Linearly Implicit Exponential Integrators for Damped Hamiltonian Pdes

We construct second-order structure-preserving two-step linearly implicit exponential integrators for Hamiltonian partial differential equations with linear constant damping combining discrete gradient methods and polarization of the polynomial Hamiltonian function. We also construct an exponential version of the well-known one-step Kahan's method by polarizing the quadratic vector field. These integrators are applied to one-dimensional damped Burger's, Korteweg-de Vries, and nonlinear Schr & ouml;dinger equations. Preservation of the dissipation rate is demonstrated for linear and quadratic conformal invariants and for the Hamiltonians by numerical experiments.