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Second order conformal symplectic schemes for damped Hamiltonian systems. (English) Zbl 1377.65165

This paper is devoted to structure-preserving numerical solutions of linearly damped Hamiltonian systems. Numerical algorithms that are based on the Störmer-Verlet and implicit midpoint methods are constructed. It is shown that under certain assumptions the methods are of second order. In addition, they preserve conformal symplecticity and angular momentum. Linear stability analysis yields explicit stepsize requirements and also shows that the damping rate of the numerical solutions are exactly preserved under the stepsize conditions. A comparison of the methods is given via illustrative numerical experiments. Finally, the time-stepping schemes are applied to finite difference discretizations of some examples of linear damped Hamiltonian partial differential equations. In these examples, dissipation in total linear momentum or in mass is also preserved by the methods. Numerical experiments and comparisons confirm the advantages of the methods proposed.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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