  We study the optimal design of numerical integrators for dissipative systems,
for which there exists an underlying thermodynamic structure known as GENERIC
(general equation for the nonequilibrium reversible-irreversible coupling). We
present a frame-work to construct structure-preserving integrators by splitting
the system into reversible and irreversible dynamics. The reversible part,
which is often degenerate and reduces to a Hamiltonian form on its symplectic
leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate
variables being left unchanged, for which an associated modified Hamiltonian
(and subsequently a modified energy) in the form of a series expansion can be
obtained by using backward error analysis. The modified energy is then used to
construct a modified friction matrix associated with the irreversible part in
such a way that a modified degeneracy condition is satisfied. The modified
irreversible dynamics can be further solved by an explicit midpoint method if
not exactly solvable. Our findings are verified by various numerical
experiments, demonstrating the superiority of structure-preserving integrators
over alternative schemes in terms of not only the accuracy control of both
energy conservation and entropy production but also the preservation of the
conformal symplectic structure in the case of linearly damped systems.
