Solving linear convection equation (wave equation) by Lax Wendroff Scheme

Hi!

I am trying to solve the problem in the text attached.

I am copying my MATLAB code to solve the Lax Wendroff scheme. I am struggling to put in the periodic boundary conditions. Maybe someone could guide?

Thanks!

% MAE 5093 Homework #5- Due November 13th, 2018
%%Problem 1
%%Code written by Saad Saleem
% % This MATLAB code solves linear convection equation
clear all
close all

N=100; %No. of grid points
Tmax=1; % time period=1
alpha=1;  %given

h=0.01; %given value of delX
delt=0.005;

maxt=Tmax/delt; %Number of time steps

c=alpha*delt/h; %c=0.5

% Initial condition
for i=1:N+1;
    x(i)=(i-1)*h;
    u(i,1)=sin(2*pi*x(i));
end

% Temp at boundaries
for k=1:maxt+1;
    u(1,k)=u(N+1,k);
end

u=zeros(maxt,N);

for k=1:maxt
    for i=1:N %Space loop
        u(i,k+1) =u(i,k)-c/2*(u(i+1,k)-u(i-1,k))+(c.^2)/2*(u(i+1,k)-2*u(i,k)+u(i-1,k));
    end
end

Answer 1

[Torsten]

N=100; %No. of grid points
Tmax=1; % time period=1
alpha=1;  %given

h=0.01; %given value of delX
delt=0.005;

maxt=Tmax/delt; %Number of time steps

c=alpha*delt/h; %c=0.5

u=zeros(N+1,maxt+1);
x=zeros(N+1,1);

% Initial condition
for i=1:N+1
  x(i)=(i-1)*h;
  u(i,1)=sin(2*pi*x(i));
end

for k=1:maxt
  u0 = u(N,k);
  u(1,k+1)=u(1,k)-c/2*(u(2,k)-u0)+c^2/2*(u(2,k)-2*u(1,k)+u0);
  for i=2:N
    u(i,k+1) =u(i,k)-c/2*(u(i+1,k)-u(i-1,k))+(c^2)/2*(u(i+1,k)-2*u(i,k)+u(i-1,k));
  end
  uNp2 = u(2,k);
  u(N+1,k+1) =u(N+1,k)-c/2*(uNp2-u(N,k))+c^2/2*(uNp2-2*u(N+1,k)+u(N,k));  
end

plot(x,u(:,10),x,u(:,20),x,u(:,30),x,u(:,40),x,u(:,50),x,u(:,60),x,u(:,70))