1,721,189 research outputs found
Physically based modeling in catchment hydrology at 50: Survey and outlook
Full text linkIntegrated, process-based numerical models in hydrology are rapidly evolving, spurred by novel theories in mathematical physics, advances in computational methods, insights from laboratory and field experiments, and the need to better understand and predict the potential impacts of population, land use, and climate change on our water resources. At the catchment scale, these simulation models are commonly based on conservation principles for surface and subsurface water flow and solute transport (e.g., the Richards, shallow water, and advection-dispersion equations), and they require robust numerical techniques for their resolution. Traditional (and still open) challenges in developing reliable and efficient models are associated with heterogeneity and variability in parameters and state variables; nonlinearities and scale effects in process dynamics; and complex or poorly known boundary conditions and initial system states. As catchment modeling enters a highly interdisciplinary era, new challenges arise from the need to maintain physical and numerical consistency in the description of multiple processes that interact over a range of scales and across different compartments of an overall system. This paper first gives an historical overview (past 50 years) of some of the key developments in physically based hydrological modeling, emphasizing how the interplay between theory, experiments, and modeling has contributed to advancing the state of the art. The second part of the paper examines some outstanding problems in integrated catchment modeling from the perspective of recent developments in mathematical and computational science
Mesh Locking Effects in the Finite Volume Solution of 2-D Anisotropic Diffusion Equations
No full textStrongly anisotropic diffusion equations require special techniques to overcome or reduce the mesh locking phenomenon. We present a finite volume scheme that tries to approximate with the best possible accuracy the quantities that are of importance in discretizing anisotropic fluxes. In particular, we discuss the crucial role of accurate evaluations of the tangential components of the gradient acting tangentially to the control volume boundaries, that are called into play by anisotropic diffusion tensors. To obtain the sought characteristics from the proposed finite volume method, we employ a second-order accurate reconstruction scheme which is used to evaluate both normal and tangential cell-interface gradients. The experimental results on a number of different meshes show that the scheme maintains optimal convergence rates in both L(2) and H(1) norms except for the benchmark test considering full Neumann boundary conditions on non-uniform grids. In such a case, a severe locking effect is experienced and documented. However, within the range of practical values of the anisotropy ratio, the scheme is robust and efficient. We postulate and verify experimentally the existence of a quadratic relationship between the anisotropy ratio and the mesh size parameter that guarantees optimal and sub-optimal convergence rate
High Order Godunov Mixed Methods on tetrahedral meshes for density driven flow simulations in porous media
No full textTwo-dimensional Godunov mixed methods have been shown to be effective for the numerical solution of density-dependent flow and transport problems in groundwater even when concentration gradients are high and the process is dominated by density effects. This class of discretization approaches solves the flow equation by means of the mixed finite element method, thus guaranteeing mass conserving velocity fields, and discretizes the transport equation by mixed finite element and finite volumes techniques combined together via appropriate time splitting. In this paper, we extend this approach to three dimensions employing tetrahedral meshes and introduce a spatially variable time stepping procedure that improves computational efficiency while preserving accuracy by adapting the time step size according to the local Courant-Friedrichs-Lewy (CFL) constraint. Careful attention is devoted to the choice of a truly three-dimensional limiter for the advection equation in the time-splitting technique, so that to preserve second order accuracy in space (in the sense that linear functions are exactly interpolated). The three-dimensional Elder problem and the salt-pool problem, recently introduced as a new benchmark for testing three-dimensional density models, provide assessments with respect to accuracy and reliability of this numerical approach
Investigation of extension of High Order Finite Volume Schemes from triangles to tetrahedra
No full textThree-dimensional mixed finite element-finite volume approach for the solution of density-dependent flow in porous media
No full textThe density dependent flow and transport problem in groundwater
on three dimensional triangulations is solved numerically by
means of a Mixed Hybrid Finite Element scheme for the flow
equation combined
with a Mixed Hybrid Finite Element-Finite Volume (MHFE-FV)
time-splitting based technique
for the transport equation. This procedure is analyzed and shown to be an
effective tool in particular when the process is advection dominated
or when density variations induce the formation of instabilities
in the flow field.
From a computational point of view, the most effective strategy
turns out to be a combination of the MHFE and a
spatially variable time splitting technique in which the FV scheme
is given by a second order linear reconstruction based on the least
square minimization and the Barth-Jespersen limiter.
