Reply on RC1 fix error
Lilong Zhou · 2025
Abstract. This study presents the High Order Prediction Environment (HOPE), an automatically differentiable, non-oscillatory finite-volume dynamical core for shallow water equations on the cubed-sphere grid. HOPE integrates four key features: (1) arbitrary high-order accuracy through genuine two-dimensional reconstruction schemes; (2) essential non-oscillation via adaptive polynomial order reduction in discontinuous regions; (3) exact mass conservation inherited from finite-volume discretization; (4) automatically differentiable and (5) GPU-native scalability through PyTorch-based implementation. Another innovation is the intensive panel boundary treatment, which eliminates numerical instability during using high order reconstruction scheme, meanwhile, simplifies the interpolation process to a matrix-vector multiplication without losing accuracy. Numerical experiments demonstrates the capabilities of HOPE: The 11th-order scheme reduces errors to near double-precision round-off levels in steady-state geostrophic flow tests on coarse 1°×1° grids. Maintenance of Rossby-Haurwitz waves over 100 simulation days without crashing. A cylindrical dam-break test case confirms the genuinely two-dimensional WENO scheme exhibits significantly better isotropy compared to dimension-by-dimension approaches. Two implementations are developed: a Fortran version for convergence analysis and a PyTorch version leveraging automatic differentiation and GPU acceleration. The PyTorch implementation maps reconstruction and quadrature operation to 2D convolution and Einstein summation respectively, achieving about 2× speedup on single NVIDIA RTX3090 GPU versus Dual Intel E5-2699v4 CPUs execution. This design enables seamless coupling with neural network parameterizations, positioning HOPE as a foundational tool for next-generation differentiable atmosphere models.