Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum.

Differential equations are frequently used for modeling systems in the physical sciences, biology, and other important real-world disciplines. Oftentimes, however, these equations cannot be solved exactly, so suitable computer algorithms are necessary to provide an approximated solution. While these computational simulations fail to exactly represent all behaviors of the true solution, they can be constructed to exactly, or very closely, reproduce certain properties which are key to the physical or scientific applications of a problem. This paper explores a computational method specifically constructed for modeling the behavior of systems with linear damping, or a reduction of energy, introduced in them. The method was designed to be conformal symplectic, and closely reproduce dissipation of physical properties such as linear and angular momentum, mass, and energy, caused by the damping. The algorithm was constructed in such a way that it maintains low computational cost to implement. Additionally, the method demonstrates favorable accuracy and stability properties in simulation. The method can also handle more complex scenarios, such as systems with forcing terms, and nonlinear systems. In these cases, it has been shown to hold advantages over other commonly used methods in particular circumstances.

Methods featured are primarily conformal symplectic exponential time differencing methods, with a focus on families of methods, the construction of methods, and the features and advantages of methods, such as order, stability, and symmetry. Methods are applied to the problem of the damped harmonic oscillator. Construction of both exponential time differencing and integrating factor methods are discussed and contrasted. It is shown how to determine if a system of equations or a method is conformal symplectic with flow maps, how to determine if a method is symmetric by taking adjoints, and how to find the stability region of a method. Exponential time differencing Stormer-Verlet is derived and is shown as the example for how to find the order of a method using Taylor series. Runge-Kutta methods, partitioned exponential Runge-Kutta methods, and their associated tables are introduced, with versions of Euler's method serving as examples. Lobatto IIIA and IIIB methods also play a key role, as a new exponential trapezoid rule is derived. A new fourth order exponential time differencing method is derived using composition techniques. It is shown how to implement this method numerically, and thus it is analyzed for properties such as error, order of accuracy, and structure preservation.