Particle methods are commonly used to obtain numerical solutions to Williams' spray equation, which describes the evolution of the droplet distribution junction (DDF) f(x,v,r,t). Accurate, efficient and numerically convergent particle methods are needed for predictive computational modeling of sprays. In this work, a simple vaporization test problem is proposed that admits analytic solutions to the spray equation and is useful for testing the accuracy of numerical solutions. This study shows that a simple particle method solution using uniform sampling of the DDF yields an accurate solution to the simple vaporization test problem. However, many spray codes, such as KIVA, use importance sampling of the mass-weighted DDF on the grounds that this is more computationally efficient. The implementation of importance sampling in KIVA results in an inaccurate numerical solution of the spray equation that does not converge to the analytic solution for the simple vaporization test, even for a very large number of computational particles. We show that importance sampling can be accurate and computationally efficient if statistical weights are correctly assigned to match the initial radius distribution. Simulations also reveal that the discontinuous evolution of statistical weights corresponding to vaporization in existing particle methods results in an numerical estimates of spray statistics that do not unconditionally converge to a continuous asymptotic limit as the time step is decreased. An algorithm of continuously evolving weights is developed that yields numerically convergent results that also match the analytic solution very well. These improvements to the particle method solution of the spray equation, which result in an excellent match of numerical predictions with the analytical solution in the test problem, can be expected to dramatically improve the accuracy of complex spray calculations at minimum computational expense.