Mass Conservative Method Research Articles

Modelling flow in the middle and lower Yangtze River, China

A one-dimensional numerical model was developed to simulate the hydrodynamic processes in a large and complex river network located in the middle and lower Yangtze River basin, China. The main stem of the river is connected to two large lakes, Dongting and Poyang, through many branches. A mass conservation method was used to determine the lateral inflows from local basins along the river. The model was verified against measured data at a large number of hydrological stations over a 2-year period. The water levels and discharges predicted by the model were found to generally agree well with the measured data. The model was then used to reproduce the flow process during a large flood event that occurred in 1998. It was found that the two large lakes could store a large volume of water and thus played an important role in attenuating the flood impact.

Joint Analysis of Cortical Area and Thickness as a Replacement for the Analysis of the Volume of the Cerebral Cortex.

Cortical surface area is an increasingly used brain morphology metric that is ontogenetically and phylogenetically distinct from cortical thickness and offers a separate index of neurodevelopment and disease. However, the various existing methods for assessment of cortical surface area from magnetic resonance images have never been systematically compared. We show that the surface area method implemented in FreeSurfer corresponds closely to the exact, but computationally more demanding, mass-conservative (pycnophylactic) method, provided that images are smoothed. Thus, the data produced by this method can be interpreted as estimates of cortical surface area, as opposed to areal expansion. In addition, focusing on the joint analysis of thickness and area, we compare an improved, analytic method for measuring cortical volume to a permutation-based nonparametric combination (NPC) method. We use the methods to analyze area, thickness and volume in young adults born preterm with very low birth weight, and show that NPC analysis is a more sensitive option for studying joint effects on area and thickness, giving equal weight to variation in both of these 2 morphological features.

Regionalização da vazão Q95: comparação de métodos para a bacia hidrográfica do Rio Taquari-Antas, RS

A baixa densidade espacial da rede hidrométrica nacional tem resultado em grandes limitações para a representação do comportamento dos recursos hídricos. Dessa forma, a regionalização de vazões é uma alternativa para melhor explorar os dados já existentes, transferindo informações de um local para outro dentro de uma área com comportamento hidrológico semelhante. Nesse sentido, o objetivo deste trabalho foi avaliar as estimativas da vazão mínima de permanência Q95 obtidas através dos métodos Tradicional e Conservação de Massas na bacia do Rio Taquari-Antas, pertencente ao Estado do Rio Grande do Sul, visando auxiliar o planejamento dos recursos hídricos, além de facilitar as tomadas de decisão em relação aos processos de outorga do direito de uso das águas. Como variável explicativa para a regressão matemática foi utilizada a área de drenagem proveniente do Modelo Digital de Elevação Hidrograficamente Condicionado (MDEHC). Foram obtidas as equações de regionalização da vazão de referência Q95 e as respectivas regiões hidrologicamente homogêneas para os dois métodos testados, sendo a magnitude dos erros relativos inerente às metodologias empregadas. Consideradas as particularidades deste trabalho, ambos os métodos avaliados apresentaram resultados satisfatórios na obtenção de modelos para estimativa da vazão Q95, entretanto destaca-se que as técnicas aplicadas não substituem as informações hidrológicas obtidas com uma rede hidrométrica suficientemente densa.

Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes

In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn–Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier–Stokes equation – for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh–Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10−8 and energy decreases in time.

ZLF (Zero Lateral Flux): a simple mass conservation method for semi‐Lagrangian‐based limited‐area models

In this paper we outline a simple method to achieve mass conservation with semi‐Lagrangian‐based limited‐area models (SL‐LAMs). Imposing mass conservation with LAMs requires knowledge of advective fluxes across their lateral boundaries. Estimation of these fluxes within a semi‐Lagrangian scheme can be problematic. Methods have been devised to achieve this, however these require extra interpolations which increase the computational cost of the model. By exploiting the way these models are designed and under some mild assumptions, a simple scheme is proposed which avoids computing the lateral fluxes altogether. The scheme achieves the same objectives as the earlier methods, but with minimal extra computational cost. This scheme is validated in a simple one‐dimensional problem and then implemented in a full atmospheric model to show its effectiveness.

