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Publication Date
1 December 2023

Time-Stepping Optimized for Ocean and Atmosphere Models

Moving towards faster large scale ocean and atmosphere modeling with optimized time-stepping schemes.
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A table showing the speedups achieved by FB-RK(3,2) as opposed to SSPRK3

We devise and optimize a new class of time-stepping methods tuned specifically for the shallow water equations (SWEs). The SWEs are the foundational equations on which all ocean models are built; our new schemes are between 1.6 and 2.2 times faster (in terms of wall-clock time) than comparable methods.


The new time-stepping scheme presented in this work provides a significant increase to the speed of a shallow water model without sacrificing solution quality. The method can be adapted for use in climate scale ocean models (a version has been recently implemented for single-layer MPAS-Ocean).


We develop and optimize a class of time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically for the shallow-water equations, we optimize a schemes that can take time-steps as large as possible while retaining solution quality. These schemes, collectively called FB-RK(3,2), use weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.

Point of Contact
Mark Petersen
Los Alamos National Laboratory
Oregon State University
Funding Program Area(s)