A Comparison of Stiff ODE Solvers for Astrochemical Kinetics Problems

Abstract

The time dependent chemical rate equations arising from astrochemical kinetics problems are described by a system of stiff ordinary differential equations (ODEs). In this paper, using three astrochemical models of varying physical and computational complexity, and hence different degrees of stiffness, we present a comprehensive performance survey of a set of well-established ODE solver packages from the ODEPACK collection, namely LSODE, LSODES, VODE and VODPK. For completeness, we include results from the GEAR package in one of the test models. The results demonstrate that significant performance improvements can be obtained over GEAR which is still being used by many astrochemists by default. We show that a simple appropriate ordering of the species set results in a substantial improvement in the performance of the tested ODE solvers. The sparsity of the associated Jacobian matrix can be exploited and results using the sparse direct solver routine LSODES show an extensive reduction in CPU time without any loss in accuracy. We compare the performance and the computed abundances of one model with a 175 species set and a reduced set of 88 species, keeping all physical and chemical parameters identical with both sets.We found that the calculated abundances using two different size models agree quite well. However, with no extra computational effort and more reliable results, it is possible for the computation to be many times faster with the larger species set than the reduced set, depending on the use of solvers, the ordering and the chosen options. It is also shown that though a particular solver with certain chosen parameters may have severe difficulty or even fail to complete a run over the required integration time, another solver can easily complete the run with a wider range of control parameters and options. As a result of the superior performance of LSODES for the solution of astrochemical kinetics systems, we have tailor-made a sparse version of the VODE solver by replacing the full numerical matrix linear algebra component of the standard VODE solver with sparse matrix solver routines. The preliminary tests from the preconditioned iterative solver package VODPK indicate very good results for one of our test models, but not for all of the models.