Taylor table method and Matlab code

Derivation of 4^th order numerical approximation for 2^nd order derivative using Taylor table method and comparison of the differencing schemes

1. AIM

To derive the 4^th order numerical approximation for 2^nd order derivative of a given function using central, skewed right and skewed left differencing schemes and to write a program in order to compare the analytical derivative with three numerical approximation derived.

2. EQUATIONS/ FORMULAE USED

 Given function, f(x) = e^x/cos(x)

 Analytical 2^nd order derivative, f'’’(x) = -2*(e^x)*sin(x), x=pi/3 is the point of consideration where the function is being solved and dx ranges from pi/2000 to pi/20 with 20 interval points.

 Taylor Table method

It’s a method in which the coefficients of the hypothesis function is calculated using Taylor series and expressed in tabular form. Each of these coefficients is then used for deriving the equation for numerical approximation of the derivative. Three types of differencing scheme are used such as central, skewed right and skewed left differencing scheme. Also the hypothesis function changes as per the differencing scheme used.

 According to the aim, the Taylor Table method is used here to derive the 4^th order numerical approximation for 2^nd order derivative of a given function using central, skewed right and skewed left differencing schemes.

 Skewed right differencing scheme

It is basically forward differencing scheme in which the derivative is calculated using the information from the point on right side of the point of consideration.

Skewed left differencing scheme

It is basically backward differencing scheme in which the derivative is calculated using the information from the point on left side of the point of consideration.

 Central differencing scheme

It’s a scheme in which the derivative is calculated using the information from the very next point on both sides of the point of consideration. It is always used for approximation of even number order.

 Number of points (P)

Number of points depends on the order of numerical approximation (n) and order of derivative (m).

For,

      Skewed differencing scheme, P = m + n

      Central differencing scheme,  P = m + n - 1

 Error is equal to the difference in numerical and analytical derivative.  

 3. OBJECTIVES & PROCEDURE

 The main objective of the challenge is to understand the Taylor table method and to derive the 4^th order numerical approximation for 2^nd order derivative of a given function using central, skewed right and skewed left differencing schemes.

• Also to compare against the analytical derivative of a function.
• In the process, firstly the hypothesis function for each scheme are formed and coefficients are calculated using Taylor series.
• Second order analytical derivative of the given function is calculated at a point, x=pi/3.
• Central, Skewed right and Skewed left differencing schemes are applied in separate function programs to calculate the difference between analytical derivative and approximation derivative respectively.
• Then main program is used to call each function programs in order to calculate the derivative over a range of dx.
• The values obtained from three approximations are compared with the analytical derivative to find the error and graph is plotted.

 4. PROGRAM

Programming language used – Octave 5.1.1

Program

-------------------------------------------------------------------------------------
MAIN PROGRAM
-------------------------------------------------------------------------------------
x = pi/3;
dx = linspace(pi/2000,pi/20,20);

for i = 1:length(dx)
  
  Central_differencing_scheme(i)  = Taylor_Central(x,dx(i));
  Skewed_Right_sided_differencing_scheme(i) = Taylor_Right(x,dx(i));
  Skewed_Left_sided_differencing_scheme(i) = Taylor_Left(x,dx(i));
  
end
figure(1)
plot(dx,Central_differencing_scheme,'*')
hold on
plot(dx,Skewed_Right_sided_differencing_scheme,'*')
hold on
plot(dx,Skewed_Left_sided_differencing_scheme,'*')

title('Comparison of Differencing Schemes - Central differencing Vs Skewed Left Vs Skewed Right')
xlabel('dx','Fontsize',18)
ylabel('Absolute error','Fontsize',18)

legend('Central differencing scheme','Skewed Right sided differencing scheme','Skewed Left sided differencing scheme')
-------------------------------------------------------------------------------------
-------------------------------------------------------------------------------------
FUNCTION PROGRAM
-------------------------------------------------------------------------------------
TAYLOR_CENTRAL

function out = Taylor_Central(x,dx)

Matrix_A = [1 1 1 1 1; -2 -1 0 1 2; 2 1/2 0 1/2 2; -8/6 -1/6 0 1/6 8/6; 16/24 1/24 0 1/24 16/24];

Matrix_B = [0 ; 0 ; 1 ; 0 ; 0];

X = Matrix_AMatrix_B;

% Given function f(x)= (e^x)*cos(x);

% Analytical second order derivative for the given function f''(x) = -2*(e^x)*sin(x);

