1D Linear Convection (Grid points)

Objective: Write a MATLAB Code and to solve 1D linear equation

Given Data:   

Length( `L` ) = 1m                    

Convective Coeffieient( `C` ) = 1

Time step( `dt` ) = 0.01

Grid point( `n` ) = [20,40,80,160]

Total simulation time( `t` ) = 0.4 seconds

intial velocity profile is 2 `m/s` between 0.1 and 0.3 and 1 `m/s` everywhere.

Matlab Code:

clear all
close all
clc

% Input Parameters

L = 1; % Length

c = 1; % Convective coefficient 

n = [20,40,80,160]; % Grid points

t = 0.4; % Total time

dt = 0.01; % Time step


x_start = 0.1;  % Start point of square wave
x_old = 0.3;    % End point of square wave
nt = t/dt; % Number of time steps (nt)

% Loop
for i = 1:4
    x = linspace(0,L,n(i));
    dx = L/(n(i)-1);
    u = ones(1,n(i));
    
    n_start(i) = round((x_start/dx)+1);
    n_end(i) = round((x_old/dx)+1);
    
    u(:,n_start(i):n_end(i)) = 2 % Defining start and end nodes where velocity is 2m/s
    uold=u;
    u_initial=u;
    
    for k = 1:nt
        for j = 2:n(i)
            u(j) = uold(j) - (c*(dt/dx)*(uold(j)-uold(j-1)))
        end
        uold=u;
        
        % Plotting
        figure(i)
        plot(x,u_initial,'color','b')
        hold on
        plot(x,u,'color','g')
        xlabel('Length')
        xlabel('Velocity')
        grid on
        title('Velocity profile for various time steps')
        axis([0 1.5 0 2.5])
    end
end 

    
    
    

  Plots

                                       Figure 1, n=20

In this plot where n=20, we can see that there is a drastic decrease in the velocity profile

the amplitude of the velocity profile keeps decreasing with length

the minimum velocity is approximately is  1.6  to 1.7`m/s`

                                               Figure 2, n=40

In this plot where n=40, we can see that there is a significant decrease in the velocity profile

the amplitude of the velocity profile keeps decreasing with length

the minimum velocity is approximately is between 1.7  to 1.8`m/s`

                                               Figure 3, n=80

In this plot where n=80, we can see that there is a gradual decrease in the velocity profile, The change in the velocity profile with respect to length is not a major change. Hence we can tell that the amplitude is almost constant.

the min and max velocity very very close 2`m/s`

                                                  Figure 4, n=160

In Figure 4, there is a disturbance in the velocity profile. We can see that there is a blow-up and it noticed that ripples are formed in the plot making it unstable.

Conclusion

From the above figures 1,2,3 & 4, we can tell that with the increase in grid spacing may lead to ripples. But at n=80 the optimum results are achieved and then decreases.