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 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 Energy-corrected FEM and explicit time-stepping for parabolic problems

2019 ◽  
Vol 53 (6) ◽  
pp. 1893-1914
Author(s):  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Computational Domain ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

Semi-Lagrangian advection models for quasi-uniform nodes on the sphere

2021 ◽  
Author(s):  
Takeshi Enomoto ◽  
Koji Ogasawara
Keyword(s):  
Radial Basis Functions ◽  
Minimum Energy ◽  
Time Integration ◽  
Basis Functions ◽  
Lagrangian Model ◽  
Time Step ◽  
Interpolation Matrix ◽  
Radial Basis ◽  
The Matrix ◽  
Lagrangian Advection

<p>Radial basis functions enable the use of unstructured quasi-uniform nodes on the sphere. Iteratively generated nodes such as the minimum energy nodes may not converge due to exponentially increasing local minima as the number of nodes grows. By contrast, deterministic nodes, such as those made with a spherical helix, are fast to generate and have no arbitrariness. It is noteworthy that the spherical helix nodes are more uniform on the sphere than the minimum energy nodes. Semi-Lagrangian and Eulerian models are constructed using radial basis functions and validated in a standard advection test of a cosine bell by the solid body rotation. With Gaussian radial basis functions, the semi-Lagrangian model found produces significantly smaller error than the Eulerian counterpart in addition to approximately three times longer time step for the same error. Moreover, the ripple-like noise away from the cosine bell found in the Eulerian model is significantly reduced in the semi-Lagrangian model. It is straightforward to parallelize the matrix–vector multiplication in the time integration. In addition, an iterative solver can be applied to calculate the inverse of the interpolation matrix, which can be made sparse.</p>

scholarly journals Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions

2019 ◽  
Vol 60 (1) ◽  
pp. 157-197 ◽  
Author(s):  
Tobias Jawecki ◽  
Winfried Auzinger ◽  
Othmar Koch
Keyword(s):  
Error Bounds ◽  
Time Integration ◽  
Matrix Exponential ◽  
Selection Strategy ◽  
Step Size ◽  
Time Step ◽  
A Posteriori Estimate ◽  
Krylov Space ◽  
The Matrix ◽  
Matrix Exponentials

Abstract An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $$\varphi $$φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

scholarly journals Statistical Uncertainty of DNS in Geometries without Homogeneous Directions

2021 ◽  
Vol 11 (4) ◽  
pp. 1399
Author(s):  
Jure Oder ◽  
Cédric Flageul ◽  
Iztok Tiselj
Keyword(s):  
Channel Flow ◽  
A Priori ◽  
Time Integration ◽  
Navier Stokes ◽  
Statistical Uncertainty ◽  
Time Step ◽  
Order Of Magnitude ◽  
Averaging Time ◽  
The Individual ◽  
Time Required

In this paper, we present uncertainties of statistical quantities of direct numerical simulations (DNS) with small numerical errors. The uncertainties are analysed for channel flow and a flow separation case in a confined backward facing step (BFS) geometry. The infinite channel flow case has two homogeneous directions and this is usually exploited to speed-up the convergence of the results. As we show, such a procedure reduces statistical uncertainties of the results by up to an order of magnitude. This effect is strongest in the near wall regions. In the case of flow over a confined BFS, there are no such directions and thus very long integration times are required. The individual statistical quantities converge with the square root of time integration so, in order to improve the uncertainty by a factor of two, the simulation has to be prolonged by a factor of four. We provide an estimator that can be used to evaluate a priori the DNS relative statistical uncertainties from results obtained with a Reynolds Averaged Navier Stokes simulation. In the DNS, the estimator can be used to predict the averaging time and with it the simulation time required to achieve a certain relative statistical uncertainty of results. For accurate evaluation of averages and their uncertainties, it is not required to use every time step of the DNS. We observe that statistical uncertainty of the results is uninfluenced by reducing the number of samples to the point where the period between two consecutive samples measured in Courant–Friedrichss–Levy (CFL) condition units is below one. Nevertheless, crossing this limit, the estimates of uncertainties start to exhibit significant growth.

scholarly journals Regional Associations of Cortical Thickness With Gait Variability: The Tasmanian Study of Cognition and Gait

2020 ◽  
Vol 4 (Supplement_1) ◽  
pp. 232-233
Author(s):  
Oshadi Jayakody ◽  
Monique Breslin ◽  
Richard Beare ◽  
Velandai Srikanth ◽  
Helena Blumen ◽  
...  
Keyword(s):  
Cortical Thickness ◽  
Step Length ◽  
Gait Variability ◽  
Thickness Ratio ◽  
Sensory Function ◽  
Time Variability ◽  
Step Width ◽  
Time Step ◽  
Cortical Regions ◽  
Regional Associations

Abstract Gait variability is a marker of cognitive decline. However, there is limited understanding of the cortical regions associated with gait variability. We examined associations between regional cortical thickness and gait variability in a population-based sample of older people without dementia. Participants (n=350, mean age 71.9±7.1) were randomly selected from the electoral roll. Variability in step time, step length, step width and double support time (DST) were calculated as the standard deviation of each measure, obtained from the GAITRite walkway. MRI scans were processed through FreeSurfer to obtain cortical thickness of 68 regions. Bayesian regression was used to determine regional associations of mean cortical thickness and thickness ratio (regional thickness/overall mean thickness) with gait variability. Smaller overall cortical thickness was only associated with greater step width and step time variability. Smaller mean thickness in widespread regions important for sensory, cognitive and motor functions were associated with greater step width and step time variability. In contrast, smaller thickness in a few frontal and temporal regions were associated with DST variability and the right cuneus was associated with step length variability. Smaller thickness ratio in frontal and temporal regions important for motor planning, execution and sensory function and, greater thickness ratio in the anterior cingulate was associated with greater variability in all measures. Examining individual cortical regions is important in understanding the relationship between gray matter and gait variability. Cortical thickness ratio highlights that smaller regional thickness relative to global thickness may be important for the consistency of gait.

scholarly journals Coupled THM and Matrix Stability Modeling of Hydroshearing Stimulation in a Coupled Fracture-Matrix Hot Volcanic System

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Xiao ◽  
Jianchun Guo ◽  
Hehua Wang ◽  
Lize Lu ◽  
John McLennan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conduction ◽  
Dominant Role ◽  
Injection Well ◽  
Time Step ◽  
Volcanic System ◽  
Induced Stress ◽  
The Matrix ◽  
Matrix Stability ◽  
The Stability

A coupled thermal-hydraulic-mechanical (THM) model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracture-matrix hot volcanic reservoir system. Fluid flows in the fracture are calculated based on the cubic law. Heat transfer within the fracture involved is thermal conduction, thermal advection, and thermal dispersion; within the reservoir matrix, thermal conduction is the only mode of heat transfer. In view of the expansion of the fracture network, deformation and thermal-induced stress model are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix. A series of results from the coupled THM model, induced stress, and matrix stability indicate that thermal-induced aperture plays a dominant role near the injection well to enhance the conductivity of the fracture. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. The induced stress has the maximum value at the injection point; the deformation-induced stress has large value with smaller affected range; on the contrary, thermal-induced stress has small value with larger affected range. Matrix stability simulation results indicate that the stability of the matrix nodes may be destroyed; this mechanism is helpful to create complex fracture networks.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

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