A scalable and implicit meshless RBF method for the 3D unsteady nonlinear Richards equation with single and multi‐zone domains

Abstract

SUMMARYThe Local Hermitian Interpolation (LHI) method is a strong-form meshless numerical technique which uses Radial Basis Function (RBF) interpolants to satisfy linear and nonlinear governing equations and boundary operators. Recent developments have shown that, for linear transport problems, applying the PDE governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions. In this work, the LHI formulation with local PDE-interpolation is extended to the nonlinear gravity-driven Richards equation, in order to solve unsteady problems involving flow in unsaturated porous media.The application of the linearised PDE-operator to the basis functions incorporates information, such as the effective velocity field, directly into the solution construction. This results in a form of 'analytical upwinding' which helps to stabilise the solution. In addition, the local interpolation itself satisfies the linearised governing equation, allowing for more accurate reconstruction of partial derivatives, and hence a more accurate solution. The procedure is tested using a 3D infiltration problem with a known analytical solution. The performance of the LHI method, both with and without local PDE interpolation, is compared to the Finite Element method via the FEMWATER software. The LHI formulation with PDE data centres shows consistent improvement over the FEMWATER solutions, reducing errors by several orders of magnitude.In addition, a procedure is introduced to model strongly heterogeneous and layered soils. The physically correct matching conditions are applied over layer interfaces, i.e. continuity of pressure and mass flux. The 'double collocation' property of the Hermitian RBF method is exploited to enforce both matching conditions at the same set of locations on the layer interface. This procedure allows the accurate capture of solutions across such interfaces, replicating the required discontinuities in the first derivatives of the pressure profile. The multi-layer formulation is validated using a transient two-layer infiltration problem, with the analytical solution replicated to a high precision in a variety of configurations.

Implicit Solar Coronal Magnetohydrodynamic (MHD) Modeling with a Low-dissipation Hybridized AUSM-HLL Riemann Solver

“…Besides, we reconstruct the magnetic field by a divergencefree radial basic function (RBF) to enhance the divergence-free constraint of the magnetic field. The RBF interpolation technique was first proposed in Kansa (1990), and has been widely used to solve partial differential equations in a finite difference frame (e.g., Lai et al 2008Lai et al , 2010Stevens & Power 2011;Yensiri & Skulkhu 2017;Shankar et al 2018;Zamolo et al 2019) and a finite volume frame (e.g., Moroney & Turner 2006Liu et al 2016Liu et al , 2017Hesthaven & Mönkeberg 2021). These RBFs have been used to interpolate scalar fields.…”

Identification of Parameters of Richards Equation Using Modified Hybrid Grey Wolf Optimizer-Genetic Algorithm (HmGWOGA)

“…The idea is to reconstruct a function defined on a continuous space from the set of discrete values taken by this function on a not connected point cloud of the physical domain. Stevens and Power [23] used a implicit RBF method to solve the pressure h formulation of Richards equation. More recently, F. Motaman and al.…”

Finite Element Model of Soil Water and Nutrient Transport with Root Uptake: Explicit Geometry and Unstructured Adaptive Meshing

“…The new variable p is called the generalized pressure. Previous applications of the Kirchhoff transformation to Richards equation can be found for example in [20,25,12,23].…”

“…Following the approach suggested in [4,23], the solution at each time step is obtained iteratively; applying Newton's method to linearize M(p n+1 ) gives the following Newton-like iteration where i is the inner iteration counter for time n + 1:…”