Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 35, Issue 5, Pages B1162-B1194Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120876034
Keywords
cloud-resolving model; compressible flow; element-based Galerkin methods; Euler; global model; IMEX; Lagrange; Legendre; mesoscale model; Navier-Stokes; nonhydrostatic; semi-implicit; spectral elements; time integration
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Funding
- Office of Naval Research [PE-0602435N]
- National Science Foundation (Division of Mathematical Sciences) [121670]
- Air Force Office of Scientific Research through the Computational Mathematics program
- Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy [DE-AC02-06CH11357]
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We derive an implicit-explicit (IMEX) formalism for the three-dimensional (3D) Euler equations that allow a unified representation of various nonhydrostatic flow regimes, including cloud resolving and mesoscale (flow in a 3D Cartesian domain) as well as global regimes (flow in spherical geometries). This general IMEX formalism admits numerous types of methods including single-stage multistep methods (e.g., Adams methods and backward difference formulas) and multistage single-step methods (e.g., additive Runge-Kutta methods). The significance of this result is that it allows a numerical model to reuse the same machinery for all classes of time-integration methods described in this work. We also derive two classes of IMEX methods, one-dimensional and 3D, and show that they achieve their expected theoretical rates of convergence regardless of the geometry (e. g., 3D box or sphere) and introduce a new second-order IMEX Runge-Kutta method that performs better than the other second-order methods considered. We then compare all the IMEX methods in terms of accuracy and efficiency for two types of geophysical fluid dynamics problems: buoyant convection and inertia-gravity waves. These results show that the high-order time-integration methods yield better efficiency particularly when high levels of accuracy are desired.
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