function [M] = upscale(M_2h)
% This function interpolates a staggered coarse grid and creates a staggered fine
% grid. We transfer the result for the U,V grid points to the finer grid.

s = size(M_2h); rows = s(1) ; cols = s(2);
new_rows = (rows - 1)*2 + 1;
new_cols = (cols - 1)*2 + 1;

M = zeros([new_rows,new_cols]);

%===========================================
% Transfer the  U
M(4:4:new_rows-1,1:4:new_cols) = M_2h(2:2:rows-1,1:2:cols);

% Interpolate I
for i = 2:4:new_rows -2
    for j= 1:4:new_cols
        if (i == 2)
            M(i,j) = .5*(M(i+2,j));
        else
            M(i,j) = .5*(M(i-2,j) + M(i+2,j));
        end
    end
end


% Interpolate II
for i=2:2:new_rows-1
    for j=3:4:new_cols-2
        M(i,j) = .5*(M(i,j-2) + M(i,j+2));
    end 
end


%===========================================
% Transfer the  V

M(1:4:new_rows,4:4:new_cols-1) = M_2h(1:2:rows,2:2:cols-1);
% Interpolate I
for i=1:4:new_rows
    for j=2:4:new_cols-2
        if (j == 2)
            M(i,j) = .5*M(i,j+2);
        else
            M(i,j) = .5*(M(i,j-2) + M(i,j+2));
        end
    end
end

%Interpolate 2
for i=3:4:new_rows-2
    for j=2:2:new_cols
        M(i,j) = .5*(M(i-2,j) + M(i+2,j));
    end
end