% To solve u_t+u_x=0
%Using 
%1  Simple upwind scheme
% u(x,t+k)=u(x,t)-r*(u(x,t)-u(x-h,t)); 
%2  Simple, Space centred
% u(x,t+k)=u(x,t)-r*(u(x+h,t)-u(x-h,t))/2; 
%3  Lax-Friedrichs
% u(x,t+k)=(u(x+h,t)+u(x-h,t))/2-r*(u(x+h,t)-u(x-h,t))/2;
%4  Lax-Wendroff  
% u(x,t+k)=u(x,t)-r*(u(x+h,t)-u(x-h,t))/2+r^2*(u(x+h,t)-2*u(x,t)+u(x-h,t))/2; 
%5  Leap-Frog 
% u(x,t+k)=u(x,t-k)-r*(u(x+h,t)-u(x-h,t));
%6 The Keller Box scheme
% u(x+h,t+k)-u(x+h,t)+u(x,t+k)-u(x,t)+r*(u(x+h,t+k)+u(x+h,t)-u(x,t+k)-u(x,t))=0;
% u(x+h,t+k)=(u(x+h,t)*(1-r)-u(x,t+k)*(1-r)+u(x,t)*(1+r))/(1+r)
 C0='Solve for -2<x<3, t>0, with h=0.1, r=k/h=0.8           ';
 C1='Method 1  Simple upwind scheme                         ';
 C2=' u(x,t+k)=u(x,t)-r*(u(x,t)-u(x-h,t))                   ';
 C3='Method 2  Simple, Space centred                        ';
 C4=' u(x,t+k)=u(x,t)-r*(u(x+h,t)-u(x-h,t))/2               ';
 C5='Method 3  Lax-Friedrichs                               ';
 C6=' u(x,t+k)=(u(x+h,t)+u(x-h,t))/2-r*(u(x+h,t)-u(x-h,t))/2';
 C7='Method 4  Lax-Wendroff                                 ';
 C8=' u(x,t+k)=u(x,t)-r*(u(x+h,t)-u(x-h,t))/2               ';
 C9='                    +r^2*(u(x+h,t)-2*u(x,t)+u(x-h,t))/2';
C10='Method 5  Leap-Frog                                    ';
C11=' u(x,t+k)=u(x,t-k)-r*(u(x+h,t)-u(x-h,t))               ';
C12='Method 6 The Keller Box scheme                         ';
C13=' u(x,t+k)=                                             ';
C14='   (u(x,t)*(1-r)-u(x-h,t+k)*(1-r)+u(x-h,t)*(1+r))/(1+r)';
C=[C0;C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12;C13;C14];
disp(C)
Method=input('Select Method: (1 - 6)! ' );
     if (Method==1)
       disp('Simple Scheme')
      elseif (Method==2)
       disp('Simple Centred Scheme')
      elseif (Method==3)
       disp('Lax-Friedrichs Scheme')    
      elseif (Method==4)
       disp('Lax-Wendroff Scheme ')
      elseif (Method==5)
       disp('Leap-Frog Scheme ')
      else
       disp('Box Scheme ')
      end
N=70; % N = number of mesh points at each time 
h=7/N;% h = mesh length
MT=30; % MT = number of time steps
OT=5;  % OT = number of steps between outputs graphs
NT = MT/OT;
r=0.8;% r = dt/h;
dt=r*h;% dt = time step
u=zeros(1,N+1); v=u; w=u;
%Initial condition u(x,0)
for m=2:N
   x=-2+(m-1)*h;
   if (abs(x)<1.0)
     u(m)=1-abs(x);
   end
end
f=u;
hold off
x=-2:h:5;
plot(x,u)
title('Initial condition')
text=input('press enter to continue ');
for i=1:NT;
hold off
M=i*OT;
plot(x+M*dt,f)
axis([-2,4.5,-0.25,1.25])
hold on
for n=1:OT
   v=u;
   for m=2:N
      if (Method==1)
       v(m)=u(m)-r*(u(m)-u(m-1)); %Simple upwind scheme
      elseif (Method==2)
       v(m)=u(m)-r*(u(m+1)-u(m-1))/2; %Simple Space centred Scheme
      elseif (Method==3)
       v(m)=(u(m+1)+u(m-1))/2-r*(u(m+1)-u(m-1))/2; %Lax-Friedrich Scheme
      elseif (Method==4)
       v(m)=u(m)+r^2*(u(m+1)-2*u(m)+u(m-1))/2-r*(u(m+1)-u(m-1))/2; %Lax-Wendroff Scheme
      elseif (Method==5)
         if (n==1&i==1)
            v(m)=u(m)-r*(u(m+1)-u(m-1));
         else 
         v(m)=w(m)-r*(u(m+1)-u(m-1)); % Leap-Frog Scheme
         end
      else 
         v(m)=(u(m)*(1-r)-v(m-1)*(1-r)+u(m-1)*(1+r))/(1+r);
%         v(m)-u(m)+v(m-1)-u(m-1)+r*(v(m)-v(m-1)+u(m)-u(m-1))=0; %The Box Scheme
      end
   end %m
   w=u;
   u=v;
end %n
plot(x,u,'b-o')
      if (Method==1)
        AX = title(['Simple Upwind Scheme at time step ' num2str(i*OT)]);
      elseif (Method==2)
       AX = title(['Simple Centred Scheme at time step ' num2str(i*OT)]);
      elseif (Method==3)
       AX = title(['Lax-Friedrichs Scheme at time step ' num2str(i*OT)]);    
      elseif (Method==4)
       AX = title(['Lax-Wendroff Scheme at time step ' num2str(i*OT)]);
      elseif (Method==5)
       AX = title(['Leap-Frog Scheme at time step ' num2str(i*OT)]);
      else
       AX = title(['Box Scheme at time step ' num2str(i*OT)]);
      end
%LEG = findobj(AX,'type','text');
%set(LEG,'FontSize',18);
input('press enter to continue for another 10 time-steps');
end %i