Detailed numerical simulation of swirling primary atomization using a mass conservative level set method

We report detailed numerical simulations of swirling liquid atomization. A recently developed mass conservative level set method is employed to capture the gas–liquid interface and a ghost fluid method is utilized to deal with the jump conditions across the interface. The swirl and atomization characteristics of two-phase annular swirling jets with the influence of turbulent inflow are investigated. Through comparing the sheet thickness, the breakup length and the cone angle, the numerical convergence of the global characteristics of the swirling two phase flow has been obtained. The numerical results show that turbulent inflow can induce liquid sheet breakup near the nozzle exit, reduce the stiffness of the liquid sheet, and lead to the statistically homogeneous distribution of small-scale liquid structures in the radial direction. Compared with the single-phase jet, the two-phase jet exhibits the chaotic velocity filed downstream that can enhance the mixing of droplets and ambient gas, and the precessing vortex core (PVC) is not observed in the center of the two-phase jet. In addition, the recirculation zone is smaller and farther from the nozzle exit for the turbulent inflow case than that from the laminar inflow case, and the preferential alignment of ωi with the intermediate strain rate indicates that the fluctuating velocity in the recirculation zone is statistically similar to isotropic turbulence. The interaction of the liquid-gas interface and vortices shows the preferential normal alignment of the vorticity and the normal of the interface, and the liquid sheet can generate high shear layers to produce anisotropic small-scale fluctuations.

Numerical modeling of subsidence in saturated porous media: A mass conservative method

In this paper, a second order accurate cell-centered finite volume method (FVM) is coupled with a finite element method (FEM) to solve the deformation of a saturated porous layer based on Biot’s consolidation model. The proposed numerical technique is applied to the fully unstructured triangular grids to simulate actual geological formations. To reconstruct the pressure gradient at control volume faces, the diamond scheme is implemented as a multipoint flux approximation method. Also the least square algorithm is used to interpolate pressure at the vertices from the cell-center values. The stability of this numerical model is studied in comparison to the different FEMs through various examples. It is shown that, although the Taylor-Hood FEM has been introduced as a remedy for violation of the inf-sup condition, it does not entirely remove the non-physical oscillations. Contrary to the linear and Taylor-Hood FEMs, the proposed discretization model provides monotonic solution without imposing any restriction on the mesh or time step size. Compared to the mixed FEM, the method achieves local mass balance with fewer degrees of freedom. To couple the flow and mechanical sub-problems, the fixed-stress operator split is implemented as an iterative sequential method, due to its unconditional stability, accuracy and high rate of convergence. The accuracy of the proposed model is verified via a range of examples including analytical and numerical solutions. The performance of this methodology is assessed through modeling of subsidence in an aquifer-interbed system. This problem illustrates the capability of the model in providing stable solution in heterogeneous domains with complicated shapes.

High-order and mass conservative methods for the conservative Allen–Cahn equation

The conservative Allen–Cahn (AC) equation has been studied analytically and numerically. Our mathematical analysis and numerical experiment, however, show that previous numerical methods are not second-order accurate in time and/or do not conserve the initial mass. The aim of this paper is to propose high-order and mass conservative methods for solving the conservative AC equation. In the methods, we discretize the conservative AC equation by using a Fourier spectral method in space and first-, second-, and third-order implicit–explicit Runge–Kutta schemes in time. We show that the methods inherit the mass conservation. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.

Mixed GMsFEM for second order elliptic problem in perforated domains

We consider a class of second order elliptic problems in perforated domains with homogeneous Neumann boundary condition. It is well-known that numerically solving these problems require a very fine computational mesh, and some model reduction techniques are therefore necessary. We will develop a new model reduction technique based on the generalized multiscale finite element method (GMsFEM). The GMsFEM has been applied successfully to second order elliptic problems in perforated domains with Dirichlet boundary conditions Chung et al. (2015). However, due to the use of multiscale partition of unity functions, the same method cannot be applied to the case with Neumann boundary conditions. The aim of this paper is to develop a new mixed GMsFEM, based on a piecewise constant approximation for pressure and a multiscale approximation for velocity, giving a mass conservative method. The method can handle the Neumann boundary condition naturally. The multiscale basis functions for velocity are constructed by some carefully chosen local snapshot spaces and local spectral decompositions. The spectral convergence of the method is analyzed. Moreover, by using some local error indicators, the basis functions can be added locally and adaptively. We also consider an online procedure for the construction of new basis functions in the online stage in order to capture the distant effects. We will present some numerical examples to show the performance of the method.

