Schwarz type domain decomposition and subcycling multi-time step approach for solving Richards equation

2016 ◽  
Author(s):  
Michal Kuraz
Keyword(s):  
Domain Decomposition ◽  
Richards Equation ◽  
Time Step

scholarly journals Algorithms for Solving Darcian Flow in Structured Porous Media

10.14311/1829 ◽  
2013 ◽  
Vol 53 (4) ◽  
Author(s):  
Michal Kuráž ◽  
Petr Mayer
Keyword(s):  
Proper Time ◽  
Time Integration ◽  
Mass Term ◽  
Step Length ◽  
Richards Equation ◽  
Parabolic Problems ◽  
Computational Domain ◽  
Wetting Front ◽  
Time Step ◽  
The Matrix

This paper presents several algorithms that were implemented in DRUtES [1], a new open source project. DRUtES is a finite element solver for coupled nonlinear parabolic problems, namely the Richards equation with the dual porosity approach (modeling the flow of liquids in a porous medium). Mass balance consistency is crucial in any hydrological balance and contaminant transportation evaluations. An incorrect approximation of the mass term greatly depreciates the results that are obtained. An algorithm for automatic time step selection is presented, as the proper time step length is crucial for achieving accuracy of the Euler time integration method. Various problems arise with poor conditioning of the Richards equation: the computational domain is clustered into subregions separated by a wetting front, and the nonlinear constitutive functions cover a high range of values, while a very simple diagonal preconditioning method greatly improves the matrix properties. The results are presented here, together with an analysis.

scholarly journals Domain Decomposition Modified with Characteristic Mixed Finite Element and Numerical Analysis for Three-Dimensional Slightly Compressible Oil-Water Seepage Displacement

2017 ◽  
Vol 9 (1) ◽  
pp. 143
Author(s):  
Yirang Yuan ◽  
Luo Chang ◽  
Changfeng Li ◽  
Tongjun Sun
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Method Of Characteristics ◽  
Mixed Finite Element ◽  
Three Dimensional ◽  
Optimal Error Estimate ◽  
Computational Domain ◽  
Time Step ◽  
Slightly Compressible ◽  
The Method Of Characteristics

A parallel algorithm is presented to solve three-dimensional slightly compressible seepage displacement where domain decomposition and characteristics-mixed finite element are combined. Decomposing the computational domain into several subdomains, we define a special function to approximate the derivative at interior boundary explicitly and obtain numerical solutions of the saturation implicitly on subdomains in parallel. The method of characteristics can confirm strong stability at the fronts, and can avoid numerical dispersion and nonphysical oscillation. It can adopt large-time step but can obtain small time truncation error. So a characteristic domain decomposition finite element scheme is put forward to compute the saturation. The flow equation is computed by the method of mixed finite element and numerical accuracy of Darcy velocity is improved one order. For a model problem we apply some techniques such as variation form, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, and the theory of priori estimates of differential equations to derive optimal error estimate in $l^2$ norm. Numerical example is given to testify theoretical analysis and numerical data show that this method is effective in solving actual applications. Then it can solve the well-known problem.

Implementation of 2D Domain Decomposition in the UCAN Gyrokinetic Particle-in-Cell Code and Resulting Performance of UCAN2

2016 ◽  
Vol 19 (1) ◽  
pp. 205-225 ◽  
Author(s):  
Jean-Noel G. Leboeuf ◽  
Viktor K. Decyk ◽  
David E. Newman ◽  
Raul Sanchez
Keyword(s):  
Domain Decomposition ◽  
Parallel Implementation ◽  
Three Dimensional ◽  
Massively Parallel ◽  
Problem Size ◽  
Time Step ◽  
Particle In Cell ◽  
Minor Radius ◽  
Efficient Charge ◽  
Cartesian Geometry

