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

Aim : To Derive the following 4th order approximations of the second-order derivative.

1. Central difference

2. Skewed right-sided difference

3. Skewed left-sided difference

Objective :

The given equation analytical function exp(x)*cos(x)

solving:

Skewed right sided diffence :

The order of accuracy P = Number of nodes in the stencil N - order of derivative Q for one-sided/skewed schemes

where N = Q + P

N = 4+2

N = 6 no of node were we calculated

the equation

`Deltax^2 frac(d^2u)(dx^2) = af(i)+bf(i+1)+cf(i+2)+df(i+3)+ef(i+4)+gf(i+5)`

Now we are using Tylor's table to find co-efficients

Tylor table for skewed right difference :

	`f(i)`	`Deltaxf^'(i)`	`Deltax^2f^''(i)`	`Deltax^3f^'''(i)`	`Deltax^4f''''(i)`	`Deltax^5f'''''(i)`
`af(i)`	`a`	`0`	`0`	`0`	`0`	`0`
`bf(i+1)`	`b`	`b`	`b/2`	`b/6`	`b/24`	`b/120`
`cf(i+2)`	`c`	`2c`	`c4/2`	`c8/6`	`c16/24`	`c32/120`
`df(i+3)`	`d`	`3d`	`d9/2`	`d27/6`	`d81/24`	`d243/120`
`ef(i+4)`	`e`	`4e`	`e16/2`	`e64/6`	`e256/24`	`e1024/120`
`gf(i+5)`	`g`	`5g`	`g25/2`	`g125/6`	`g625/24`	`g3125/120`

`af(i)`= `af(i)`

`bf(i+1)`= `b[f(i)+Deltaxf^'(i)+Deltafrac(x^2)2f^''(i)+Deltafrac(x^3)3f^'''(i)+Deltafrac(x^4)4f^''''(i)+Deltafrac(x^5)5f^'''''(i)]`

`cf(i+2)`= `c[f(i)+2Deltaxf^'(i)+2Deltafrac(x^2)2f^''(i)+2Deltafrac(x^3)3f^'''(i)+2Deltafrac(x^4)4f^''''(i)+2Deltafrac(x^5)5f^'''''(i)]`

`df(i+3)`= `d[f(i)+3Deltaxf^'(i)+3Deltafrac(x^2)2f^''(i)+3Deltafrac(x^3)3f^'''(i)+3Deltafrac(x^4)4f^''''(i)+3Deltafrac(x^5)5f^'''''(i)]`

`ef(i+4)`=`e[f(i)+4Deltaxf^'(i)+4Deltafrac(x^2)2f^''(i)+4Deltafrac(x^3)3f^'''(i)+4Deltafrac(x^4)4f^''''(i)+4Deltafrac(x^5)5f^'''''(i)]`

`gf(i+5)=g[f(i)+5Deltaxf^'(i)+5Deltafrac(x^2)2f^''(i)+5Deltafrac(x^3)3f^'''(i)+5Deltafrac(x^4)4f^''''(i)+5Deltafrac(x^5)5f^'''''(i)]`

`a+b+c+d+e+g = 0`

`b+2c+3d+4e+5g = 0`

`b/2+c4/2+d9/2+e16/2+g25/2 = 1`

`b/6+c8/6+d27/6+e64/6+g125/6 =0`

`b/24+c16/24+d81/24+e256/24+g625/24 = 0`

`b/120+c32/120+d243/120+e1024/120+g3125/120 = 0`

Using matrix method and computing required co-efficients

`frac(d^2u)(dx^2) = frac(af(i)+bf(i+1)+cf(i+2)+df(i+3)+ef(i+4)+gf(i+5))(Deltax^2)`

substitute coefficients in above equation

in form of `f(x) = e^xcosx`

Skewed left sided difference :