Multiscale Simulations for Coupled Flow and Transport Using the Generalized Multiscale Finite Element Method

In this paper, we develop a mass conservative multiscale method for coupled flow and transport in heterogeneous porous media. We consider a coupled system consisting of a convection-dominated transport equation and a flow equation. We construct a coarse grid solver based on the Generalized Multiscale Finite Element Method (GMsFEM) for a coupled system. In particular, multiscale basis functions are constructed based on some snapshot spaces for the pressure and the concentration equations and some local spectral decompositions in the snapshot spaces. The resulting approach uses a few multiscale basis functions in each coarse block (for both the pressure and the concentration) to solve the coupled system. We use the mixed framework, which allows mass conservation. Our main contributions are: (1) the development of a mass conservative GMsFEM for the coupled flow and transport; (2) the development of a robust multiscale method for convection-dominated transport problems by choosing appropriate test and trial spaces within Petrov-Galerkin mixed formulation. We present numerical results and consider several heterogeneous permeability fields. Our numerical results show that with only a few basis functions per coarse block, we can achieve a good approximation.

Grounding line retreat of Totten Glacier, East Antarctica, 1996 to 2013

AbstractTotten Glacier, East Antarctica, a glacier that holds a 3.9 m sea level change equivalent, has thinned and lost mass for decades. We map its grounding line positions in 1996 and 2013 using differential radar interferometry (InSAR) data and develop precise, high‐resolution topographies of its ice surface and ice draft using NASA Operation IceBridge data, InSAR data, and a mass conservation method. We detect a 1 to 3 km retreat of the grounding line in 17 years. The retreat is asymmetrical along a two‐lobe pattern, where ice is only grounded a few 10 m above sea level, or ice plain, which may unground further with only modest amounts of ice thinning. The pattern of retreat indicates ice thinning of 12 m in 17 years or 0.7±0.1 m/yr at the grounding line on average. Sustained thinning will cause further grounding line retreat but may not be conducive to a marine instability.

A comparison of finite volume formulations and coupling strategies for two-phase flow in deforming porous media

In this paper a locally mass conservative finite volume method is employed to model the one-dimensional, two-phase immiscible flow in a poroelastic media. Since, an appropriate choice of primary variables is critical in simulating multiphase subsurface flow, depending on such a choice, the governing equations can be expressed in different forms. By implementing Picard iteration to a highly nonlinear system of equations, three numerical models including pressure form, mixed form and mixed form with a modified Picard linearization are developed in this study. These models have been evaluated in terms of stability, convergence and mass conservation in various one-dimensional test cases. Selecting water saturation in the mixed form as a primary variable, which is not frequent in the geotechnical engineering, could produce convergence problems in transition from saturated to unsaturated regimes, but in other conditions show good convergence and also mass balance properties. The pressure form and the mixed form with a modified Picard linearization converge in all test cases even near the fully saturated conditions. The pressure form suffers from poor mass balance and the mixed form with a modified Picard linearization poses superior mass balance property than the pressure form. In order to solve the coupled multiphase flow and geomechanics, two coupling strategies are used, first the fully coupled approach and second the iterative algorithm based on the fixed-stress operator split. Comparison between the total number of iterations and the total execution time of the fully coupled method and the fixed-stress schemes are presented through different one-dimensional examples. The accuracy, robustness and efficiency of the fixed-stress method have been demonstrated due to the reduced CPU time and low values of error for different variables.

A new posteriori error estimates for a discontinuous Galerkin method for a reaction–diffusion problem

In this paper, a new posteriori error estimates and an error indicator for locally mass conservative methods for subsurface flow are given. The methods are based on discontinuous approximation spaces, and the penalty terms on the normal fluxes across the element interfaces are added to the bilinear form to obtain the nonsymmetric interior penalty discontinuous Galerkin methods. The global explicit estimates of the error in the L2 norm are derived, and an implicit indicator of the error in the energy norm is numerically studied.

Hydrochemical characteristics and genesis analysis of the Jifei hot spring in Yunnan, southwestern China

The Jifei hot spring is a typical travertine-depositing hot spring with many travertine cones. The spring water is meteoric in origin and is heated when it undergoes deep circulation of 1792m through the well-developed faults. Based on the hydrochemical and isotopic analysis, the estimated residence time of the deep circulating meteoric water is 35–50 years and the estimated reservoir temperature of the thermal groundwater is approximate 110.9°C. The mixing ratio of the cold groundwater is around 44% in the Jifei hot spring calculated by mass conservation method of enthalpy and silica dioxide concentration. Finally, a conceptual diagram of the genesis of the Jifei hot spring is given. In addition, it has been inferred that the travertine deposited from the beginning of the Holocene Epoch.

