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 sided difference and skewed left sided difference with the help of taylor table method and to compare the analytical derivative with obtained values for the given function

Object :-

Derive the following 4th order approximations of the second-order derivative.

1. Central difference

2. Skewed right-sided difference

3. Skewed left-sided difference

After deriving these schemes, prove that your skewed schemes are fourth-order accurate i.e the thumb rule given in HINT (ii).

Once you have done this, write a program in Matlab to evaluate the second-order derivative of the analytical function exp(x)*cos(x) and compare it with the 3 numerical approximations that you have derived.

Provide a plot that compares the absolute error between the above-mentioned schemes

In your own words, describe why a skewed scheme is useful? What can a skewed scheme do that a CD scheme cannot?

MATLAB CODE:

MAIN CODE:

 

MATLAB CODE:

MAIN CODE:

clear all
close all
clc

%value of x
x= pi/3;

%range of dx
dx= linspace(pi/4,pi/4000,20);

%for loop
for i=1:length(dx)
    
    central_differencing(i) = fourth_order_central_differencing(x,dx(i));
    
    right_sided_skew(i) = fourth_order_right_sided_skew(x,dx(i));
    
    left_sided_skew(i) = fourth_order_left_sided_skew(x,dx(i));
    
end

%plotting
figure(1)
loglog(dx,central_differencing,'r')
hold on
loglog(dx,right_sided_skew,'g')
hold on
loglog(dx,left_sided_skew,'b')
xlabel('range of dx')
ylabel('truncation error')
title('range of dx vs error')
legend('central differencing','skewed right sided difference','skewed left s

FUNCTION CODE FOR CENTRAL DIFFERENCING:

function Error_central_differencing = fourth_order_central_differencing(x,dx)

% analytic function = exp(x)*cos(x)

% analytic derivative
%f''(x) = -2*exp(x)*sin(x)

analytical_derivative = -2*exp(x)*sin(x);

%numerical derivative

%central differencing
central_differencing = (((-0.0833)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))+((1.3333)*((exp(x-dx))*(cos(x-dx))))-((2.5000)*((exp(x))*(cos(x))))+((1.3333)*((exp(x+dx))*(cos(x+dx))))-((0.0833)*((exp(x+(2*dx)))*(cos(x+(2*dx))))))/(dx^2);

%error
Error_central_differencing = abs(central_differencing- analytical_derivative);
end

FUNCTION CODE FOR SKEWED RIGHT SIDED DIFFERENCE:

function Error_right_sided_skew = fourth_order_right_sided_skew(x,dx)

% analytic function = exp(x)*cos(x)

% analytic derivative
%f''(x) = -2*exp(x)*sin(x)

analytical_derivative = -2*exp(x)*sin(x);

%numerical derivative

%central differencing
right_sided_skew = (((3.7500)*((exp(x))*(cos(x))))-((12.8333)*((exp(x+dx))*(cos(x+dx))))+((17.8333)*((exp(x+(2*dx)))*(cos(x+(2*dx)))))-((13.0000)*((exp(x+(3*dx)))*(cos(x+(3*dx)))))+((5.0833)*((exp(x+(4*dx)))*(cos(x+(4*dx)))))-((0.8333)*((exp(x+(5*dx)))*(cos(x+(5*dx))))))/(dx^2);

%error
Error_right_sided_skew = abs(right_sided_skew- analytical_derivative);
end

FUNCTION CODE FOR SKEWED LEFT SIDED DIFFERENCE:

function Error_left_sided_skew = fourth_order_left_sided_skew(x,dx)

% analytic function = exp(x)*cos(x)

% analytic derivative
%f''(x) = -2*exp(x)*sin(x)

analytical_derivative = -2*exp(x)*sin(x);

%numerical derivative

%central differencing
left_sided_skew = (((-0.8333)*((exp(x-(5*dx)))*(cos(x-(5*dx)))))+((5.0833)*((exp(x-(4*dx)))*(cos(x-(4*dx)))))-((13.0000)*((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))))+((3.7500)*((exp(x))*(cos(x)))))/(dx^2);

%error
Error_left_sided_skew = abs(left_sided_skew- analytical_derivative);
end

OUTPUT:

CONCLUSION:

As seen from the plot, the central differencing scheme has very less error compared to other schemes. Therefore, central differencing is better than other schemes because it uses information from both the sides from the point of consideration to give approximative and accurate values.

As the left and right skewed difference methods uses data only from one side of point of interest it gives less approximative and accurate values about the point.