The order of accuracy P = Number of nodes in the stencil N - order of derivative Q for one-sided/skewed schemes

where N = Q + P

N = 4+2

N = 6 no of nodes were we calculated

the equation

`Deltax^2 frac(d^2u)(dx^2) = af(i)+bf(i-1)+cf(i-2)+df(i-3)+ef(i-4)+gf(i-5)`

Now we are using Tylor's table to find co-efficients

Tylor table for skewed left sided difference :

	`f(i)`	`Deltaxf^'(i)`	`Deltax^2f^''(i)`	`Deltax^3f^'''(i)`	`Deltax^4f^''''(i)`	`Deltax^5f^'''''(i)`
`af(i)`	`a`	`0`	`0`	`0`	`0`	`0`
`bf(i-1)`	`b`	`-b`	`b/2`	`-b/6`	`b/24`	`-b/120`
`cf(i-2)`	`c`	`-2c`	`c4/2`	`-c8/6`	`c16/24`	`-c32/120`
`df(i-3)`	`d`	`-3d`	`d9/2`	`-d27/6`	`d81/24`	`-d243/120`
`ef(i-4)`	`e`	`-4e`	`e16/2`	`-e64/6`	`e256/24`	`-e1024/120`
`gf(i-5)`	`g`	`-5g`	`g25/2`	`g125/6`	`g625/24`	`-g3125/120`

`af(i)= af(i)`

`bf(i-1)=b[f(i)-Deltaxf^'(i)+Deltafrac(x^2)2f^''(i)-Deltafrac(x^3)3f^'''(i)+Deltafrac(x^4)4f^''''(i)-Deltafrac(x^5)5f^'''''(i)]`

`cf(i-2)=c[f(i)-2Deltaxf^'(i)+2Deltafrac(x^2)2f^''(i)-2Deltafrac(x^3)3f^'''(i)+2Deltafrac(x^4)4f^''''(i)-2Deltafrac(x^5)5f^'''''(i)]`

`df(i-3)=d[f(i)-3Deltaxf^'(i)+3Deltafrac(x^2)2f^''(i)-3Deltafrac(x^3)3f^'''(i)+3Deltafrac(x^4)4f^''''(i)-3Deltafrac(x^5)5f^'''''(i)]`

`ef(i-4)=e[f(i)-4Deltaxf^'(i)+4Deltafrac(x^2)2f^''(i)-4Deltafrac(x^3)3f^'''(i)+4Deltafrac(x^4)4f^''''(i)-4Deltafrac(x^5)5f^'''''(i)]`

`gf(i-5)=g[f(i)-5Deltaxf^'(i)+5Deltafrac(x^2)2f^''(i)-5Deltafrac(x^3)3f^'''(i)+5Deltafrac(x^4)4f^''''(i)-5Deltafrac(x^5)5f^'''''(i)]`

`a+b+c+d+e+g = 0`

`b-2c-3d-4e-5g = 0`

`b/2+c4/2+d9/2+e16/2+g25/2 = 1`

`-b/6-c8/6-d27/6-e64/6-g125/6 =0`

`b/24+c16/24+d81/24+e256/24+g625/24 = 0`

`-b/120-c32/120-d243/120-e1024/120-g3125/120 = 0``

Using matrix method and computing required co-efficients

`frac(d^2u)(dx^2) = frac(af(i)+bf(i-1)+cf(i-2)+df(i-3)+ef(i-4)+gf(i-5))(Deltax^2)`

substitute coefficients in above equation

in form of `f(x) = e^xcosx`

Central difference :

The order of accuracy P = Number of nodes in the stencil N - order of derivative Q + 1 for symmetric/central schemes

N = P+Q-1

N = 4+2-1

N = 5

no of nodes were we calculated

the equation `Deltax^2 frac(d^2u)(dx^2) = af(i-2)+bf(i-1)+cf(i)+df(i+1)+ef(i+2)`

Now we are using Tylor's table to find co-efficients`Deltax^4f''''(i)`

Tylor table for Central difference :

