A Fully Discrete, Mass-Conservative Adaptive Discontinuous Spectral-Element Method for the One-Dimensional Richards Equation in Mixed Form

Abstract

We present and analyze a spatially and temporally adaptive high-order method for the one-dimensional Richards equation describing vertical water transport in unsaturated soil, with an emphasis on resolving sharp wetting fronts while satisfying a fully discrete water mass balance. The spatial discretization uses an arbitrary-order local discontinuous Galerkin spectral-element formulation with collocated Legendre-Gauss-Lobatto quadrature, and the time integration is based on a high-order Rosenbrock-Wanner method. We focus primarily on the mixed formulation of the Richards equation, which advances water content as the conserved variable while evaluating the nonlinear diffusion operator in terms of pressure head; the resulting fully discrete scheme satisfies the water mass balance in exact arithmetic, independent of spatial resolution and time step size, including under spatial and temporal adaptivity. In contrast, the pressure-head formulation advances pressure head through a nonlinear capacity function and satisfies the corresponding conservation identity only at the semi-discrete level. Numerical experiments examine convergence through spatial and temporal refinement studies, with mass conservation assessed under spatial and temporal adaptivity using the solver flux output method (SFOM). The SFOM results confirm machine-precision mass conservation for the mixed formulation, whereas the pressure-head formulation exhibits mass balance errors above roundoff. Moreover, in the temporal refinement studies, the mixed formulation yields more accurate solutions than the pressure-head formulation for a prescribed time step size. The combination of high-order accuracy, spatial and temporal adaptivity, and fully discrete mass conservation makes the proposed mixed-form discontinuous spectral-element method a promising approach for simulating unsaturated flow in next-generation land models.