1D LINEAR CONVECTION SIMULATION IN MATLAB

AIM:

To solve 1 dimension linear convection equation in matlab by finite difference method and observe the propagation of wave with increasing time and also compare the effect of grids in the stability of the wave propagation.

INTRODUCTION:

1D Linear Wave Equation:

`∂u/∂t+c ∂u/∂x=0`

The above equation is discretized using forward differencing for time derivative and backward differencing for space derivative.

where 'n' is the current time step and 'i' is the current point.

Δt=time interval between two time step

Δx=step length

where,

L=1 m (domain length)

C=1

Δt=0.01

u(t=0)=2m/s for x=0.1 to 0.3

The effect of grid points on the stability of the flow is being observed in this project, hence we have taken the number of grid points as 20,40,80,160.

The total time for the solution to run is taken as T=0.4 seconds.

The program snippet for 1D linear convection is :

clear all 
close all
clc


%inputs for linear convection
L=1;                                      %domain length
c=1;                                      %constant
n=[21 41 81 161];                         %number of grid points
dt=1e-2;                                  %time interval between two time steps
%loop for the number of grid points

for g=1:4

    %calculations
     x=linspace(0,L,n(g));
     dx=x(2)-x(1);
     n_start=(0.1/dx)+1;                  %start and end point of the square wave
     n_end=(0.3/dx)+1;
     
     %Initialize and apply boundary conditions
     u=ones(1,n(g));
     u(n_start:n_end)=2;
    
     %Boundary Conditions
     u(1)=1;
     uold=u;
     u_initial=u;

     nt=40;                               %no of time steps

     %Time loop
     for k=1:nt
         
         %space loop
         for i=2:n(g)
             
             u(i)=uold(i)-(c*dt/dx)*(uold(i)-uold(i-1));
        
         end
         
         %updating values
         uold=u;
        
         %plotting the velocity
         figure(g)
         hold all
         plot(x,u_initial,'r','linewidth',3)
         
         plot(x,u,'b','linewidth',1)
         
         xlabel('domain length')
         ylabel('velocity')
         axis([0 1.2 0.8 2])
         pause(0.3)
      end
 end

PLOTS:

1.For grid size= 20.

The red square wave is the initial velocity profile and the blue lines indicate the manner of propagation of wave.

2.For grid size= 40.

The red square wave is the initial velocity profile and the blue lines indicate the manner of propagation of wave.

3.For grid size= 80.

The red square wave is the initial velocity profile and the blue lines indicate the manner of propagation of wave.

4.For grid size= 160.

The red square wave is the initial velocity profile and the blue lines indicate the manner of propagation of wave.

CONCLUSION OF THE RESULTS:

1.With the incresing number of grid points, the diffusion which can be seen in blue wave gradually slows down.

2.Also with increasing time step, the solution gets more stable but with a much finer mesh like with n=160, the solution blew up.

Hence with increasing number of grid points the diffusion slows down but at higher values it gets distorted.

3.As the grid point increses, the time of computation also increases.