Numerical weather prediction (NWP) involves solving equations that govern the evolution of the atmosphere over the time intervals of interest, in other words an initial value problem for evolution equations is solved with the current state of the atmosphere treated as the initial condition. The computational time and spatial scale of NWP models involve large scale atmospheric dynamics such as planetary waves, tropical cyclones, gravity waves, acoustic waves, and more. One of the major computational difficulties in NWP is the wide varying propagation speeds of the atmospheric waves. The resulting stiffness from the fast-slow waves encourages the development of more efficient time integrators for efficient and accurate weather forecasting.

The majority of weather centers use a semi-implicit semi-Lagrangian (SISL) time integrator. The semi-Lagrangian method is unconditionally stable for the transport equation, and therefore allows for stable integration of the atmospheric model with time steps larger than Eulerian methods. However, SISL methods develop resonance in the presence of mountains. Many tried to address this issue with additional damping, diffusion terms, or adjusting the coefficient of the time integrator, but the techniques can lead to order reduction in the time integration and can affect the overall accuracy of the solution. The presence of resonance has re-incentivized research directions back to an Eulerian formulation of the atmospheric models.

In this dissertation, we explore two directions of research for improving the time integration of NWP models. First, exponential integrators have demonstrated to be a computationally efficient and accurate method for the shallow water equations and the Euler equations. Therefore, exponential integrators are a viable choice for NWP. We improve upon the parallel performance of exponential integrators on high performance computing platforms for large-scale, massively parallel environments, consequently making them a more practical choice for NWP. Second, it will take time to transition to a new operational model. Many forms of validation with other models such as ocean, air quality, climate, and more, must be cleared in order to fully adopt a new proposed scheme. Therefore, it is still beneficial to continue to improve upon the current SISL methods. Thus, we address the problem of resonance by replacing the Crank-Nicolson time integrator in the Canadian Global Environmental Multiscale model with a second order backward difference formula time integrator.