Help indexing: How to index at the interfaces of cells and not the cells

I am struggling to understand how to setup some indexing problem in MATLAB. If my entire domain is defined from 1 to N with a padding of two cells. So say my internal domain (domain without padded cells) from 2 to N-2, then if I want to calculate some values at the "interfaces" of the cells instead at the cells, how can I setup this on MATLAB? For example I have the following code for the internal domain that I would like to change my indicies to the interfaces?

My set up is like:

%   calculate Flux at j+1/2
% 
%     j+1/2         Cell's grid:
%   | wL|   |
%   |  /|wR |           1   2   3   4        N-2 N-1  N
%   | / |\  |   {x=0} |-o-|-o-|-o-|-o-| ... |-o-|-o-|-o-| {x=L}
%   |/  | \ |         1   2   3   4   5        N-1  N  N+1
%   |   |  \|
%   |   |   |       NC: Here cells 1 and N are ghost cells% 
%     j  j+1

I keep getting the error:

Index in position 2 exceeds array bounds 

The code for example:

function [ U_L, U_R] = MUSCL(U,N,NEQ )
size_U = size(U);
U_L = zeros(size_U); 
U_R = zeros(size_U);
delta = 1e-6; % epsilon 
for i= 1:NEQ
    for j = 3:N-1
        denom_L = (U(i,j) - U(i,j-1)); 
        denom_R = (U(i,j+2) - U(i,j+1)); % error here since index exceeds the bounds of the array N (1001 here)
        denom_L = sign(denom_L+1e-64)*max(delta, abs(denom_L)); 
        denom_R = sign(denom_R+1e-64)*max(delta, abs(denom_R));
        r_L = (U(i,j) - U(i,j-1))/denom_L;
        r_R = (U(i,j+2) - U(i,j+1))/denom_R; 
        theta_L = Limiter(r_L);
        theta_R = Limiter(r_R);
    end
end
for j = 2:N-2
    U_L(:,j) = U(:,j)+0.5*theta_R.*(U(:,j)-U(:, j-1)); 
    U_R(:,j) = U(:,j+1)-0.5*theta_L.*(U(:,j+2)-U(:,j+1));
end
end

How can I find U_L, U_R, r_L, r_R at j+1/2 ??

Answer 1

[per isakson]

Index in position 2 exceeds array bounds 

U(:,j+2) is most likely the problem.

I assume that the size of U is NEQxN. The maximum value of j is N-1 (for j = 3:N-1). Thus, the maximum value of j+2 is N+1, which exceeds the array bound, N

[Jamie Al]

Yes so I changed j to go from 2:N-2 and the code works but I am still getting wrong results for UL and UR which suggest maybe I am not really caclulating those two at the interfaces of the cells

[per isakson]

Your code is about "flux", "cells" and "interfaces". The name, MUSCL, does that stand for "muscle"? However, I neither understand the physics nor the code. Futhermore, I cannot run the code, since I neither have the function, Limiter(), nor a realistic sample of U.

It looks strange to me that none of the left-hand-side variables in the double loop is indexed.

I've made a little demo in response to your answer to @DGM.

%%  A sample matrix
M = 40 .* membrane( 1, 25 );
%%  
to_the_interfaces = @(row) interp1( (1:numel(row)), row, (1.5:1:numel(row)-0.5) );
row = (1:10);
to_the_interfaces(row)

ans = 1×9

1.5000 2.5000 3.5000 4.5000 5.5000 6.5000 7.5000 8.5000 9.5000

%%
M_at_the_interfaces = nan(size(M)-[0,1]);
for jj = 1 : size(M,1)
    M_at_the_interfaces(jj,:) = to_the_interfaces(M(jj,:));
end
imagesc( M_at_the_interfaces )

Answer 2

[DGM]

Just interpolate.

x = 0:10;
y = x.^2

xhalf = 0.5:9.5;
yhalf = interp1(x,y,xhalf,'linear')

plot(x,y,'rd',xhalf,yhalf,'bo')

[Jamie Al]

I have no clue how to apply this in my example. Can you show something for my for loop?