How I write periodic BCs

clear all;

xl=0; xr=1; %domain[xl,xr]

J=40; % J: number of dividion for x

dx=(xr-xl)/J; % dx: mesh size

tf=0.5; % final simulation time

Nt=50; %Nt:number of time steps

dt=tf/Nt;

c=50; % paremeter for the solution

mu=dt/dx;

if mu>1.0 %make sure dt satisfy stability condition

error('mu should<1.0!')

end

%Evaluate the initial conditions

x=xl:dx:xr; %generate the grid point

f=exp(-c*(x-0.2).^2); %dimension f(1:J+1)

%store the solution at all grid points for all time steps

u=zeros(J+1,Nt);

%find the approximate solution at each time step

for n=1:Nt

t=n*dt; % current time

gl = exp(-c*(xl-t-0.2)^2); % BC at left side

gr = exp(-c*(xr-t-0.2)^2); % BC at right side

if n==1 % first time step

for j=2:J % interior nodes

u(j,n)=f(j)-0.5*mu*(f(j+1)-f(j-1))+0.5*mu^2*(f(j+1)-2*f(j)+f(j-1));

end

u(1,n) = gl; % the left-end point

u(J+1,n) = gr; % the right-end point

else

for j=2:J % interior nodes

u(j,n)=u(j,n-1)-0.5*mu*(u(j+1,n-1)-u(j-1,n-1))+0.5*mu^2*(u(j+1,n-1)-2*u(j,n-1)+u(j-1,n-1));

end

u(1,n)=gl; % the left-end point

u(J+1,n)=gr; % the right-end point

end

%calculate the analytical solution

for j=1:J+1

xj=xl+(j-1)*dx;

u_ex(j,n)=exp(-c*(xj-t-0.2)^2);

end

end

%plot the analytical and numerical solution at different times

figure;

hold on;

n=10;

plot(x,u(:,n),'r.',x, u_ex(:,n),'r-');

n=30;

plot(x,u(:,n),'g.',x, u_ex(:,n),'g-');

n=50;

plot(x,u(:,n),'b.',x, u_ex(:,n),'b-');

legend('Numerical t=0.1','Analytical t=0.1','Numerical t=0.3','Analytical t=0.3','Numerical t=0.5','Analytical t=0.5');

title('Numerical and Analytical Solutions at t=0.1,0.3,0.5');

[KALYAN ACHARJYA]

Please specify the issue?

[lulu]

i want to change the boundary conditions from exp to periodic boundary conditions

[KALYAN ACHARJYA]

Are you talking about this line

f=exp(-c*(x-0.2).^2);     %dimension f(1:J+1)

Make this periodic?? Like sine and cosine or more??

[lulu]

No, f is the initial condition

I want to change gl and gr to periodic boundary conditions. If the initial conditions leaves the domain at one end, and re-enters it at the other end then the Periodic BCs are correct.