Welcome to the World of Modelling and Simulation

What is Modelling?

This blog is all about system dynamics modelling and simulation applied in the engineering field, especially mechanical, electrical, and ...

Adams-Bashforth Method with RK2 as a Start-up Scheme in MATLAB

Let's consider the following differential equation:

y'' + 10 y' + 500y = 0

Where, the initial conditions are,

y(0) = -0.025, and y'(0) = -1

We will solve this differential equation using a multi-step methodAdams-Bashforth, where second order Runge-Kutta approach (RK2) is added as a start-up scheme in the algorithm. 

MATLAB Program:
The following MATLAB program implements the Adams-Bashforth method with initialization with RK2 method.

close all;

h = 0.02;     % Step size
Tmax = 0.5;    % Maximum time
N = Tmax / h;  % Maximum number of steps
t = linspace(0,0.5,N+1);  % Time range

% Analytical solution of the differential equation
y_real = -(((9.*sqrt(19))/760).*exp(-5.*t).*sin(5.*sqrt(19).*t)) - ((1/40).*exp(-5.*t).*cos(5.*sqrt(19).*t));
hold on

%Numerical solution
f=@(t,y) [y(2); -500*y(1)-10*y(2)]; % Governing system of equations

% Initial Conditions
Y = [-0.025; -1];
% Initialization with second order Runge-Kutta method
k1 = h.*f(t(1),Y(:,1));
    k2 = h.*f(t(1)+alpha.*h, Y(:,1)+alpha.*k1);
    Y(:,2) = Y(:,1) + (1-1/2/alpha).*k1 + k2/2/alpha;

% Second Order Adams-Bashforth method steps
for i=2:N
    Y(:,i+1) = Y(:,i) + 3/2*h*f(t(i),Y(:,i)) - h/2*f(t(i-1),Y(:,i-1));

legend('Exact Solution','Adams-Bashforth Solution','Location','NorthEast')
title('When h = 0.02')

Program Output:
The following plot shows the numerical and analytical solution of the differential equation with two different step sizes with respect to time.

Showing results for Adams-Bashforth method when step is 0.02

Showing results for Adams-Bashforth method when step is 0.01

No comments:

Post a Comment