Midpoint Method on an SHO to a Velocity Verlet Method
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t(1,1) = 0; % Starting time (arbitrary units)
u(1,1) = 1; % Initial position (arbitrary units)
u(1,2) = 0; % Initial velocity (arbitrary units)
tend = 4*pi; % Simulation run time.
Nsteps = 200; % Number of integration steps;
dt = (tend-t(1))/Nsteps; % Time step size (arbitrary units)
%% Set up the differential equation
% This is the "right hand side" function of the differential equation, T'
k = 1; m = 1; omega = sqrt(k/m); % Physical parameters of oscillator
dudt = @(u) [ u(2) -omega^2 * u(1) ]; % Right-hand side function
% Midpoint Method
%for idx = 1:tend/dt
% t(idx+1,1) = t(idx,1) + dt;
% utmp = u(idx,:) + (dt/2)*dudt(u(idx,:));
% u(idx+1,:) = u(idx,:) + dt*dudt(utmp);
%end
%Velocity Verlet Method
for idx = 1:tend/dt
t(idx+1,1) = t(idx,1) + dt;
uv = u(:,idx) - (dt/2)*dudt(u(idx,:));
u(idx+1,:) = u(idx,:) + dt*dudt(uv);
u(:,idx+1) = uv - (dt/2)*dudt(u(idx+1,:));
end
Here is my code I am try to fix. I have the midpoint method for a simple harmonic oscillator, which I know runs and works. I am curently trying to edit it and make it into a velocity verlet method but I can not figure out what I have done wrong, and if I am not even close please let me know as well. Any and all help will be appreciated.
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Antworten (1)
Dongyue
am 17 Nov. 2022
The following commands will lead to dimension inconsistency. Could you please verify whether they are correct? You may exchange row indexing and column indexing for the matrix u.
u(:,idx) - (dt/2)*dudt(u(idx,:));
u(:,idx+1) = uv - (dt/2)*dudt(u(idx+1,:));
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