Stochastic reaction-di usion systems frequently exhibit behavior that is not predicted by deterministic simulation models. Stochastic simulation methods, however, are computationally expensive. We present a more efficient stochastic reaction-di usion simulation algorithm that samples realizations from the exact solution of the reaction-di usion master equation. The present algorithm, called Partial-propensity Stochastic Reaction-Di usion (PSRD) method, uses an on-lattice discretization of the reaction-di usion system and relies on partial-propensity methods for computational efficiency. We describe the algorithm in detail, provide a theoretical analysis of its computational cost, and demonstrate its computational performance in benchmarks. We then illustrate the application of PSRD to two- and three-dimensional pattern-forming Gray-Scott systems, highlighting the role of intrinsic noise in these systems.