AIM:
To derive the fourth order approximations of a second order derivative using central differencing scheme, skewed right sided difference and skewed left sided difference with the help of taylor table method and to compare the analytical derivative with obtained values for the given function.
SCHEME DERIVATIONS:
CENTRAL DIFFERENCING SCHEME:
NO OF NODES:
The number of nodes = p+q-1
Where, p is order of derivative.
q is order of approximations.
The no of nodes = 2+4-1 =>5.
DERIVATION:
To derive a fourth order approximation for the second order derivative,we start with these approximation,
∂2u∂x2=af(i−2)+bf(i−1)+cf(i)+df(i+1)+ef(i+2)dx2
..........(1)
taylor series derivation for the above terms,
af(i−2)=af(i)−2af'(i)dx1!+4af''(i)dx22!+8af'''(i)dx33!+16af''''(i)dx44!
the linear equation obtained from the above table are solved using matrix inversion method,
A.X = B
a=-0.0833, b=1.3333, c=-2.5000, d=1.3333 e=-0.0833
substituting these coefficient in equation(1) we get,
∂2u∂x2=(−0.0833)f(i−2)+(1.3333)f(i−1)+(−2.5000)f(i)+(1.3333)f(i+1)+(−0.0833)f(i+2)dx2
SKEWED RIGHT HAND DIFFERENCE SCHEME:
NO OF NODES:
The number of nodes = p+q
Where, p is order of derivative.
q is order of approximations.
The no of nodes = 2+4 =>6.
DERIVATION:
To derive a fourth order approximation for the second order derivative,we start with these approximation,
∂2u∂x2=af(i)+bf(i+1)+cf(i+2)+df(i+3)+ef(i+4)+gf(i+5)dx2
..........(2)
the linear equation obtained
MAT CODE
clear all
close all
clc
%value of x
x= pi/3;
%range of dx
dx= linspace(pi/4,pi/4000,20);
%for loop
for i=1:length(dx)
central_differencing(i) = fourth_order_central_differencing(x,dx(i));
right_sided_skew(i) = fourth_order_right_sided_skew(x,dx(i));
left_sided_skew(i) = fourth_order_left_sided_skew(x,dx(i));
end
%plotting
figure(1)
loglog(dx,central_differencing,'r')
hold on
loglog(dx,right_sided_skew,'g')
hold on
loglog(dx,left_sided_skew,'b')
xlabel('range of dx')
ylabel('truncation error')
title('range of dx vs error')
legend('central differencing','skewed right sided difference','skewed left sided difference','location','northwest')
CODE FOR CENTRAL DIFFRENCING
function Error_central_differencing = fourth_order_central_differencing(x,dx)
% analytic function = exp(x)*cos(x)
% analytic derivative
%f''(x) = -2*exp(x)*sin(x)
analytical_derivative = -2*exp(x)*sin(x);
%numerical derivative
%central differencing
central_differencing = (((-0.0833)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))+((1.3333)*((exp(x-dx))*(cos(x-dx))))-((2.5000)*((exp(x))*(cos(x))))+((1.3333)*((exp(x+dx))*(cos(x+dx))))-((0.0833)*((exp(x+(2*dx)))*(cos(x+(2*dx))))))/(dx^2);
%error
Error_central_differencing = abs(central_differencing- analytical_derivative);
end
FUNCTION CODE FOR SKEWED RIGHT SIDED DIFFERENCE:
function Error_right_sided_skew = fourth_order_right_sided_skew(x,dx)
% analytic function = exp(x)*cos(x)
% analytic derivative
%f''(x) = -2*exp(x)*sin(x)
analytical_derivative = -2*exp(x)*sin(x);
%numerical derivative
%central differencing
right_sided_skew = (((3.7500)*((exp(x))*(cos(x))))-((12.8333)*((exp(x+dx))*(cos(x+dx))))+((17.8333)*((exp(x+(2*dx)))*(cos(x+(2*dx)))))-((13.0000)*((exp(x+(3*dx)))*(cos(x+(3*dx)))))+((5.0833)*((exp(x+(4*dx)))*(cos(x+(4*dx)))))-((0.8333)*((exp(x+(5*dx)))*(cos(x+(5*dx))))))/(dx^2);
%error
Error_right_sided_skew = abs(right_sided_skew- analytical_derivative);
end
FUNCTION CODE FOR SKEWED LEFT SIDED DIFFERENCE:
function Error_left_sided_skew = fourth_order_left_sided_skew(x,dx)
% analytic function = exp(x)*cos(x)
% analytic derivative
%f''(x) = -2*exp(x)*sin(x)
analytical_derivative = -2*exp(x)*sin(x);
%numerical derivative
%central differencing
left_sided_skew = (((-0.8333)*((exp(x-(5*dx)))*(cos(x-(5*dx)))))+((5.0833)*((exp(x-(4*dx)))*(cos(x-(4*dx)))))-((13.0000)*((exp(x-(3*dx)))*(cos(x-(3*dx)))))+((17.8333)*((exp(x-(2*dx)))*(cos(x-(2*dx)))))-((12.8333)*((exp(x-dx))*(cos(x-dx))))+((3.7500)*((exp(x))*(cos(x)))))/(dx^2);
%error
Error_left_sided_skew = abs(left_sided_skew- analytical_derivative);
end
OUTPUT GRAPH AND PLOT
CONCLUSION:
As seen from the plot, the central differencing scheme has very less error compared to other schemes. Therefore, central differencing is better than other schemes because it uses information from both the sides from the point of consideration to give approximative and accurate values.
As the left and right skewed difference methods uses data only from one side of point of interest it gives less approximative and accurate values about the point.