AbstractThe massively parallel, nonlinear, three-dimensional (3D), toroidal, electrostatic, gyrokinetic, particle-in-cell (PIC), Cartesian geometry UCAN code, with particle ions and adiabatic electrons, has been successfully exercised to identify non-diffusive transport characteristics in present day tokamak discharges. The limitation in applying UCAN to larger scale discharges is the 1D domain decomposition in the toroidal (or z-) direction for massively parallel implementation using MPI which has restricted the calculations to a few hundred ion Larmor radii or gyroradii per plasma minor radius. To exceed these sizes, we have implemented 2D domain decomposition in UCAN with the addition of the y-direction to the processor mix. This has been facilitated by use of relevant components in the P2LIB library of field and particle management routines developed for UCLA's UPIC Framework of conventional PIC codes. The gyro-averaging specific to gyrokinetic codes is simplified by the use of replicated arrays for efficient charge accumulation and force deposition. The 2D domain-decomposed UCAN2 code reproduces the original 1D domain nonlinear results within round-off. Benchmarks of UCAN2 on the Cray XC30 Edison at NERSC demonstrate ideal scaling when problem size is increased along with processor number up to the largest power of 2 available, namely 131,072 processors. These particle weak scaling benchmarks also indicate that the 1 nanosecond per particle per time step and 1 TFlops barriers are easily broken by UCAN2 with 1 billion particles or more and 2000 or more processors.

scholarly journals It rains and then? Numerical challenges with the 1D Richards equation in kilometer-resolution land surface modelling

2021 ◽  
Author(s):  
Daniel Regenass ◽  
Linda Schlemmer ◽  
Elena Jahr ◽  
Christoph Schär
Keyword(s):  
Surface Runoff ◽  
Land Surface ◽  
Richards Equation ◽  
Wetting Front ◽  
Time Step ◽  
Infiltration Process ◽  
Numerical Convergence ◽  
Surface Modelling ◽  
Regular Grids ◽  
Land Surface Modelling

Abstract. Over the last decade kilometer-scale weather predictions and climate projections have become established. Thereby both the representation of atmospheric processes, as well as land-surface processes need adaptions to the higher-resolution. Soil moisture is a critical variable for determining the exchange of water and energy between the atmosphere and the land surface on hourly to seasonal time scales, and a poor representation of soil processes will eventually feed back on the simulation quality of the atmosphere. Especially the partitioning between infiltration and surface runoff will feed back on the hydrological cycle. Several aspects of the coupled system are affected by a shift to kilometer-scale, convection-permitting models. First of all, the precipitation-intensity distribution changes to more intense events. Second, the time-step of the numerical integration becomes smaller. The aim of this study is to investigate the numerical convergence of the one-dimensional Richards Equation with respect to the soil hydraulic model, vertical layer thickness and time-step during the infiltration process. Both regular and non-regular (unequally spaced) grids typical in land surface modelling are considered, using a conventional semi-implicit vertical discretization. For regular grids, results from a highly idealized experiment on the infiltration process show poor numerical convergence for layer thicknesses larger than approximately 5 cm and for time steps greater than 40 s, irrespective of the soil hydraulic model. The velocity of the wetting front decreases systematically with increasing time step and decreasing vertical resolution. For non-regular grids, a new discretization based on a coordinate transform is introduced. In contrast to simpler vertical discretizations, it is able to represent the solution second-order accurate. The results for non-regular grids are qualitatively similar, as a fast increase in layer thickness with depth is equivalent to a lower vertical resolution. It is argued that the sharp gradients in soil moisture around the propagating wetting front must be resolved properly in order to achieve an acceptable numerical convergence of the Richards Equation. Furthermore, it is shown that the observed poor numerical convergence translates directly into a poor convergence of infiltration-runoff partitioning for precipitation time series characteristic of weather and climate models. As a consequence, soil simulations with low resolution in space and time may produce almost twice the amount of surface runoff within 24 hours than their high-resolution counterparts. Our analysis indicates that the problem is particularly pronounced for kilometer-resolution models.

