HOPE: An Automatically Differentiable High-Order Non-Oscillatory Finite-Volume Shallow-Water Dynamic Core

Abstract

An automatically differentiable, high-order non-oscillatory finite volume shallow water dynamic core has been constructed on a cubed sphere grid. This dynamic core has four advantageous properties: high order accuracy, essential non-oscillation, mass conservation, and scalability. Besides, the code development is based on PyTorch, enabling the model to run seamlessly on both CPU and GPU, while naturally possessing the capability of automatic differentiation. We named the new dynamic core as High Order Prediction Environment (HOPE). The spatial reconstruction is based on the two-dimensional tensor product polynomial (TPP) and the genuine two-dimensional Weighted Essentially Non-Oscillatory scheme. A novel panel boundary approach ensures that the accuracy can reach arbitrary order. These algorithms have very high degree of compatibility with GPU architecture, allowing the computational overhead to be mitigated through the utilization of GPU. The Low Mach number Approximate Riemann Solver scheme is adopted as Riemann solvers to determine fluxes on the Gaussian points on edges. Flux across the interface between each cell edge is computed using Gaussian quadrature, and the tendencies of prognostic variables are obtained by integration all the source terms and the fluxes across the cell boundaries. This shallow water dynamic core exhibits outstanding performance in ideal shallow water test cases. In the steady-state geostrophic flow, the 11th order scheme reduces errors to nearly double precision round-off error even on coarse grids. Furthermore, HOPE maintains the Rossby-Haurwitz wave over 100 days without collapse. To test the non-oscillation property, we designed a cylinder dam break case, the WENO approach effectively suppresses non-physical oscillation, and the genuine two-dimensional reconstruction exhibits significantly better isotropy than the dimension-by-dimension scheme.