Structure-preserving Exponential Runge-Kutta Methods

Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) 4 methods are developed for solving initial value problems with vector fields that can be split into con5 servative and linear non-conservative parts. The focus is on linearly damped ordinary differential 6 equations, that possess certain invariants when the damping coefficient is zero, but, in the presence of 7 constant or time-dependent linear damping, the invariants satisfy linear differential equations. Simi8 lar to the way that Runge-Kutta and partitioned Runge-Kutta methods preserve quadratic invariants 9 and symplecticity for Hamiltonian systems, ERK and PERK methods exactly preserve conformal 10 symplecticity, as well as decay (or growth) rates in linear and quadratic invariants, under certain 11 constraints on their coefficient functions. Numerical experiments illustrate the higher order accu12 racy and structure-preserving properties of various ERK methods, demonstrating clear advantages 13 over classical conservative Runge-Kutta methods, as well as usefulness for solving a wide range of 14 differential equations. 15