f(i+1) +(-0.0833)f(i+2)`

SKEWED RIGHT HAND DIFFERENCE SCHEME

NO OF NODES:

N= p+q;

where p is order derivative & q is order of approximation
n= 2+4 =6

DERIVATION

Data obtained will solved using matrix

A.X=B
X= Inv(A)*B
we get coefficients value

a= 3.7500, b= -12.8333, c= 17.8333, d= -13.0000, e = 5.0833 & g=-0.8333

substituting in equation we get,
`(del^2u)/(delx^2)=(3.7500) f(i) +(-12.8333)f(i+1) +(17.8333)f(i+2) +(-13.0000)f(i+3) +(5.0833)f(i+4)+ (-0.8333)f(i+5)`

SKEWED LEFT HAND DIFFERENCE

NO OF NODES:

n= p+q
p is order of derivative and q is order of approximations.
n= 2+4=6

DERIVATION

Equation obtained from above is solved using matrix inversion method

A.X=B
X=inv(A)B

we get coefficients and we put tem in equation.

a= -.8333, b= 5.0833, c= -13.0000, d =17.8333, e = -12.8333 & g = 3.7500

Equation:
`(del^2u)/(delx^2)=(-0.8333) f(i-5) +(5.0833)f(i-4) +(-13.0000)f(i-3) +(17.8333)f(i-2) +(-12.8333)f(i-1)+ (3.7500)f(i)`

MAIN CODE

close all
clear all
clc

% value of x & dx
x= pi/3;
dx= linspace(pi/4,pi/4000,200);

% loop for calulating errors
for i= 1:length(dx)
    
    Error_central_differencing(i)= fourth_central(x,dx(i));
    Error_Right_skewed_differencing(i)= fourth_right_skewed(x,dx(i));
    Error_Left_skewed_differencing(i)= fourth_left_skewed(x,dx(i));
    
end
%plotting Errors
 figure(1)
 loglog(dx,Error_central_differencing,'r<-','markersize',2)
 hold on
 loglog(dx,Error_Right_skewed_differencing,'b<-','markersize',2)
 hold on
 loglog(dx,Error_Left_skewed_differencing,'g->','markersize',2)
 xlabel('Range of dx')
 ylabel('Truncataion errors')
 legend('Central Differencing','Right Skewed differencing','Left Skewed differencing','location','best')
 
 
 

CENTRAL FUNCTION

function Error_central_differencing= fourth_central(x,dx)

% analytic function = exp(x)*cos(x)
analytic_derivative= -2*(exp(x))*(sin(x));
% numerical derivative
central_differencing=  (((-0.08333)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))+ ((1.33333)*((exp(x-dx))*(cos(x-dx)))) - ((2.5000)*((exp(x))*(cos(x))))+ ((1.33333)*((exp(x+dx))*(cos(x+dx))))- ((0.08333)*((exp(x+(2*dx)))*(cos(x+(2*dx))))))/(dx^2);
% Error
Error_central_differencing= abs(central_differencing - analytic_derivative);
end

RIGHT SKEWED FUNCTION

function Error_Right_skewed_differencing= fourth_right_skewed(x,dx)

% analytic function = exp(x)*cos(x)
analytic_derivative= -2*(exp(x))*(sin(x));
% numerical derivative
right_skewed=  (((3.75000)*((exp(x))*(cos(x)))) -((12.8333)*((exp(x+dx))*(cos(x+dx))))+ ((17.8333)*((exp(x+(2*dx)))*(cos(x+(2*dx))))) -((13.000)*((exp(x+(3*dx)))*(cos(x+(3*dx)))))+ ((5.08333)*((exp(x+(4*dx)))*(cos(x+(4*dx))))) -((0.8333)*((exp(x+(5*dx)))*(cos(x+(5*dx))))))/(dx^2);
% Error
Error_Right_skewed_differencing= abs(right_skewed - analytic_derivative);
end

LEFT SKEWED FUNCTION

function Error_Left_skewed_differencing= fourth_left_skewed(x,dx)

% analytic function = exp(x)*cos(x)
analytic_derivative= -2*(exp(x))*(sin(x));
% numerical derivative
left_skewed=  (((-0.8333)*((exp(x-(5*dx)))*(cos(x-(5*dx)))))+ ((5.0833)*((exp(x-(4*dx)))*(cos(x-(4*dx)))))- ((13.000)*((exp(x-(3*dx)))*(cos(x-(3*dx)))))+ ((17.8333)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))- ((12.8333)*((exp(x-dx))*(cos(x-dx))))+((3.75)*((exp(x))*(cos(x)))))/(dx^2);
% Error
Error_Left_skewed_differencing= abs(left_skewed - analytic_derivative);
end

RESULTS

CONCLUSION

As the plot shows the central scheme is more eefficient as it has less error compared to both skewed schemes as it uses information from both sides to give approximate and accurate result.

On the other hand skewed scheme is succesfull in places where there is no sufficient information on both sides eg, a case where we dont know previous points or either forward points we can use left handed skewed or right handed skewed accordingly in that case.