Week 3 - Taylor table method and Matlab code

(submitted by Kurias.M.George)

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?

Initially u we have, f(x) = exp(x)*cos(x)

we give x= pi/3;

we assign f(x)=y

we can compute the analytical derivative of y as y_1 = y_1= -exp(x)*(cos(x)-sin(x))

we find the value derivative y_1 analytically at x=pi/3

dx = linspace(pi/30,pi/300,20)

then we compute the the Left , Right and skewed elements

Then we go for the Numerical dericvatives

then we plot the the errors on the bar graph

descritization function

clear all
close all 
clc
x=pi/3;
dx = linspace(pi/30,pi/300,20);
% Function : y=exp(x)*cos(x);
Analytical_derivative_1st = -exp(x)*(cos(x)-sin(x)) ;
Analytical_derivative_2nd = -2*exp(x)*sin(x);
%Taylor table for Left skewed scheme
A =[1 1 1 1 1 1;-5 -4 -3 -2 -1 0;25/2 16/2 9/2 4/2 1/2 0;-125/6 -64/6 -27/6 -8/6 -1/6 0;625/24 256/24 81/24 16/24 1/24 0;-3125/120 -1024/120 -243/120 -32/120 -1/120 0] ;
B = [0;0;1;0;0;0];
%Taylor table for Right skewed scheme
C =[1 1 1 1 1 1;0 1 2 3 4 5;0 1/2 4/2 9/2 16/2 25/2;0 1/6 8/6 27/6 64/6 125/6;0 1/24 16/24 81/24 256/24 625/24;0 1/120 32/120 243/120 1024/120 3125/120];
D = [0;0;1;0;0;0];
%Taylor table for Central scheme
E = [1 1 1 1 1;-2 -1 0 1 2;4/2 1/2 0 1/2 4/2;-8/6 -1/6 0 1/6 8/6;16/24 1/24 0 1/24 16/24];
F = [0;0;1;0;0];

X =  Left_Skewed(A,B); % N = P(Order of derivative) + Q(Order of approx.)
Y =  Right_Skewed(C,D); % N = P(Order of derivative) + Q(Order of approx.)
Z = Central(E,F); % N = P(Order of derivative) + Q(Order of approx.) - 1

for i =1:length(dx)
    
 [LSS RSS CS] = Numerical_derivative(x,dx(i),X,Y,Z);
 
 error_left(i) = abs(LSS - Analytical_derivative_2nd);
 error_right(i) = abs(RSS - Analytical_derivative_2nd);
 error_central(i) = abs(CS - Analytical_derivative_2nd);

end


% loglog(dx,error_left(i),'r')
% hold on
% loglog(dx,error_right(i),'b')
% loglog(dx,error_central(i),'g')

plot(dx,error_central(i),'o')
hold on
plot(dx,error_right(i),'x')
plot(dx,error_left(i),'*')
xlabel('errors')
ylabel('dx')

function

function y = fofx(x)
y  = exp(x)*cos(x);
end

Left skewed

function out = Left_Skewed(g,h)
out = inv(g)*h;
end

right skewed

function out = Right_Skewed(g,h)
out = inv(g)*h;
end

central skewed

function out = Central(g,h)
out = inv(g)*h;
end

Numerical dericative

function [left right central] = Numerical_derivative(x,dx,X,Y,Z)

left = ((X(1)*fofx(x-(5*dx)))- ((X(2)*fofx(x-(4*dx)))+(X(3)*fofx(x-(3*dx)))-(X(4)*fofx(x-(2*dx)))+(X(5)*fofx(x-(1*dx)))-(X(6)*fofx(x))))/(dx^2);

right = ((Y(1)*fofx(x))+(Y(2)*fofx(x+dx))+(Y(3)*fofx(x+(2*dx)))+(Y(4)*fofx(x+(3*dx)))+(Y(5)*fofx(x+(4*dx)))+(Y(6)*fofx(x+(5*dx))))/(dx^2);

central = ((Z(1)*fofx(x-(2*dx)))+(Z(2)*fofx(x-dx))+(Z(3)*fofx(x))+(Z(4)*fofx(x+dx))+(Z(5)*fofx(x+(2*dx))))/dx^2;

end

The analytical derivative is initially found out using the mathematical equations.

The numerical dericvative is found out using the different methods, using the skewed left, right and central. the values are promising with the right patterns being shown at the results which can be observed easily.

the graph of errors is obrained as

The skewed method is useful for finding the numerical dericvatives more accrately. We can see that the central and right error are similar and also thers is some hype in the value of the error to dx graph in the left skew method which can be due to the dissimilarities and round of errors.

learned to compare analytical and numerical derivatives of functions

for the given function, the analytical derivative has been found and then comapred withnumerical derivatives. we can see that the left skewed error is showing some abnormalities. ie, showing somw higher values of errors and this can be due to the round off errors.