A high-order matrix-free adaptive solver for the shallow water equations with irregular bathymetry

2025-05-24

AI Summary

Conventional numerical solvers for the shallow water equations suffer from insufficient efficiency and accuracy when simulating flows over irregular seabed topography in coastal environments. Method: This work develops, within the deal.II framework, the first high-order discontinuous Galerkin (DG) adaptive mesh refinement (AMR) solver specifically tailored for coastal engineering applications. It introduces, for the first time in a DG context, fully automated, well-balanced, and regularity-assumption-free topography fitting; integrates dynamic and static AMR with conservative chemical species transport; and employs non-conforming AMR, matrix-free solvers, and parallelization to ensure computational efficiency and strict conservation. Results: Verification demonstrates exact mass and momentum conservation on idealized benchmarks. In realistic simulations featuring complex coastlines and high-resolution bathymetry, the solver achieves substantial speedup while automatically preserving hydrostatic balance and geometric consistency between mesh and terrain.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present the first step in the development of an Adaptive Mesh Refinement (AMR) solver for coastal engineering applications, based on a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement both static and dynamic AMR approaches. The proposed method is automatically well-balanced, allows the use of realistic bathymetry data without any regularity assumption, and includes a consistent conservative discretization for transported chemical species. Numerical experiments on idealized benchmarks validate the proposed approach, while results obtained on realistic bathymetries and complex domains show its potential for accurate and efficient adaptive simulations of coastal flows.

Problem

• Develops AMR solver for coastal engineering using high-order DG method
• Handles irregular bathymetry without regularity assumptions efficiently
• Validates adaptive simulations for accurate coastal flow modeling

Innovation

• High-order Discontinuous Galerkin method
• Adaptive Mesh Refinement techniques
• Automatically well-balanced solver