The application of the preconditioned biconjugate gradient algorithm to NLTE rate matrix equations (Q1201048)

The solution of the system of ordinary differential equations (1) \(\partial u/\partial t=R\cdot u\) where \(u\in \mathbb{R}^ N\), \(R\) is a (time dependent) rate matrix, is considered with \(N\) very large (e.g. \(N\sim 1000)\). In equation (1) \(R\) is linearized about \(u\) and the resulting linear problem is solved with a simple one-step difference equation. The preconditioned biconjugate gradient algorithm and its variant, the preconditioned conjugate gradient squared algorithm, are applied to the linear system arising from a simple backward Euler differencing of (1). It is found that the iterations were rapidly convergent when (1) describes nonlocal thermal equilibria (NLTE) problems.