Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Stormer-Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissipation of angular momentum of an N-body system with pairwise distance dependent interactions. Necessary and sufficient conditions for second order accuracy are derived. Analysis for linear equations gives explicit relationships between the damping parameter and the step size to reveal when the methods are most advantageous; essentially, the damping rate of the numerical solution is exactly preserved under these conditions. The methods are applied to several model problems, both ODEs and PDEs. Additional structure preservation is discovered for the discretized PDEs, in one case dissipation in total linear momentum and in another dissipation in mass are preserved by the methods. The numerical results, along with comparisons to standard Runge-Kutta methods and another structure-preserving method, demonstrate the usefulness and strengths of the methods.