VERLET: Symplectic integration for Hamiltonian systems and adjoint solvers

Purpose

• The package "Verlet" solves the second-order differential equation (or a Hamiltonian system) with the velocity Verlet scheme. It is second-order accurate in both position and velocity. In addition, the velocity Verlet scheme almost preserves the Hamiltonian (oscillating around a constant mean). The package also provides the corresponding adjoint solver.

Specifications

• Name: Verlet.
• Author: Feng Chen.
• Finishing date: 07/24/2013.
• Languages: MATLAB.

Simple Example

• Equation: $$\begin{equation} \begin{aligned} & u''=\lambda u, \quad t \in (0,T],\\ & u(0) = 0, \quad u'(0) = \sqrt{-\lambda}. \end{aligned} \end{equation}$$
• Parameters: $$\begin{equation} T=10, \quad \lambda=-400, \quad \delta t = 0.0001. \end{equation}$$
• Exact solution: $$\begin{equation} u(t) = \sin(\sqrt{-\lambda} t). \end{equation}$$

Quick Start

• Compiling and running:

  matlab Verlet_Driver
  

• Output:

  maximum error of position and velocity =
       5.821930990741464e-04
  

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Release: MATLAB R2012a.

References

• Verlet integration. Link.

Code Highlights

% the velocity-Verlet time stepping.

K = -lambda;

r = u0(1);
v = u0(2);
a = -K*r;

for i = 1:nt
 rt = r; 
 r = (1-K*dt*dt/2)*r + dt*v; 
 v = (1-K*dt*dt/2)*v + dt*(K*dt*dt/4-1)*K*rt; 
end
u = [r; v];