We have developed an alternative to the one‐dimensional partial differential equation (PDE) attributed to Richards (1931) that describes unsaturated porous media flow in homogeneous soil layers. Our solution is a set of three ordinary differential equations (ODEs) derived from unsaturated flux and mass conservation principles. We used a hodograph transformation, the Method of Lines, and a finite water‐content discretization to produce ODEs that accurately simulate infiltration, falling slugs, and groundwater table dynamic effects on vadose zone fluxes. This formulation, which we refer to as “finite water‐content”, simulates sharp fronts and is guaranteed to conserve mass using a finite‐volume solution. Our ODE solution method is explicitly integrable, does not require iterations and therefore has no convergence limits and is computationally efficient. The method accepts boundary fluxes including arbitrary precipitation, bare soil evaporation, and evapotranspiration. The method can simulate heterogeneous soils using layers. Results are presented in terms of fluxes and water content profiles. Comparing our method against analytical solutions, laboratory data, and the Hydrus‐1D solver, we find that predictive performance of our finite water‐content ODE method is comparable to or in some cases exceeds that of the solution of Richards' equation, with or without a shallow water table. The presented ODE method is transformative in that it offers accuracy comparable to the Richards (1931) PDE numerical solution, without the numerical complexity, in a form that is robust, continuous, and suitable for use in large watershed and land‐atmosphere simulation models, including regional‐scale models of coupled climate and hydrology.

We have developed a one‐dimensional numerical method to simulate infiltration and redistribution in the presence of a shallow dynamic water table. This method builds upon the Green‐Ampt infiltration with Redistribution (GAR) model and incorporates features from the Talbot‐Ogden (T‐O) infiltration and redistribution method in a discretized moisture content domain. The redistribution scheme is more physically meaningful than the capillary weighted redistribution scheme in the T‐O method. Groundwater dynamics are considered in this new method instead of hydrostatic groundwater front. It is also computationally more efficient than the T‐O method. Motion of water in the vadose zone due to infiltration, redistribution, and interactions with capillary groundwater are described by ordinary differential equations. Numerical solutions to these equations are computationally less expensive than solutions of the highly nonlinear Richards' (1931) partial differential equation. We present results from numerical tests on 11 soil types using multiple rain pulses with different boundary conditions, with and without a shallow water table and compare against the numerical solution of Richards' equation (RE). Results from the new method are in satisfactory agreement with RE solutions in term of ponding time, deponding time, infiltration rate, and cumulative infiltrated depth. The new method, which we call “GARTO” can be used as an alternative to the RE for 1‐D coupled surface and groundwater models in general situations with homogeneous soils with dynamic water table. The GARTO method represents a significant advance in simulating groundwater surface water interactions because it very closely matches the RE solution while being computationally efficient, with guaranteed mass conservation, and no stability limitations that can affect RE solvers in the case of a near‐surface water table.

A total variation diminishing Runge-Kutta discontinuous Galerkin finite element method for the solution of one-dimensional (1D) shallow water flow equations for natural channels is presented. The hydrostatic pressure force and the wall pressure force terms are combined to simplify the calculations and prevent unphysical flow attributable to improper treatment of the bottom slope term. The treatment of the combined term that appropriately accounts for the momentum flux is given. HLL and Roe Riemann solvers are assessed for the mass and momentum flux terms. Numerical tests are conducted using prismatic rectangular and nonrectangular channels as well as non prismatic channels and natural channel for dam break, supercritical flow, transcritical flow, and dry-bed problems. Slope limiters based on flow cross section area, water surface, and water depth are evaluated. The tests show that HLL and Roe solvers provide similar accuracy. However, the slope limiter based on flow area provides more accurate solutions for tests in nonrectangular and natural channels.