	`f(i)`	`Deltaxf'(i)`	`Deltax^2f''(i)`	`Deltax^3f'''(i)`	`Deltax^4f''''(i)`
`af(i-2)`	`a`	`-2a`	`a4/2`	`-a8/6`	`a16/24`
`bf(i-1)`	`b`	`-b`	`b1/2`	`-b/6`	`b/24`
`cf(i)`	`c`	`0`	`0`	`0`	`0`
`df(i+1)`	`d`	`d`	`d1/2`	`d/6`	`d/24`
`ef(i+2)`	`e`	`2e`	`e4/2`	`e8/6`	`e16/24`

`af(i-2)=a[f(i)-2Deltaxf^'(i)+2Deltafrac(x^2)2f^''(i)-2Deltafrac(x^3)3f^'''(i)+2Deltafrac(x^4)4f^''''(i)]`

`bf(i-1)=b[f(i)-Deltaxf^'(i)+Deltafrac(x^2)2f^''(i)-Deltafrac(x^3)3f^'''(i)+Deltafrac(x^4)4f^''''(i)]`

`cf(i) = cf(i)`

`df(i+1)=d[f(i)+Deltaxf^'(i)+Deltafrac(x^2)2f^''(i)+Deltafrac(x^3)3f^'''(i)+Deltafrac(x^4)4f^''''(i)]`

`ef(i+2)=e[f(i)+2Deltaxf^'(i)+2Deltafrac(x^2)2f^''(i)+2Deltafrac(x^3)3f^'''(i)+2Deltafrac(x^4)4f^''''(i)]`

`a+b+c+d+e=0`

`-2a-b+0+d+2e =0`

`4/2+b/2+0+d/2+4e/2=1`

`-a8/6-b/6+0+d/6+e8/6=0`

`16/24+b/24+0+d/24+16e/24=0`

Using matrix method and computing required co-efficients

`frac(d^2u)(dx^2) = frac(af(i-2)+bf(i-1)+cf(i)+df(i+1)+ef(i+2))(Deltax^2)`

substitute coefficients in above equation

in form of `f(x) = e^xcosx`

Now we are coding a program to know errors,

Code :

Main code

close
clear
clc

% inputs
x = pi/3;
dx = linspace(pi/4,pi/4000,20);
analytical_derivative = -2*exp(x)*sin(x);

for i=1:length(dx)
    
    central_error(i)=central_differ(x,dx(i));
    skewed_right_error(i)=forward_differ(x,dx(i));
    skewed_left_error(i)=backward_differ(x,dx(i));
    
end
% plotting to know error 
loglog(dx,central_error,'color','g')
hold on
loglog(dx,skewed_right_error,'color','b')
loglog(dx,skewed_left_error,'color','r')
legend('central_error','skewed_right_error','skewed_left_error')
xlabel('dx')
ylabel('absolute error')
title('consistency of different scheme')

Function code for skewed right sided difference

function skewed_right_error = forward_differ(x,dx)
         %deriving eqution
        analytical_derivative = -2*exp(x)*sin(x);
        % computing co-efficients with numerical equation
        Numerical_derivative = ((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;
        skewed_right_error = abs(Numerical_derivative-analytical_derivative);
end

Function code for skewed left sided difference

function skewed_left_error = backward_differ(x,dx)
          % deriving equation
        analytical_derivative = -2*exp(x)*sin(x);
        % computing co-efficients with numerical equation
        Numerical_derivative = ((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);
        skewed_left_error = abs(Numerical_derivative-analytical_derivative);
end

Function code for central diffence

function central_error = central_differ(x,dx)
         %deriving eqution
        analytical_derivative = -2*exp(x)*sin(x);
        % computing co-efficients with numerical equation
        Numerical_derivative = ((-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;
        central_error = abs(Numerical_derivative-analytical_derivative);
end

Result :

Central diffence<Right skewed difference~Left skewed difference

Why Skewed scheme is useful ?

Ans - The skewed scheme is useful because it is necessary to compute the grids at the corners also which central difference scheme is unable to do.

Conclusion :

Therefore finded the coefficients and plotted the graph to know erros ,here central difference is less erros compare to other two difference.