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

ANSWER FOR THE FIRST 3 QUESTIONS ARE WRITTEN AND UPLOADED IN THE PDF ATTACHED TO THIS.

MAIN PROGRAM

clear all
close all
clc

x= pi/3;
dx = linspace(pi/4,pi/4000,40);

for i = 1:length(dx)
central_difference_error(i) = central_differencing(x,dx(i));
skewed_right_side_error(i) = skewed_right_sided(x,dx(i));
skewed_left_side_error(i) = skewed_left_sided(x,dx(i));
end

%plotting

figure(1)
plot(dx,central_difference_error,'linewidth',2)
hold on
plot(dx,skewed_right_side_error,'r','linewidth',2)
plot(dx,skewed_left_side_error,'g','linewidth',2)
xlabel('dx')
ylabel('error')
legend('central difference error','skewed right sided error','skewed left sided error')

figure(2)
loglog(dx,central_difference_error,'linewidth',2)
hold on
loglog(dx,skewed_right_side_error,'r','linewidth',2)
loglog(dx,skewed_left_side_error,'g','linewidth',2)
xlabel('dx')
ylabel('error')
legend('central difference_error','skewed_right_side_error','skewed_left_side_error')

CENTRAL DIFFERENCING

function error_central_differencing = central_differencing(x,dx)
%analytical function f(x) = exp(x)*cos(x);
%f''(x) = -2*exp(x)*sin(x);

analytical_derivative = -2*exp(x)*sin(x);
%values of coefficients calculated from our taylor table;

a = -0.083333;
b=1.33333;
c=-2.50000;
d=1.33333;
e=-0.083333;

%numerical derivative;
%central differencing
%((a*f(x-2*dx))+(b*f(x-dx))+(c*f(x))+(d*f(x+dx))+(e*f(x+2*dx)))/dx^2;
%central difference fourth order approximaton;

central_differencing_fourth_order_approx = ((a*exp(x-2*dx)*cos(x-2*dx))+(b*exp(x-dx)*cos(x-dx))+(c*exp(x)*cos(x))+(d*exp(x+dx)*cos(x+dx))+(e*exp(x+2*dx)*cos(x+2*dx)))/dx^2;

error_central_differencing = abs(central_differencing_fourth_order_approx-analytical_derivative);
end

SKEWED RIGHT SIDED

function error_skewed_right_sided = skewed_right_sided(x,dx)

%analytical_function f(x) = exp(x)*cos(x);
%f''(x) = -2*exp(x)*sin(x);
analytical_derivative = -2*exp(x)*sin(x);

%values of coefficients calculated from our taylor table;

a = 3.750000;
b = -12.8333;
c = 17.83333;
d = -13.0000;
e = 5.083333;
g = -0.833333;

%numerical derivative;
%skewed right sided differencing =
%((a*f(x))+(b*f(x+dx))+(c*f(x+2*dx))+(d*f(x+3*dx))+(e*f(x+4*dx))+(g*f(x+5*dx)))/dx^2;

skewed_right_sided_fourth_order_approx = ((a*exp(x)*cos(x))+(b*exp(x+dx)*cos(x+dx))+(c*exp(x+2*dx)*cos(x+2*dx))+(d*exp(x+3*dx)*cos(x+3*dx))+(e*exp(x+4*dx)*cos(x+4*dx))+(g*exp(x+5*dx)))/dx^2;
error_skewed_right_sided = abs(skewed_right_sided_fourth_order_approx - analytical_derivative);
end

SKEWED LEFT SIDED

function error_skewed_left_sided = skewed_left_sided(x,dx)

%analytical_function f(x) = exp(x)*cos(x);
%f''(x) = -2*exp(x)*sin(x);
analytical_derivative = -2*exp(x)*sin(x);

%values of coefficients calculated from our taylor table;

a = -0.83333;
b = 5.08333;
c = -13.0000;
d = 17.83333;
e = -12.83333;
g = 3.750000;

%numerical derivative;
%skewed left sided differencing =((a*f(x-5*dx))+(b*f(x-4*dx))+(c*f(x-3*dx))+(d*f(x-2*dx))+(e*f(x-dx))+(g*f(x)))/dx^2;

skewed_left_sided_fourth_order_approx = ((a*exp(x-5*dx)*cos(x-5*dx))+(b*exp(x-4*dx)*cos(x-4*dx))+(c*exp(x-3*dx)*cos(x-3*dx))+(d*exp(x-2*dx)*cos(x-2*dx))+(e*exp(x-dx)*cos(x-dx))+(g*exp(x)))/dx^2;
error_skewed_left_sided = abs(skewed_left_sided_fourth_order_approx - analytical_derivative);
end

The central Differencing scheme can give more perfect solutions with lesser errors. But it is not always recommended for evaluating solutions, for example, if we want to compute a solution for the node at the corner or boundary, then we cannot use a central difference scheme, because, for a central difference scheme, data is being calculated from both the sides. So Skewed difference method can be very useful in this kind of situations