Discretization and derivation of fourth order approximations of second order derivative using taylor table and matlab

Aim: To derive 4th order approximations of second order derivative for central difference, right skewed and left skewed schemes and write a matlab program to compare error between them

Method

The code for comparison of error between different schemes is shown below

clc
clear all
close all
dx=linspace(1/40,1/20,25);
x=(pi/40);
analytical_function=exp(x)*cos(x);
for i=1:length(dx)
CDS_approx(i)=(-0.0834*(-2*exp(x-2*dx(i))*sin(x-2*dx(i)))+1.3337*(-2*exp(x-dx(i))*sin(x-dx(i)))-2.5005*(-2*exp(x)*sin(x))+1.3337*(-2*exp(x+dx(i))*sin(x+dx(i)))-0.0834*(-2*exp(x+2*dx(i))*sin(x+2*dx(i))))/(dx(i)^2);
second_order_derivative(i)=-2*exp(x)*sin(x);
error_CDS(i)=abs(second_order_derivative(i)-CDS_approx(i));
FDS_approx(i)=(2.9280*(-2*exp(x)*sin(x))-8.7103*(-2*exp(x+dx(i))*sin(x+dx(i)))+9.5624*((-2*exp(x+2*dx(i))*sin(x+2*dx(i))))-4.7063*(-2*exp(x+3*dx(i))*sin(x+3*dx(i)))+0.9260*(-2*exp(x+4*dx(i))*sin(x+4*dx(i))))/(dx(i)^2);
error_FDS(i)=abs(second_order_derivative(i)-FDS_approx(i));
BDS_approx(i)=(2.9336*(-2*exp(x)*sin(x))-8.7327*(-2*exp(x-dx(i))*sin(x-dx(i)))+9.5961*((-2*exp(x-2*dx(i))*sin(x-2*dx(i))))-4.7287*(-2*exp(x-3*dx(i))*sin(x-3*dx(i)))+0.9317*(-2*exp(x-4*dx(i))*sin(x-4*dx(i))))/(dx(i)^2);
error_BDS(i)=abs(second_order_derivative(i)-BDS_approx(i));
end
subplot(2,2,1)
plot(dx,error_CDS,\'k\')
xlabel(\'dx values\')
ylabel(\'Error values CDS\')
title(\'CDS Error values Vs dx\')
subplot(2,2,2)
plot(dx,error_FDS,\'r\')
xlabel(\'dx values\')
ylabel(\'Error values FDS\')
title(\'FDS Error values Vs dx\')
subplot(2,2,3)
plot(dx,error_BDS,\'g\')
xlabel(\'dx values\')
ylabel(\'Error values BDS\')
title(\'BDS Error values Vs dx\')

Derivation:

If the information used in forming the finite difference equation comes from the left of
grid point then it is called backward differencing. In such differencing no information
from the right side of the grid is used. (i.e. i,i-1). In the above fig. Dashed line represent
Backward differencing.If the information used in forming the finite difference equation come from the right sideof the grid point, Then it is called forward differencing. In such differencing no information from the left side of the grid is used. (i.e. i,i+1). In the above fig. Dotted line
represent Forward differencing.If the information is used in forming the finite difference equation come from both left and right side of the grid, that it is called Central differencing. (i.e. i-1,i,i+1). In the above fig. Continoues line represent Central differencing.

The above equation is solved using wolfram alpha to get values of coefficients a,b,c d and e which is given as follows

a=-0.0834

b=1.337

c=-2.5005

d=1.337

e=0.0834

The above equation is solved using wolfram alpha to get values of coefficients a,b,c d and e which is given as follows

a=2.9280

b=-8.7103

c=9.5624

d=-4.7063

e=0.9260

.

The above equation is solved using wolfram alpha to get values of coefficients a,b,c d and e which is given as follows

a=2.9336

b=-8.7327

c=9.5961

d=-4.7287

e=0.9317

Results

The image below shows comparison between different schemes

Since the CDS (central difference scheme) is of second order, while backward and forward differencing schemes are of first order, as exptected, the CDS is more accurate as compared to other two schemes

Need for right hand skewed and left hand skewed schemes

The disadvantage with CDS is that it requires information both both sides of node i.e one from left hand side and other from right hand side. This may not always available in all applications . In such cases, backward or forward differencing schemes can be used which take information from backward or forward nodal points respectively. Moreover, the skewd difference schemes can be used to approximate functions with steep derivatives and hence will limit the function from blowing up .For CDS scheme, there are two errors - discretization error and floating point error both of which should balance each other for CDS to be accurate . This is not the case with skewed schemes