A multiple-time-step technique for coupled free flow and porous medium systems

A mass conservative multirate time integration method is developed for a time-dependent isothermal single-phase free flow system coupled with an isothermal single-fluid-phase porous medium system. Long time stability of the proposed numerical scheme is proved under a time step restriction which depends on the physical parameters of the flow systems and the ratio between the time steps applied in the free flow and porous medium domains. Convergence analysis is carried out and a priori error estimates are obtained. Numerical results are presented for two model problems and a realistic setup. Advantages of using multiple-time-step techniques for modeling coupled systems where the processes evolve on different time scales are demonstrated and discussed.

Optimal control for mass conservative level set methods

This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach.

Development of a cross-section based streamflow routing package for MODFLOW

It is challenging to simulate stream-aquifer interactions for the wide channel streams with the existing stream routing packages of MODFLOW. To overcome this limitation, a Cross-Section streamflow Routing (CSR) package is developed to simulate the streamflow and the interaction between streams and aquifers for the stream with a width larger than the MODFLOW grid size. In the CSR package, streams are divided into stream segments which are formed by two consecutive cross-sections. A cross-section is described by a number of streambed points that determine the geometry and hydraulic properties of the streambed. The stream water depth and streamflow at the cross-sections are related by the Single Channel method, the Divided Channel method, a data table or a power function. A rapid algorithm is used to compute the submerged area of the MODFLOW grid. The streambed conductance of a grid cell is computed based on its submerged area, streambed hydraulic conductivity and thickness. Stream-aquifer seepage is subsequently estimated as the product of the streambed conductance and difference between the stream stage and groundwater hydraulic head. Stream-aquifer seepage is treated as lateral flow in the streamflow routing computation with the Muskingum-Cunge method or mass conservation method. A hypothetical problem is established to test the capabilities of the CSR package with steady- and transient-state models. The results compare favorably with the SFR2 package and the HEC-RAS model. However, significant difference in flood wave attenuation is observed between the CSR package and the SFR2 package. It proves that the CSR package is capable of simulating the variation of stream-aquifer interactions in both space and time efficiently. The CSR package represents a certain improvement over previous MODFLOW streamflow packages by providing the efficient cross-section based computation and the unique capability of simulating streambed heterogeneity in longitudinal and transverse directions.

Convergence analysis of a coupled method for time‐dependent convection‐diffusion equations

The coupling of two locally mass conservative methods is formulated and analyzed for the time‐dependent convection‐diffusion problem. Finite volume method is used in some subdomains and interior penalty discontinuous Galerkin method is used in other subdomains. Numerical examples show the advantages of the proposed hybrid method, namely an accurate approximation obtained at a reduced computational cost. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 133–157, 2014

An Extension of the Talbot–Ogden Hydrology Model to an Affine Multi-Dimensional Moisture Content Domain

The Talbot–Ogden hydrology model provides a fast mass conservative method to compute infiltration in unsaturated soils. As a replacement for a model based on Richards equation, it separates the groundwater movement into infiltration and redistribution for every time step. The typical feature making this method fast is the discretization of the moisture content domain rather than the spatial one. The Talbot-Ogden model rapidly determines how well ground water and aquifers are recharged only. Hence, it differs from models based on advanced reservoir modeling that are uniformly far more expensive computationally since they determine where the water moves in space instead, a completely different and more complex problem. According to the pore-size distribution curve for many soils, this paper extends the one dimensional moisture content domain into a two dimensional one by keeping the vertical spatial axis. The proposed extension can describe any pore-size or porosity distribution as an important soil feature. Based on this extension, infiltration and redistribution are restudied. The unconditional conservation of mass in the Talbot–Ogden model is inherited in this extended model. A numerical example is given for the extended model.

Measuring and comparing brain cortical surface area and other areal quantities

Structural analysis of MRI data on the cortical surface usually focuses on cortical thickness. Cortical surface area, when considered, has been measured only over gross regions or approached indirectly via comparisons with a standard brain. Here we demonstrate that direct measurement and comparison of the surface area of the cerebral cortex at a fine scale is possible using mass conservative interpolation methods. We present a framework for analyses of the cortical surface area, as well as for any other measurement distributed across the cortex that is areal by nature. The method consists of the construction of a mesh representation of the cortex, registration to a common coordinate system and, crucially, interpolation using a pycnophylactic method. Statistical analysis of surface area is done with power-transformed data to address lognormality, and inference is done with permutation methods. We introduce the concept of facewise analysis, discuss its interpretation and potential applications.