scholarly journals Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed-form of Richards' equation

2016 ◽  
Author(s):  
Fadji Hassane Maina ◽  
Philippe Ackerer
Keyword(s):  
Adaptive Algorithm ◽  
Water Saturation ◽  
The State ◽  
Richards Equation ◽  
Time Step ◽  
Mixed Form ◽  
Convergence Error ◽  
Time Stepping ◽  
State Variable ◽  
Newton Raphson

Abstract. The solution of the mathematical model for flow in variably saturated porous media described by Richards equation (RE) is subject to heavy numerical difficulties due to its highly non-linear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed-form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the time step magnitude within the iteration procedure while the state variable is kept constant. The Ross method is an example of this type of scheme, and we show that it is equivalent to the Newton-Raphson method with a time-adaptive algorithm. Both algorithms are coupled to different time stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change of water saturation. Three different test cases are used to evaluate the efficiency of these algorithms. The numerical results highlight the necessity of implementing two types of errors: the iterative convergence error (maximum difference of the state variable between two iterations) and an estimate of the time truncation errors. The algorithms using these two types of errors together were found to be the most efficient when highly accurate results are required.

scholarly journals A Moving Mesh Finite Difference Method for Non-Monotone Solutions of Non-Equilibrium Equations in Porous Media

2017 ◽  
Vol 22 (4) ◽  
pp. 935-964 ◽  
Author(s):  
Hong Zhang ◽  
Paul Andries Zegeling
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Moving Mesh ◽  
Richards Equation ◽  
Time Step ◽  
Central Difference ◽  
Monitor Function ◽  
Non Equilibrium

AbstractAn adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure effect in porous media. One is the non-equilibrium Richards Equation and the other is the modified Buckley-Leverett equation. The governing equations are discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive monitor function with directional control is applied to redistribute the mesh grid in every time step, then a diffusive mechanism is used to smooth the monitor function. The behaviors of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction are investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, good mesh quality and high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the proposed method.

scholarly journals Stability Analysis of the Explicit Difference Scheme for Richards Equation

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 352
Author(s):  
Fengnan Liu ◽  
Yasuhide Fukumoto ◽  
Xiaopeng Zhao
Keyword(s):  
Difference Scheme ◽  
Numerical Experiments ◽  
Richards Equation ◽  
Induction Method ◽  
Explicit Difference Scheme ◽  
Time Step ◽  
The Difference ◽  
The Stability ◽  
Forward Euler ◽  
Parabolic Term

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

Improving the Numerical Solution of Soil Moisture–Based Richards Equation for Land Models with a Deep or Shallow Water Table

2009 ◽  
Vol 10 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Xubin Zeng ◽  
Mark Decker
Keyword(s):  
Boundary Condition ◽  
Soil Moisture ◽  
Water Table ◽  
Moisture Distribution ◽  
Richards Equation ◽  
Time Step ◽  
Bottom Boundary ◽  
Bottom Boundary Condition ◽  
Free Drainage ◽  
Soil Moisture Distribution

Abstract The soil moisture–based Richards equation is widely used in land models for weather and climate studies, but its numerical solution using the mass-conservative scheme in the Community Land Model is found to be deficient when the water table is within the model domain. Furthermore, these deficiencies cannot be reduced by using a smaller grid spacing. The numerical errors are much smaller when the water table is below the model domain. These deficiencies were overlooked in the past, most likely because of the more dominant influence of the free drainage bottom boundary condition used by many land models. They are fixed here by explicitly subtracting the hydrostatic equilibrium soil moisture distribution from the Richards equation. This equilibrium distribution can be derived at each time step from a constant hydraulic (i.e., capillary plus gravitational) potential above the water table, representing a steady-state solution of the Richards equation. Furthermore, because the free drainage condition has serious deficiencies, a new bottom boundary condition based on the equilibrium soil moisture distribution at each time step is proposed that also provides an effective and direct coupling between groundwater and surface water.

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