Magnus integrators on multicore CPUs and GPUs

In the present paper we consider numerical methods to solve the discrete Schr\"odinger equation with a time dependent Hamiltonian (motivated by problems encountered in the study of spin systems). We will consider both short-range interactions, which lead to evolution equations involving sparse matrices, and long-range interactions, which lead to dense matrices. Both of these settings show very different computational characteristics. We use Magnus integrators for time integration and employ a framework based on Leja interpolation to compute the resulting action of the matrix exponential. We consider both traditional Magnus integrators (which are extensively used for these types of problems in the literature) as well as the recently developed commutator-free Magnus integrators and implement them on modern CPU and GPU (graphics processing unit) based systems. We find that GPUs can yield a significant speed-up (up to a factor of $10$ in the dense case) for these types of problems. In the sparse case GPUs are only advantageous for large problem sizes and the achieved speed-ups are more modest. In most cases the commutator-free variant is superior but especially on the GPU this advantage is rather small. In fact, none of the advantage of commutator-free methods on GPUs (and on multi-core CPUs) is due to the elimination of commutators. This has important consequences for the design of more efficient numerical methods.