Comparison of adaptive mesh refinement techniques for numerical weather prediction

TL;DR

The paper addresses the challenge of capturing multi-scale atmospheric phenomena in numerical weather prediction by comparing two AMR paradigms: a level-based approach using AMReX with Arakawa C-grid discretization, and a tree-based approach using NebulaSEM with dGSEM on polyhedral cells. Both frameworks solve the fully compressible Euler equations for dry dynamics, and the study analyzes accuracy, conservation (via mortar methods and flux corrections), scalability, and practicality on spherical geometry. Across benchmarks including isentropic vortex, linear advection, rising thermal bubble, and spherical acoustic/oceanic transport, both AMR strategies achieve high fidelity while dramatically reducing time-to-solution compared to uniform grids; conservation is maintained with carefully designed transfer operators, especially on non-conformal interfaces and curved surfaces. The results support adopting AMR frameworks in NWP to dynamically track localized phenomena like tropical cyclones, with level-based AMR favored for scalability and portability, and tree-based AMR offering strong high-order capability on global grids when integrated with suitable conservation strategies.

Abstract

This paper examines the application of adaptive mesh refinement (AMR) in the field of numerical weather prediction (NWP). We implement and assess two distinct AMR approaches and evaluate their performance through standard NWP benchmarks. In both cases, we solve the fully compressible Euler equations, fundamental to many non-hydrostatic weather models. The first approach utilizes oct-tree cell-based mesh refinement coupled with a high-order discontinuous Galerkin method for spatial discretization. In the second approach, we employ level-based AMR with the finite difference method. Our study provides insights into the accuracy and benefits of employing these AMR methodologies for the multi-scale problem of NWP. Additionally, we explore essential properties including their impact on mass and energy conservation. Moreover, we present and evaluate an AMR solution transfer strategy for the tree-based AMR approach that is simple to implement, memory-efficient, and ensures conservation for both flow in the box and sphere. Furthermore, we discuss scalability, performance portability, and the practical utility of the AMR methodology within an NWP framework -- crucial considerations in selecting an AMR approach. The current de facto standard for mesh refinement in NWP employs a relatively simplistic approach of static nested grids, either within a general circulation model or a separately operated regional model with loose one-way synchronization. It is our hope that this study will stimulate further interest in the adoption of AMR frameworks like AMReX in NWP. These frameworks offer a triple advantage: a robust dynamic AMR for tracking localized and consequential features such as tropical cyclones, extreme scalability, and performance portability.

Figures (16)

• Figure 1: Illustration of level-based and tree-based AMR. The left figure depicts level-based AMR as used in AMReX (Courtesy of Almgren et al.) amrex2021. Three overlapping levels are depicted: level-0 (gray), level-1 (blue) and level-2 (red). Note that the boxes within a given level are non-overlapping even though the different levels themselves overlap. The right figure depicts tree-based AMR as used in NebulaSEM NebulaSEM. Here, the blue, green and red boxes all lie within the same level and are also non-overlapping.
• Figure 2: Supported face refinement patterns. Quadrilateral and triangular faces are sub-divided into 4 elements, while $n$-sided polygon is sub-divided to $n$ quadrilaterals.
• Figure 3: Illustration of how ghost cells are filled in AMReX's block-structured multi-level grid hierarchy (Courtesy of Almgren et al). This figure shows two levels, with three grids at the finer level. The colored regions here show the ghost cells of grid 1 and the different ways in which the data are filled in those regions. Cells in the green region are "valid" cells of grids 2 and 3; values in this region are copied from grids 2 and 3 to level 1. Cells in the red region do not live in any other fine grid; data here is interpolated from the next coarser level. Finally, grids in the blue region are outside the domain and are filled according to the domain boundary conditions.
• Figure 4: Mortar configuration for h-refinement of the right element.
• Figure 5: Evolution of the isentropic vortex for 3 seconds (snapshots at 1, 2 and 3 seconds). The potential temperature is dipicted with three different configurations: a) tree-based AMR with a buffer zone of 2 cells b) level-based AMR with grid efficiency of 0.7. Notably, only one rectangular patch is created in this case. c) level-based AMR with grid efficiency of 0.9, resulting in the creation of multiple patches.