Numerical treatment of models arising in nuclear magnetic resonance spectroscopy

Abstract

The simulation of experiments in nuclear magnetic resonance spectroscopy for k interacting nuclei with spin one half is formulated as a system of 4 k − 1 linear ordinary differential equations (ODE's) when the Liouville representation is in use. Some components of the solution vector of such a system of ODE's are highly oscillatory. Moreover, the systems are often very large. This indicates that one should be careful both in the process of selecting a numerical method for solving the system of ODE's and in the handling of long sequences of systems of linear algebraic equations obtained after the time-discretisation. The salient properties of problems of this type can be exploited if (i) the numerical method chosen in the time-discretisation has good stability properties (AN-stability being desirable), (ii) a sparse matrix version of the code is available, and (iii) the linearity of the problem is exploited. A package of sub-routines, in which these three requirements are satisfied, has been developed. The package is fully documented. Several demonstration programs, typical for the class of problems for which the package is designed, are available. The numerical results show clearly that the package can successfully be used in the simulation of different situations appearing in nuclear magnetic resonance spectroscopy. The package can also be applied in the treatment of models from other fields if these are expresssed as large systems of ODE's whose solutions are oscillatory. The package can be att attached to large programs for solving some types of partial differential equations and used in the time-integration part when the systems of ODE's obtained after the space discretization satisfy the above conditions.