1D Linear equation by time marching , Finite difference method.

Objective : To solve 1D linear wave equation by time marching method in finite difference using matlab. Observing how the equation diffuses and Analyzing results .

1D linear Wave equation :        `(∂u)/(∂t) = c*(∂u)/(∂x)`

Method :  Explicit form  , using first order forward differencing for time derivative , and 1st oer backward differencing for the space term in the eqn .

Program : 

%%  1-D Linear eqn  model by time marching method of (Finite difference Method )
% Code by Aadil Shaikh 

close all 
clear all
clc

%% Specifying parameters
L = 1;                              % domain length
n = input('Enter number (n)  : ');  % Numbers
                   
nx = linspace(0,L,n);               % Number of steps in grid
T = 0.4 ; 
c = 1;                              % velocity
dt = 1e-2;                          % time step
dx = nx(2)-nx(1);                   % width of space 
nt = T/dt;                          % number of steps in time
 u = zeros(1,n);                    % preallocating space



%% define start and end points for square vave 
 for i = 1:n
     
    if ((0.1<=nx(i))&&(nx(i)<=0.3))
       u(i) = 2;
     else 
         u(i) = 1;
         
    end
 end


uold = u;
u_initial = u;

%%
for k = 1:nt
    
    %space loop
    for i = 2:n
       u(i) = uold(i) - (c*dt/dx)*(uold(i)-uold(i-1));
    end

    %update old velocities 
    uold = u;
end

%% plotting the velocities
    figure(1)
    hold all
    plot(nx,u_initial,'r','linewidth',1)
    plot(nx,u,'-o','linewidth',.5)
    axis([0 1.2 0.8 2])
    title({['1-D Linear Convection at C = 1 & n = ',num2str(n)  ];['time \it = ',num2str(dt*k)]})
    xlabel('Grid Co-ordinate (x) \rightarrow')
    ylabel('Velocity (u) \rightarrow')

1. The parameters are specified as required in the beginning . 
2. The 'n' (Number) which denotes the number of steps created in the grid between 0-1 , needs to be plotted at different numbers, have been assigned an input to simply ease the process without creating any arrays. 
3. Basically the results can be plotted now for any number of steps in grid required and is not limited . 
4. c , time step and dx (width of space) is defined . 
5. nt is the number of steps in time is a variable utilized to produce the plots at 0.4 seconds .
6. a for loop is created to get the velocity profile as the step function , and create the wave , it automatically changes the velocity (u) to 2m/s as required , between the nx value of 0.1 and 0.3 
7. Another for loop is created , first for loop is the number of steps in time loop and the second one is space loop . 
8. Where the time marching soln for explicit form is used . using first order forward differencing for time derivative , and 1st oer backward differencing for the space term in the eqn .
9. u is the new velocity to be found . 
10. Plot is created for the initial square wave at the beginning , and then the time marching solution is observed on it . at the req conditions . 
11. appropriate title and labels are given . 

Plot : 

n = 20 : 

The red square wave is the initial velocity profile . 

The blue wave is the final velocity wave profile at time 0.4 seconds and at Space steps number 20

n = 40 

The red square wave is the initial velocity profile . 

The blue wave is the final velocity wave profile at time 0.4 seconds and at Space steps number 40

n = 80 

The red square wave is the initial velocity profile . 

The blue wave is the final velocity wave profile at time 0.4 seconds and at Space steps number 80

n = 160 

The red square wave is the initial velocity profile . 

The blue wave is the final velocity wave profile at time 0.4 seconds and at Space steps number 160 . we can see that wave is distorted and feels like an error . A result cant be obtained from such a wave . 

Explanation of plot behaviour : 

1. As the number of grid spacing is increased one can observe the diffusion of blue wave slows down . 
2. The reason for square wave not being maintained is  , the First order backward differencing scheme used in the space term creates a false diffusion. False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection diffusion equations . False diffusion with the upwind scheme is reduced by increasing the mesh density. 
3. Basically increasing the number slows down the diffusion . but at higher values it distorts as its result . 
4. The wave flows to right side , as its being convected by constant linear wave speed c = 1 . 

------------------------------------- THE END ------------------------------------------------