Siddhartha Bishnu | Searcharxiv

arXiv2025

High-level, high-resolution ocean modeling at all scales with Oceananigans

We describe the user interface, governing equations, and numerical methods underpinning the community ocean modeling software called "Oceananigans". Oceananigans development has been lead by the Climate Modeling Alliance to build a trainable climate model with quantifiable uncertainty. Oceananigans is written in the Julia programming language, which, like similar recent efforts based on modern programming languages, distinguishes it from usual software based on Fortran. Oceananigans can efficiently simulate all scales of ocean motion, ranging from millimeter-scale turbulence in a small box to planetary-scale ocean circulation. Oceananigans design combines (i) a basic structured finite volume algorithm (ii) optimized for high-resolution simulations on GPUs which is (iii) exposed behind a high-level, programmable user interface. This design negotiates a dual mandate for highest-possible performance (to support state-of-the-art applications) and enhanced accessibility (to facilitate adoption and development). The dual mandate aims ultimately to accelerate the progress of Earth system science. Achieving this aim, however, requires a substantial and sustained increase in the collective effort of Oceananigans development.

physics.ao-ph↗

arXiv2022

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss the practical implications of this analysis. If we employ a stable numerical scheme and the orders of accuracy of the global solution error and the global truncation error agree, we make the following asymptotic observations: (a) the order of convergence at constant ratio of $Δt$ to $Δx$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. An implication of (a) is that it is impractical to invest in a time-stepping method of order higher than the spatial discretization. In addition to (b), we demonstrate that under certain circumstances, the error can even monotonically increase with refinement only in space or only in time, and explain why this phenomenon occurs. To verify our theoretical findings, we conduct convergence studies of linear and non-linear advection equations using finite difference and finite volume spatial discretizations, and predictor-corrector and multistep time-stepping methods. Finally, we study the effect of slope limiters and monotonicity-preserving strategies on the order of accuracy.

math.NA↗