Publications

Steyer, Andrew S.

Abstract not provided.

Goldberg, Nathaniel N.; Demsky, Sarah D.; Youssef, Abdelrahman A.; Carter, Steven; Fowler, Deborah M.; Jackson, Nathan J.; Kuether, Robert J.; Steyer, Andrew S.

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Steyer, Andrew S.; Kuether, Robert J.

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Guba, Oksana G.; Taylor, Mark A.; Bradley, Andrew M.; Bosler, Peter A.; Steyer, Andrew S.

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Steyer, Andrew S.; Krause, Cassidy F.

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Krause, Cassidy F.; Steyer, Andrew S.

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Geoscientific Model Development

Guba, Oksana G.; Taylor, Mark A.; Bradley, Andrew M.; Bosler, Peter A.; Steyer, Andrew S.

We present a new evaluation framework for implicit and explicit (IMEX) Runge-Kutta time-stepping schemes. The new framework uses a linearized nonhydrostatic system of normal modes. We utilize the framework to investigate the stability of IMEX methods and their dispersion and dissipation of gravity, Rossby, and acoustic waves. We test the new framework on a variety of IMEX schemes and use it to develop and analyze a set of second-order low-storage IMEX Runge-Kutta methods with a high Courant-Friedrichs-Lewy (CFL) number. We show that the new framework is more selective than the 2-D acoustic system previously used in the literature. Schemes that are stable for the 2-D acoustic system are not stable for the system of normal modes.

Steyer, Andrew S.

Abstract not provided.

Kuether, Robert J.; Steyer, Andrew S.

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Kuether, Robert J.; Steyer, Andrew S.

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Hillman, Benjamin H.; Caldwell, Peter C.; Salinger, Andrew G.; Bertagna, Luca B.; Beydoun, Hassan B.; Peter, Bogenschutz.P.; Bradley, Andrew M.; Donahue, Aaron D.; Eldred, Christopher; Foucar, James G.; Golaz, Chris G.; Guba, Oksana G.; Jacob, Robert J.; Johnson, Jeff J.; Keen, Noel K.; Krishna, Jayesh K.; Lin, Wuyin L.; Liu, Weiran L.; Pressel, Kyle P.; Singh, Balwinder S.; Steyer, Andrew S.; Taylor, Mark A.; Terai, Chris T.; Ullrich, Paul A.; Wu, Danqing W.; Yuan, Xingqui Y.

Abstract not provided.

Journal of Advances in Modeling Earth Systems

Taylor, Mark A.; Guba, Oksana G.; Steyer, Andrew S.; Ullrich, Paul A.; Hall, David M.; Eldred, Christopher

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal, and potential energy budgets, ensuring an energy-consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit-explicit time stepping algorithm.

Steyer, Andrew S.

Abstract not provided.

Steyer, Andrew S.; Krause, Cassidy F.

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Steyer, Andrew S.

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Steyer, Andrew S.

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Steyer, Andrew S.

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BIT Numerical Mathematics

Steyer, Andrew S.; Van Vleck, Erik S.

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

Discrete and Continuous Dynamical Systems - Series B

Steyer, Andrew S.; Van Vleck, Erik S.

We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [34, 35, 36] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.

Steyer, Andrew S.; Taylor, Mark A.; Guba, Oksana G.

Abstract not provided.

Steyer, Andrew S.

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Steyer, Andrew S.; Van Vleck, Erik S.

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Steyer, Andrew S.

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