This project addresses a crucial but largely overlooked source of error in the Energy Exascale Earth System Model (E3SM) and similar global models, namely numerical issues in the atmospheric physics parameterizations. The objectives are to (1) understand the cause of strong time-step sensitivity and poor time-step convergence in the E3SM atmosphere model, (2) develop new time integration and process coupling methods to improve convergence and accuracy for physics parameterizations related to boundary-layer clouds, and (3) develop new parameterization and discretization methods for complex atmospheric processes using model reduction methods and the theory of stochastic differential equations. Our overarching goal is to achieve a better balance between cost and accuracy in time integration related to parameterizations in the E3SM atmosphere model. As such, the project complements the E3SM project’s efforts on improving the fidelity and computational efficiency of the coupled Earth system model.