% Computing Analytical second order derivative at x = pi/3;
Analytical_derivative = -2*(e^x)*sin(x);

% Central differencing scheme = (-0.083333*f(x-2*dx)+ 1.333333*f(x-dx)-2.500000*f(x)+1.333333*f(x+dx)-0.083333*f(x+2*dx))/(dx^2);
Central_differencing_scheme = ((X(1)*e^(x-2*dx)*cos(x-2*dx))+(X(2)*e^(x-dx)*cos(x-dx))+(X(3)*e^x*cos(x))+(X(4)*e^(x+dx)*cos(x+dx))+(X(5)*e^(x+2*dx)*cos(x+2*dx)))/(dx^2);
% Second Order Central differencing scheme Error
out = abs (Analytical_derivative - Central_differencing_scheme);

end

---------------------------------------------------------------------------------------

TAYLOR_SKEWED_RIGHT

function out = Taylor_Right(x,dx)

Matrix_A = [1 1 1 1 1 1; 0 1 2 3 4 5; 0 1/2 2 9/2 8 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];

Matrix_B = [0 ; 0 ; 1 ; 0 ; 0; 0];

X = Matrix_AMatrix_B;

% Given function f(x)= (e^x)*cos(x);

% Analytical second order derivative for the given function f''(x) = -2*(e^x)*sin(x);

% Computing Analytical second order derivative at x = pi/3;
Analytical_derivative = -2*(e^x)*sin(x);

% Skewed Right sided differencing scheme = (3.75000*f(x)-12.83333*f(x+dx)+17.83333*f(x+2*dx)-13.00000*f(x+3*dx)+5.08333*f(x+4*dx)-0.83333*f(x+5*dx))/(dx^2);
Skewed_Right_sided_differencing_scheme=((X(1)*(e^x)*cos(x))+(X(2)*e^(x+dx)*cos(x+dx))+(X(3)*e^(x+2*dx)*cos(x+2*dx))+(X(4)*e^(x+3*dx)*cos(x+3*dx))+(X(5)*e^(x+4*dx)*cos(x+4*dx))+(X(6)*e^(x+5*dx)*cos(x+5*dx)))/(dx^2);
% Second Order Skewed right sided differencing scheme Error
out = abs (Analytical_derivative - Skewed_Right_sided_differencing_scheme);

end
-------------------------------------------------------------------------------------
TAYLOR_SKEWED_LEFT

function out = Taylor_Left(x,dx)

Matrix_A = [1 1 1 1 1 1; -5 -4 -3 -2 -1 0; 25/2 8 9/2 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];

Matrix_B = [0 ; 0 ; 1 ; 0 ; 0; 0];

X = Matrix_AMatrix_B;

% Given function f(x)= (e^x)*cos(x);

% Analytical second order derivative for the given function f''(x) = -2*(e^x)*sin(x);

% Computing Analytical second order derivative at x = pi/3;
Analytical_derivative = -2*(e^x)*sin(x);

% Skewed Left sided differencing scheme = (-0.83333*f(x-5*dx)+5.08333*f(x-4*dx)-13.00000*f(x-3*dx)+17.83333*f(x-2*dx)-12.83333*f(x-dx)+3.75000*f(x))/(dx^2);;
Skewed_Left_sided_differencing_scheme = ((X(1)*e^(x-5*dx)*cos(x-5*dx))+(X(2)*e^(x-4*dx)*cos(x-4*dx))+(X(3)*e^(x-3*dx)*cos(x-3*dx))+(X(4)*e^(x-2*dx)*cos(x-2*dx))+(X(5)*e^(x-dx)*cos(x-dx))+(X(6)*(e^x)*cos(x)))/(dx^2);
% Second Order Skewed Left sided differencing scheme Error
out = abs (Analytical_derivative - Skewed_Left_sided_differencing_scheme);

end
-------------------------------------------------------------------------------------

 5. RESULTS

Calculation

1. Central Approximation

Graph

6. CONCLUSION

From the above graph, it can be clearly seen that the error obtained from central differencing scheme is minimal compared to Skewed Right and Left differencing scheme.

By using the Taylor table method we can calculate numerical approximation for any given order of derivative and differencing schemes.

But central differencing scheme cannot be used everywhere for better results, as the scheme has the requirement of information from both sides of the point of consideration.

So only skewed differencing schemes can be used if the solution is needed at the end nodes of the domain. It will be useful if the information needs to be taken from multiple points but from only one side of the point of consideration.

Finally the choice of differencing scheme depends on the direction of flow of information.