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

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

Aim:

To derive the 4th order approximation for the second-order derivative of a given function using following 3 schemes.

1. central difference
2. skewed right difference
3. skewed left difference schemes

Working:

• MATLAB is used for coding to achieve the above-mentioned objective.

1. Central difference

• For central difference, 4th order approximation of second-order derivative, 
• number of nodes = 2+4-1 = 5 nodes, Therefore 2 nodes on both side of x.
• It is mathematically represented by 

• equating the first five columns we get five equations, from that we get the values of coefficients" a,b,c,d, and e'

2. Skewed right-sided difference

• For right-skewed difference, 4th order approximation of second-order derivative, 
• number of nodes = 2+4 = 6 nodes, Therefore 5 nodes on right side of x.
• It is mathematically represented by 

• number of nodes = 2+4 = 6 nodes, Therefore 5 nodes on right side of x.
• It is mathematically represented by 

•  Equating the first 6 columns we get six equations, from that we get the values of coefficients" a,b,c,d,e and g'

3. Skewed left-sided difference

• For left-skewed difference, 4th order approximation of second-order derivative, 
• number of nodes = 2+4 = 6 nodes, Therefore 5 nodes on left side of x.
• It is mathematically represented by 

• Equating the first six columns we get six equations, from that we get the values of coefficients" a,b,c,d,e and g'

Code

clear all
close all
clc 

% to calculate a value of a function at a point x  =pi/3
x= pi/3;

% coeffecient matrix derived for 4th order approximation
% a,b,c,d,e,f,g respectively
% calculated separately

% f''(x) = af(x-2dx)+bf(x-dx)+cf(x)+df(x+dx)+ef(x+2dx)/dx^2
central_scheme = [-0 + 1/(-12);1 + 1/(3);-3 + 1/(2);1 + 1/(3);-0 + 1/(-12)];

% f''(x) = af(x)+bf(x+dx)+cf(x+2dx)+df(x+3dx)+ef(x+4dx)+gf(x+5dx)/dx^2
right_skewed_scheme = [4 + 1/(-4);-13 + 1/(6);18 + 1/(-6);-13;5 + 1/(12);-1 + 1/(6)];

% f''(x) = af(x)+bf(x-dx)+cf(x-2dx)+df(x-3dx)+ef(x-4dx)+gf(x-5dx)/dx^2
left_skewed_scheme = [4 + 1/(-4);-13 + 1/(6);18 + 1/(-6);-13;5 + 1/(12);-1 + 1/(6)];

%given function =exp(x)*cos(x)
%second order derivative of this term
fun_x = exp(x)*cos(x);
second_order_derivative = -2*exp(x)*sin(x);

%discretization length dx
dx = linspace(pi/30,pi/30000,30);

for i = 1:length(dx)
%function at the node points with increment and decrement of dx
fun_xplusdx = exp(x+dx(i))*cos(x+dx(i));
fun_xplus2dx = exp(x+(2*dx(i)))*cos(x+(2*dx(i)));
fun_xplus3dx = exp(x+(3*dx(i)))*cos(x+(3*dx(i)));
fun_xplus4dx = exp(x+(4*dx(i)))*cos(x+(4*dx(i)));
fun_xplus5dx = exp(x+(5*dx(i)))*cos(x+(5*dx(i)));
fun_xminusdx = exp(x-dx(i))*cos(x-dx(i));
fun_xminus2dx = exp(x-(2*dx(i)))*cos(x-(2*dx(i)));
fun_xminus3dx = exp(x-(3*dx(i)))*cos(x-(3*dx(i)));
fun_xminus4dx = exp(x-(4*dx(i)))*cos(x-(4*dx(i)));
fun_xminus5dx = exp(x-(5*dx(i)))*cos(x-(5*dx(i)));

%cental difference scheme of 4 the order
C_1 = central_scheme(1)*fun_xminus2dx;
C_2 = central_scheme(2)*fun_xminusdx;
C_3 = central_scheme(3)*fun_x;
C_4 = central_scheme(4)*fun_xplusdx;
C_5 = central_scheme(5)*fun_xplus2dx;

central_fourth(i) = (C_1+C_2+C_3+C_4+C_5)/(dx(i)^2);
central_error(i) = abs(central_fourth(i)-second_order_derivative);

%for right skewed scheme of 4th order approximation
R_1 = right_skewed_scheme(1)*fun_x;
R_2 = right_skewed_scheme(2)*fun_xplusdx;
R_3 = right_skewed_scheme(3)*fun_xplus2dx;
R_4 = right_skewed_scheme(4)*fun_xplus3dx;
R_5 = right_skewed_scheme(5)*fun_xplus4dx;
R_6 = right_skewed_scheme(6)*fun_xplus5dx;

R_skewed_fourth(i) = (R_1+R_2+R_3+R_4+R_5+R_6)/(dx(i)^2);
Rskewed_error(i) = abs(R_skewed_fourth(i)-second_order_derivative);

%for left skewed scheme of 4th order approximation
L_1 = left_skewed_scheme(1)*fun_x;
L_2 = left_skewed_scheme(2)*fun_xminusdx;
L_3 = left_skewed_scheme(3)*fun_xminus2dx;
L_4 = left_skewed_scheme(4)*fun_xminus3dx;
L_5 = left_skewed_scheme(5)*fun_xminus4dx;
L_6 = left_skewed_scheme(6)*fun_xminus5dx;

L_skewed_fourth(i) = (L_1+L_2+L_3+L_4+L_5+L_6)/(dx(i)^2);
Lskewed_error(i) =  abs(L_skewed_fourth(i)-second_order_derivative);

end

loglog(dx,central_error,'linewidth',3,'color','g')
hold on
loglog(dx,Rskewed_error,'linewidth',3,'color','r')
hold on
loglog(dx,Lskewed_error,'linewidth',3,'color','y')
legend('central scheme error','right skewed scheme error','left skewed scheme error')
xlabel('range of dx')
ylabel('Truncation error')
title('Comparision of central, right skewed and left skewed 4th order approximation for second order derivative')

 Results

• rom the result, it can be seen that central difference scheme error values are less compared to skewed difference scheme error values.
• As the value of dx decreases, the truncation error value also decreases.
• Due to the computer storage capacity, roundoff error is observed after a certain value of dx, it is indicated by the region representing increasing error values with a decrease in dx.