Verification of particle-in-cell simulations with Monte Carlo collisions against exact solutions of the Boltzmann-Poisson equations

Summary form only given.In recent years, the integrity of scientific computation has been called into question by demonstrations that errors in computer codes are common, even when those codes are professionally maintained. This has led to an interest in techniques for demonstrating the correctness of codes. The most powerful of these so-called verification techniques involves showing that the solutions calculated by the computer code under test converge to an exact solution of the physical model in question, at the rate anticipated from theoretical understanding of the solution procedure employed. Of course, this method assumes that a relevant exact solution is known, which is normally not the case. For some categories of problem, the so-called method of manufactured solutions permits a (not necessarily physical) exact solution to be constructed. However, at the present time, this method appears, for various reasons, unsuitable for testing codes with Monte Carlo elements, such as particle-in cell simulations with Monte Carlo collisions. In the present work, we show that the applied mathematics literature contains a number of exact solutions of the Boltzmann-Poisson system that are useful for code verification. No single one of these solutions is of a sufficiently general character to provide a comprehensive test of all the features of particle-in-cell code with Monte Carlo collisions, but each combines some features in a nontrivial way, and together they cover a large proportion of the functionality of the code under test. Thus one might call these “mezzanine solutions” in the sense that they are more than elementary unit tests, but less than fully comprehensive solutions. The solutions in question include the Child-Langmuir diode problem, the neutron criticality problem, and various other transport problems. No doubt the literature contains further such solutions not yet identified. We will discuss salient features of these solutions in the context of their use as verification test problems, and show how they may be used to verify elements of a particle-in-cell code with Monte Carlo collisions. This procedure increases confidence in the correctness of previously published benchmark solutions.