Bound-Preserving Flux-Corrected Transport Methods for Solving Richards' Equation

TL;DR

This paper develops and verifies a flux-corrected transport method for Richards' equation that preserves physical bounds, achieves second-order accuracy, and is applicable to stormwater management.

Contribution

It extends flux-corrected transport schemes to nonlinear Richards' equation, ensuring bound preservation and second-order convergence on unstructured meshes.

Findings

Achieves second-order convergence in numerical simulations.

Successfully applies the method to stormwater infrastructure modeling.

Preserves physical bounds on water pressure and saturation.

Abstract

Simulating infiltration in porous media using Richards' equation remains computationally challenging due to its parabolic structure and nonlinear coefficients. While a wide range of numerical methods for differential equations have been applied over the past several decades, basic higher-order numerical methods often fail to preserve physical bounds on water pressure and saturation, leading to spurious oscillations and poor iterative solver convergence. Instead, low-order, bound-preserving methods have been preferred. The combination of mass lumping and relative permeability upwinding preserves bounds but degrades accuracy to first order in space. Flux-corrected transport is a high-resolution numerical technique designed for combining the bound-preserving property of low-order schemes with the accuracy of high-order methods, by blending the two methods through limited anti-diffusive fluxes. In this work, we extend flux-corrected transport schemes to the nonlinear, degenerate parabolic structure of Richards' equation, verify attainment of second-order convergence on unstructured meshes, and demonstrate applications to stormwater management infrastructure.