4th order approximations of the second order derivative

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clear all;
close all;
clc;
% analytical function =(e.^x)*cos(x)

% analytical_derivative
%f\'\'(x)= (-)*2*(e.^x)*sin(x)
x=linspace(pi/3,pi/2,10);
for i=1:10
analytical_derivative(i)= (-2)*(exp(x(i)))*(sin(x(i)));
end

dx=pi/40;

e=exp(1);
%fourth order approximation
for i=1:10
centeral_differincing =((0.0833*(exp(x(i)-(2*dx))*cos(x(i)-(2*dx))))+(1.333*(exp(x(i)-dx)*cos(x(i)-dx)))+(2.5*(exp(x(i))*cos(x(i))))+(1.333*(exp(x(i)+dx)*cos(x(i)+dx)))+(0.0833*(exp(x(i)+(2*dx))*cos(x(i)+(2*dx)))))/(dx.^2);

%fourth order approximation
skewed_right_differincing =((0.0833*(e.^x(i))*cos(x(i)))+(1.333*(e.^(x(i)+dx)*cos(x(i)+dx)))+(2.5*(e.^(x(i)+(2*dx))*cos(x(i)+(2*dx))))+(1.333*(e.^(x(i)+(3*dx))*cos(x(i))+(3*dx))))+(0.0833*(e.^(x(i)+(4*dx))*cos(x(i)+(4*dx))))/(dx^.2);
% fourth_order approximation
skewed_left_differincing= (2.9167*(e.^x(i))*cos(x(i)))+(8.667*(e.^(x(i)-dx)*cos(x(i)-dx)))+(9.5*(e.^(x(i)-(2*dx))*cos(x(i)-(2*dx))))+(4.667*(e.^(x(i)-(3*dx))*cos(x(i)-(3*dx))))+(0.9167*(e.^(x(i)+(4*dx))*cos(x(i)+(4*dx))))/(dx.^2);
end
error_first_order = abs(skewed_right_differincing-analytical_derivative);

second_order = abs(centeral_differincing-analytical_derivative);

error_fourth_order = abs(skewed_left_differincing-analytical_derivative);

axis([-2 2 -8 -2]);
plot(error_first_order,analytical_derivative );
hold on;
plot (second_order,analytical_derivative);
hold on;
plot (error_fourth_order,analytical_derivative);
hold on;
grid on
xlabel(\'numerical\');
ylabel(\'analytical solution\');
legend(\'error_first_order\',\'second_order\',\'error_fourth_order\');

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The above graph represents the plot that compares the absolute error between the above-mentioned schemes. The graph shows that the higher the values of the order lower the error values. The forward difference values give single step-predictor values of the finite method. In the forward difference value, a large number of iteration gives the high accuracy with the reference. similarly, the backward differencing value gives the accuracy values with the reference value.The values for FDS and BDS increases with the increase in the values of dx of the solution.

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A common idea that arises from the CFD  asserts that the precision order of numerical schemes degrades as soon as control volumes. A skewed scheme is useful because high skewness of mesh elements involves consistency problems in the discretization of second derivative or diffusive flux.The skew schemed can give the accurate value with the help of the reference grid point values of the approximations whereas the CD can\'t give the accurate value as the problem.

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