Numerical approximation of the time-ordered exponential for spin dynamic simulation

Abstract

We describe, discuss, and compare classes of methods for the numerical solution of non-autonomous linear ODEs using problems coming from Nuclear magnetic resonance (NMR) spectroscopy with Magic-angle spinning (MAS) as case study. The newly intro- duced ⋆-product approach uses a convolution-like product to express the time-ordered exponential to then expand it in a Legendre polynomials basis so to transform the orig- inal ODE problem into a linear algebra problem. The aim is to compare the numerical performance of this method with other commonly used methods. Therefore, we take into account geometric numerical integrators. This group of integrators are frequently used in many different areas of research as, for example, quantum mechanics, molecular dynamics and particle accelerators physics. Their approach can either approximate the solution of the non-autonomous ODE by means of a single time-independent exponential (Magnus Integrators) or by means of a product of time-independent exponentials (Splitting Meth- ods, Commutator-Free Exponential Integrators). Finally, numerical experiments on the NMR MAS case study are performed to test the numerical behaviour of the ⋆-process and compare it with the already established alternatives. 1