Euler method error and step size code

%Program 6.1 Euler?s Method for Solving Initial Value Problems
%Use with ydot.m to evaluate rhs of differential equation
% Input: interval inter, initial value y0, number of steps n
% Output: time t, solution y
% Example usage: euler([0 1],1,10);
function [t,y]=euler2(inter,y0,h)
inter = [0,1];
y0 = 1;
n = 10;
t(1)=inter(1); y(1)=y0;
%step size
h=(inter(2)-inter(1))/n;
for i=1:n
%establising t-axis with equal step size h
t(i+1)=t(i)+h;
y(i+1)=eulerstep(t(i),y(i),h);
end
%Solution of the IVP
    yt = @(t) 3.*exp(t.^2/2)-t.^2-2;
close all
plot(t,y)
hold on
fplot(yt,[0 1])
xlabel('t')
ylabel('y')
legend('Approximate solution','Original solution')
%setting up the Euler Method
function y=eulerstep(t,y,h)
%one step of Euler's Method
%Input: current time t, current value y, stepsize h
%Output: approximate solution value at time t
y=y+h.*ydot(t,y);
end
function z=ydot(t,y)
%right-hand side of differential equation
z=t.*y+t.^3;
end
end

I have this code to plot the approximate and exact solution on a graph but I need to add a for loop to plot another graph that shows step size h and error which is difference between exact and approx solution. For some reason, I can't seem to plot it. Please help me plot it using loglog command

Answer 1

[Aashray]

Hello @Priyadarshini Challa,

I understand that you are trying to visualize how the error behaves with respect to the step size when using the Euler method.

I extended your original code by making the following additions:

1. A loop over different step sizes (n values like 5, 10, 20, ...).
2. Computation of the global error at the final time point (t = 1) for each step size.
3. A log-log plot to visualize the relationship between the step size (‘h’) and the error.

You can find the code below for better understanding of the changes:

clc; clear; close all; 

% Define the exact solution function for your reference 
yt = @(t) 3.*exp(t.^2/2) - t.^2 - 2; 

% Interval and initial condition calculations 
inter = [0, 1]; 
y0 = 1; 

% Vector of different n (number of steps) 
n_values = [5, 10, 20, 40, 80, 160, 320]; 
h_values = zeros(size(n_values)); 
error_values = zeros(size(n_values)); 

% Now, we'll iterate over different step sizes 
for k = 1:length(n_values) 
    n = n_values(k); 
    h = (inter(2) - inter(1)) / n; 
    h_values(k) = h; 
    
    % using Euler method  
    t(1) = inter(1); 
    y(1) = y0; 
    for i = 1:n 
        t(i+1) = t(i) + h; 
        y(i+1) = y(i) + h * ydot(t(i), y(i)); 
    end 
    
    % Compute error at final point 
    exact = yt(inter(2)); 
    approx = y(end); 
    error_values(k) = abs(approx - exact); 
end 

% Plot error vs step size on log-log scale 
figure; 
loglog(h_values, error_values, 'o-r', 'LineWidth', 2, 'MarkerSize', 8); 
grid on; 
xlabel('Step size (h)'); 
ylabel('Error at t = 1'); 
title('Error vs Step Size (Euler Method)'); 
legend('Error'); 

% Plot approximate vs exact solution for the finest step 
figure; 
plot(t, y, '-', 'DisplayName', 'Euler Approximation'); 
hold on; 
fplot(yt, [inter(1), inter(2)], 'k--', 'DisplayName', 'Exact Solution'); 
xlabel('t'); 
ylabel('y'); 
title('Euler Approximation vs Exact Solution'); 
legend; 
grid on; 

function z = ydot(t, y) 
z = t .* y + t.^3; 
end

I am also attaching the documentation links of the functions used for reference:

1. “loglog”: for plotting data on a logarithmic scale for both x and y axes. https://www.mathworks.com/help/matlab/ref/loglog.html
2. “abs”: for computing the absolute values. https://www.mathworks.com/help/matlab/ref/double.abs.html
3. “legend”: for adding legends to the plots. https://www.mathworks.com/help/matlab/ref/legend.html