The recent saltpool problem introduced as a benchmark test for
density dependent solvers is used to
verify the accuracy and reliability of this approach
Parallel finite element Laplace transform method for the non-equilibrium groundwater transport equation
No full textGroundwater transport of contaminants undergoing rate-limited or non-equilibrium sorption onto the solid matrix is often described by the dual-porosity or two-domain model, whereby the rate-limited reaction occurs between a mobile and an immobile region. When the sorption reaction is represented by a first-order kinetic relationship, the equation takes the form of a convection–dispersion partial differential equation with an integral term describing the mass transfer between the two regions. An efficient solution algorithm for this type of problems consists in the transformation of the original equation into the Laplace space and subsequent numerical solution of the resulting steady-state equation in the complex space. The exploitation of the Laplace transform to solve the time dependency restricts the application of the technique to linear advection and dispersion terms, while non-linear reactions can be accommodated in particular cases only. This approach has the advantage that it is easily parallelizable, and is therefore proposed in this paper as an efficient algorithm for the parallelization in time of these types of integrodifferential equations.
The parallel efficiency of PFELT has been tested on a Cray T3D parallel computer for three sample problems of size N=1071, 3721, and 15 275, respectively, where N is the number of nodal mesh points. The speed-ups obtained vary from 1·98, with two processors, to 39·93 with 64 processors, for the most favorable case. This corresponds to a percentage of parallel work greater than 98 per cent, and a parallel efficiency of more than 60 per cent in the best case, showing the good performance achievable with this algorith
Numerical comparison of iterative eigensolvers for large sparse symmetric matrices
No full textThe Jacobi-Davidson (JD) method has been recently proposed
for the evaluation of the partial eigenspectrum of large sparse
matrices. In this work we report a set of numerical experiments
that compare this method with other previously
proposed techniques; DACG (Deflation Accelerated Conjugate Gradient)
and Lanczos (ARPACK), on large sparse symmetric matrices.
The results obtained by JD and DACG are benchmarked against
those obtained with ARPACK in terms ofcomputational time for the evaluation of a number of the leftomost eigenpairs
of large and sparse matrice
Finite element approximation of the diffusion operator on tetrahedra
Full text linkLinear Galerkin finite element discretizations of the Laplace operator produce nonpositive stiffness coefficients for internal element edges of two-dimensional Delaunay triangulations. This property, also called the positive transmissibility (PT) condition, is a prerequisite for the existence of an M-matrix and ensures that nonphysical local extrema are not present in the solution. For tetrahedral elements, it has already been shown that the linear Galerkin approach does not in general satisfy the PT condition. We propose a modification of the three-dimensional Galerkin scheme that, if a Delaunay triangulation is used, satisfies the PT condition for internal edges and, if further conditions on the boundary are specified, yields an M-matrix. The proposed approach can also be extended to the general diffusion operator with nonconstant or anisotropic coefficients
Accuracy of Galerkin finite elements for groundwater flow simulations in two and three-dimensional triangulations
No full textIn standard finite element simulations of groundwater flow the correspondence between hydraulic head gradients and groundwater fluxes is represented by the stiffness matrix. In two-dimensional problems the use of linear triangular elements on Delaunay triangulations guarantees a stiffness matrix of type M. This implies that the local numerical fluxes are physically consistent with Darcy's law. This condition is fundamental to avoid the occurrence of local maxima or minima, and is of crucial importance when the calculated flow field is used in contaminant transport simulations or pathline evaluation. In three spatial dimensions, the linear Galerkin approach on tetrahedra does not lead to M-matrices even on Delaunay meshes. By interpretation of the Galerkin approach as a subdomain collocation scheme, we develop a new approach (OSC, orthogonal subdomain collocation) that is shown to produce M-matrices in three-dimensional Delaunay triangulations. In case of heterogeneous and anisotropic coefficients, extra mesh properties required for M-stiffness matrices will also be discussed
Mixed Finite Elements and Newton-type Linearization for the Solution of Richard's Equation
No full textWe present the development of a two-dimensional Mixed-Hybrid Finite Element (MHFE) model for the solution of the non-linear equation of variably saturated flow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest-order Raviart-Thomas (RT0) elements and is 'exactly' mass conserving. Hybridization is used to overcome the ill-conditioning of the mixed system. The scheme is globally first-order in space. Nevertheless, numerical results employing non-uniform meshes show second-order accuracy of the pressure head and normal fluxes on specific grid points. The non-linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE-Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation technique
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