Classical and quantum phase space evolution: fixed-lattice and trajectory solutions

Recently, a new trajectory method for advancing solutions of evolutionary partial differential equations was proposed: the derivative propagation method (DPM) evolves the solution and its spatial derivatives concurrently and eliminates the need for function fitting, finite-differences (FDs), etc. This trajectory method and a stationary lattice FD algorithm are applied in phase space to solve the classical Klein–Kramers and quantum modified Caldeira–Leggett equations for several examples: a double-well oscillator in contact with a thermal bath and the decay of a metastable state. For the latter potential, the trajectory and fixed-grid solutions are compared and any discrepancies are noted.