## STUDY ON EFFECT OF GRID POINTS IN 1D-CONVECTION DISTRIBUTION

DESCRIPTION:

In this report, we will be studying the effects of the grid points in the first order convection equation describing one dimensional velocity distribution.

frac{delu}{delt} = c*frac{delu}{delx}

CODE:

clear all
close all
clc

%inputs
c = 1;                   %convective coefficient
dt = 0.01;
l = 1;                   %length of domain
n = 160;                 %grid points
x = linspace(0,l,n);     %discretisation of space
dx = x(2)-x(1);
time = 0.4;

%dt*tstep = time;
tstep = time/dt;

%xstart = 0.1
% xend = 0.2
ct = 1;
for i = 1:n
if x(i) >= 0.1 && x(i)<=0.3
ustart(ct) = i;
ct = ct+1;
end
end

%initial vel = 1
u = ones(1,n);
u(ustart(1:end)) = 2;

uold = u;

%plotting initial velocity
figure(1)
plot (x,u,'linewidth',3,'color','b')
grid on
axis([0,1,1,2])
title(sprintf('1D-convection n n = %d',n))
xlabel('space - x','FontWeight','b')
ylabel('velocity - u','FontWeight','b')
hold on

%time marching
for r = 1:tstep

%space marching
for i = 2 : n

u(i) = uold(i) - ((c*dt/dx)*(uold(i) - uold(i-1)));

end

%update velocity field
uold = u;

figure(1)
plot(x,u,'linewidth',1,'color','g')
axis([0,1,1,2])

end

% plotting the final profile
plot(x,u,'linewidth',3,'color','r')

PLOT:

1. n=20

we can see that as the grid points are less there is a steep wave instead of a square wave to be formed. Also, there is a huge loss in the maximum value when time marching.

2. n=40

When we increase the grid points to 40, we can see the difference in the initial and final plots clearly.

3. n=80

If we increase the grids to 80, we are achieving the maximum velocity and we cannot even notice the velocity drop due to truncation losses.

4. n=160

When the number of grid points has been changed to 160, we can see that the initial velocity square wave is well clear but the final profile looks destructive. This is due to the instability which is described by Von Neumenn and the analysis is termed as Von neumenn stability analysis. That states the instability occurance when the number of grid points has been increased beyond the saturation level.

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