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

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,

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{af(i-2)+bf(i-1)+cf\left(i\right)+df(i+1)+ef(i+2)}{{dx}^2}$

..........(1)

taylor series derivation for the above terms,

$af(i-2)=af\left(i\right)-\frac{2af\text{'}\left(i\right)dx}{1!}+\frac{4af\text{'}\text{'}\left(i\right){dx}^2}{2!}+\frac{8af\text{'}\text{'}\text{'}\left(i\right){dx}^3}{3!}+\frac{16af\text{'}\text{'}\text{'}\text{'}\left(i\right){dx}^4}{4!}$

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,

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{(-0.0833)f(i-2)+\left(1.3333\right)f(i-1)+(-2.5000)f\left(i\right)+\left(1.3333\right)f(i+1)+(-0.0833)f(i+2)}{{dx}^2}$

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,

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{af\left(i\right)+bf(i+1)+cf(i+2)+df(i+3)+ef(i+4)+gf(i+5)}{{dx}^2}$

$\frac{{}^{}}{}$

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.