Discretization for range of dx - Comparing the first, second and fourth order approximations of the first order derivative against the analytical or exact derivative.

Objective:-

1. For a range of dx, plot the first order, second and fourth order error.

2. Make sure that you plot dx vs error in a way that makes sense.

3. Explain the results in the space provided. What can you say about the slope of the curves in this plot?

Formula for the fourth order approximation of the first derivative.

Software used:- MATLAB

Solution:-

Matlab code for relation between range of dx and first order, second and fourth order error.

clear all
close all
clc

% analytical function f(x)  = sin(x)/x^3

x = pi/3;
dx = linspace(0,1,20000); % range of dx
f = sin(x)/x^3;

% for loop for calculating the values of error of the range of error
for i = 1:length(dx)
    
    first_order_error(i) = first_order(x,dx(i)); % calling the function for calculating 1st order error
    second_order_error(i) = second_order(x,dx(i)); % calling the function for calculating 2nd order error
    fourth_order_error(i) = fourth_order(x,dx(i)); % calling the function for calculating 4th order error
     
end
% plotting all the figures
figure(1);
plot(dx,first_order_error,'color','r','linewidth',2);
xlabel('dx');
ylabel('error');
legend('1st order error','location','northwest');
title('Error between analytical derivative and 1st order approximation')
figure(2)
plot(dx,second_order_error,'color','b','linewidth',2);
xlabel('dx');
ylabel('error');
legend('2nd order error','location','northwest');
title('Error between analytical derivative and 2nd order approximation')
figure(3);
plot(dx,fourth_order_error,'color','g','linewidth',2);
xlabel('dx');
ylabel('error');
legend('4th order error','location','northwest');
title('Error between analytical derivative and 4th order approximation')

% plotting the graph on log scale so that relation between dx and error can
% be easily visualized
figure(4)
loglog(dx,first_order_error,'color','r','linewidth',2);
hold on;
loglog(dx,second_order_error,'color','b','linewidth',2);
hold on;
loglog(dx,fourth_order_error,'color','g','linewidth',2);
xlabel('dx');
ylabel('error');
legend('1st order error','2nd order error','4th order error','location','northwest');
title('Error between analytical derivative and 1st order, 2nd order and 4th order approximation')

Matlab code for Function to calculate the 1st order approximation error

function out = first_order(x,dx)

analytical_sol = cos(x)/x^3 - (3*sin(x))/x^4;

numerical_sol = ((sin(x+dx)/(x+dx)^3) -( sin(x)/(x)^3))/(dx);

out = abs(analytical_sol - numerical_sol); % out is giving the absolute error value

end

Matlab code for Function to calculate the 2nd order approximation error

function out = second_order(x,dx)

analytical_sol = cos(x)/x^3 - (3*sin(x))/x^4;

numerical_sol = ((sin(x+dx)/(x+dx)^3) - (sin(x-dx)/(x-dx)^3))/(2*dx);

out = abs(analytical_sol - numerical_sol); % out is giving the absolute error value

end

Matlab code for Function to calculate the 4th order approximation error

function out = fourth_order(x,dx)

analytical_sol = cos(x)/x^3 - (3*sin(x))/x^4;

numerical_sol = ((sin(x-2*dx)/(x-2*dx)^3)-(8*(sin(x-dx)/(x-dx)^3))+(8*(sin(x+dx)/(x+dx)^3))-(sin(x+2*dx)/(x+2*dx)^3))/(12*dx);

out = abs(analytical_sol - numerical_sol); % out is giving the absolute error value
end

Report:

• First we defined the value of x and defined the range of dx using linspace command.
• We define the for loop to calculate all the values of error over the range of dx.
• Inside the for loop we called the three functions, these three functions were defined in advance, to calculate the 1st order, 2nd order and 4th order approximation error.
• using plot command to plot the relation between range of dx and 1st order, 2nd order and 4th order approximation error respectively, in three different figures.
• Now plotting the all three graphs on one signle log scale graph to better visualize the relation between range of dx and error.

• The above graph is for 1st order approximation, here we can see that even for very small value of dx the error is large, and it increases further with the increase dx.

• The above graph is for 2nd order approximation, here we can see that it is much less than 1st order approximation. Only for large value of dx the error is significant, and for small value of dx the error is very small.

• The above graph is for error value between 4th order approximation and analytical derivative. here we can see that the error is almost negligible. 4ht order approximation is much more accurate that the 1st and 2nd order approximation. There is a sudden spike in error value at dx=0.52, but the overall, we can say that the error is negligible as the value for dx increases. 
• To better visualize the above three graphs, we can use the log scale graph and compare the three graph in one single graph. Below is the log scale graph.

• The above graph is plotted on log scale to better visualize the relation between dx and the error for 1st order, 2nd order and 4ht order approximation error values respectively.
• From the above graph we can clearly see that the value for error in 1st order, 2nd order and 4ht order approximation is decreasing as the value of dx is decreasing. The highest error value is for 1st order approximation, 4th order error value being the lowest.