i got this error "Index exceeds the number of array elements (1)." at the first line starting " U(j, n + 1, dt_index)" when the first loop completed for one time

i am trying to solve coutte flow problem using dufort-frankel Finite difference method and i got this error "Index exceeds the number of array elements (1)." at the first line starting " U(j, n + 1, dt_index)" when the first loop completed for one time .

why it's happen and how to solve ?

Defining Variables

clc,clear;
v = 0.000217; % Kinematic Viscosity
h = 0.04; dy = 0.001; % Y Dimension
Ny = length(linspace(h,0,round(h/dy))) ;
% Ny = 5 ;
dt_values = [0.002, 0.003]; % Time Dimension
Nt_values = [541, 361];

% Initial Condition

ic = 0; 

% Boundary Conditions

BC_Top = 0; 

BC_Bottom = 40;



for dt_index = 1:length(dt_values)
    dt = dt_values(dt_index);
    Nt = Nt_values(dt_index);
    
    % Diffusion Number
    d = (v * dt) / (dy^2);
    
    % Defining Domain
    U = zeros(Ny + 1, Nt, length(dt_values));
    % Initial Condition
    U(:, 1, :) = ic;
    % Boundary Conditions
    U(1, :, :) = BC_Top;
    U(end, :, :) = BC_Bottom;
    % Solution
    for k = 1:length(dt_values)
        for n = 1:Nt - 1
            if n == 1
                for j = 2:Ny
                    U(j, n + 1, dt_index) = U(j, n, dt_index) + d(dt_index) * (U(j + 1, n, dt_index) - 2 * U(j, n, dt_index) + U(j - 1, n, dt_index));
                end
            else
                for j = 2:Ny
                    U(j, n + 1, dt_index) = (U(j, n - 1, dt_index) + 2 * d(dt_index) * (U(j + 1, n, dt_index) - U(j, n - 1, dt_index) + U(j - 1, n, dt_index))) / (1 + 2 * d(dt_index));
                end
            end
        end
    end
end

Index exceeds the number of array elements. Index must not exceed 1.

Answer 1

[Walter Roberson]

for dt_index = 1:length(dt_values)
    dt = dt_values(dt_index);
    Nt = Nt_values(dt_index);
    

You extract particular scalars from dt_values and Nt_values, getting scalar dt and Nt

    d = (v * dt) / (dy^2);
    

and use them to calculate scalar d

    U(j, n + 1, dt_index) = U(j, n, dt_index) + d(dt_index) * (U(j + 1, n, dt_index) - 2 * U(j, n, dt_index) + U(j - 1, n, dt_index));

and

    U(j, n + 1, dt_index) = (U(j, n - 1, dt_index) + 2 * d(dt_index) * (U(j + 1, n, dt_index) - U(j, n - 1, dt_index) + U(j - 1, n, dt_index))) / (1 + 2 * d(dt_index));

in both of those statements you have d(dt_index) -- but d is scalar and so should not be indexed at d_index