Nonlinear Multilevel Solution Strategies for Diffusive Wave Flood Models in Perforated Domains

TL;DR

This paper develops and evaluates multilevel nonlinear solution strategies for the Diffusive Wave equation in complex perforated domains, crucial for urban flood modeling, demonstrating robustness and scalability through numerical experiments.

Contribution

It introduces a multiscale coarse space-based nonlinear preconditioning framework tailored for highly perforated domains, enhancing robustness and efficiency in solving nonlinear flood models.

Findings

The multiscale coarse space remains effective for linearized problems within Newton iterations.

Combined Schwarz-based nonlinear preconditioning strategies improve robustness.

Numerical experiments validate the approach on realistic urban flood scenarios.

Abstract

This article investigates the numerical solution of the Diffusive Wave equation posed on domains containing a large number of polygonal perforations, motivated by urban flood modeling. Such geometries induce strong multiscale effects driven by geometric complexity, which significantly challenge the robustness of standard nonlinear and linear solvers. The work builds on a multiscale coarse space previously introduced by the authors for linear Poisson problems on perforated domains. This low-dimensional space, constructed on a coarse polygonal partition and spanned by locally discrete harmonic (Trefftz-type) basis functions, is shown to remain effective for the linearized Diffusive Wave problems arising within Newton iterations. This enables the construction of robust two-level preconditioners for the resulting sequence of linear systems. Beyond linearization, the main focus of this work is on the effective solution of the fully nonlinear problem. We assess and combine several Schwarz-based nonlinear preconditioning strategies, including a two-level RASPEN method and a two-step nonlinear method, using the same multiscale coarse space to ensure scalability. While the individual components are drawn from the existing literature, their combination provides a robust and practical solution strategy for a challenging nonlinear problem posed on highly perforated domains. A systematic comparison of the methods and a discussion of algorithmic complexity are presented. The proposed approaches are validated through numerical experiments, including a realistic test case based on topographical data from the city of Nice.