Linearly implicit exponential integrators for damped Hamiltonian PDEs

Structure-preserving two-step linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping combining exponential integrators 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 Schrödinger equations. Preservation of the dissipation rate is demonstrated for linear, quadratic conformal invariants and of the Hamiltonians by numerical experiments.