Development of 1D-2D Coupled Hp-Adaptive Discontinuous Galerkin Model for Nonlinear Shallow Water Equations With Wetting and Drying

Abstract

Nonlinear shallow-water modeling for tsunamis, river flooding, and storm surges faces persistent challenges in computational speed and numerical stability near wet/dry fronts. To address these issues, this study develops Systematic Grid, an object-oriented simulation framework that integrates 1D 2D coupling and hp-adaptive methods to accelerate computations while incorporating stabilization techniques for the Discontinuous Galerkin (DG) method. Systematic Grid provides a hierarchical structure consisting of Domains, Blocks, SubBlocks, and Elements. This organization allows flexible control of dimensions, polynomial orders, and spatial resolutions at each level. Object-oriented programming further allows consistent treatment of elements with different shapes and polynomial orders, naturally supporting hp-adaptive refinement on arbitrary grids. The paper first describes the design of Systematic Grid and the implementation of each numerical technique. It then introduces the stabilization methods and explains how they are applied near wet/dry fronts. Finally, the proposed framework is validated through the Carrier Greenspan problem, the dam-break flow in an L-shaped channel, and the Okushiri benchmark for the 1993 Hokkaido Southwest Offshore Earthquake tsunami. The results demonstrate clear improvements in both computational efficiency and numerical stability.