Annamaria Mazzia

On the Reliability of Numerical Solutions of Brine Transport in Groundwater: Analysis of Infiltration from a Salt Lake

The density dependent flow and transport problem in groundwater is solved numerically by means of a mixed finite element scheme for the flow equation and a mixed finite element-finite volume time-splitting based technique for the transport equation. The proposed approach, spatially second order accurate, is used to address the issue of grid convergence by solving on successively refined grids the salt lake problem, a physically unstable downward convection with formation of fingers. Numerical results indicate that achievement of grid convergence is problematic due to ill-conditioning arising from the strong nonlinearities of the mathematical model.

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.

Mixed-finite element and finite volume discretization for heavy brine simulations in groundwater

Recently, a new theory of high-concentration brine transport in groundwater has been developed. This approach is based on two nonlinear mass conservation equations, one for the fluid (flow equation) and one for the salt (transport equation), both having nonlinear diffusion terms. In this paper, we present and analyze a numerical technique for the solution of such a model. The approach is based on the mixed hybrid finite element method for the discretization of the diffusion terms in both the flow and transport equations, and a high-resolution TVD finite volume scheme for the convective term. This latter technique is coupled to the discretized diffusive flux by means of a time-splitting approach. A commonly used benchmark test (Elder problem) is used to verify the robustness and nonoscillatory behavior of the proposed scheme and to test the validity of two different formulations, one based on using pressure head ψ and concentration c as dependent variables, and one using pressure p and mass fraction ω as dependent variables. It is found that the latter formulation gives more accurate and reliable results, in particular, at large times. The numerical model is then compared against a semi-analytical solution and the results of a laboratory test. These tests are used to verify numerically the performance and robustness of the proposed numerical scheme when high-concentration gradients (i.e., the double nonlinearity) are present.

Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov-Galerkin method

A crucial point in the implementation of meshless methods such as the meshless local Petrov-Galerkin (MLPG) method is the evaluation of the domain integrals arising over circles in the discrete local weak form of the governing partial differential equation. In this paper we make a comparison between the product Gauss numerical quadrature rules, which are very popular in the MLPG literature, with cubature formulas specifically constructed for the approximation of an integral over the unit disk, but not yet applied in the MLPG method, namely the spherical, the circularly symmetrical and the symmetric cubature formulas. The same accuracy obtained with 64x64 points in the product Gauss rules may be obtained with symmetric quadrature formulas with very few points.

Godunov mixed methods on triangular grids for advection-dispersion equations

A time-splitting approach for advection–dispersion equations is considered. The dispersive and advective fluxes are split into two separate partial differential equations (PDEs), one containing the dispersive term and the other one the advective term. On triangular elements a triangle-based high resolution Finite Volume (FV) scheme for advection is combined with a Mixed Hybrid Finite Element (MHFE) technique to solve dispersion. This approach introduces an error proportional to the time step and the overall scheme is only first order accurate if special care is not taken in the definition of the numerical flux approximation for advection. By incorporating the diffusive effects into the definition of this numerical flux, near second order accuracy (up to a log h factor) can be proved theoretically and validated by numerical experiments in both one- and two-dimensional cases.

High-order transverse schemes for the numerical solution of PDEs

Many existing numerical schemes for the solution of initial-boundary value problems for partial differential equations can be derived by the method of lines. The PDEs are converted into a system of ordinary differential equations either with initial conditions (longitudinal scheme) or with boundary conditions (transverse scheme). In particular, this paper studies the performance of the transverse scheme in combination with boundary value methods. Moreover, we do not restrict the semi-discretization by the usual first- or second-order finite-difference approximations to replace the derivative with respect to time, but we use high-order formulae.

Bad behavior of Godunov mixed methods for strongly anisotropic advection-dispersion equations

We study the performance of Godunov mixed methods, which combine a mixed-hybrid finite element solver and a Godunov-like shock-capturing solver, for the numerical treatment of the advection-dispersion equation with strong anisotropic tensor coefficients. It turns out that a mesh locking phenomenon may cause ill-conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes. This problem may be partially alleviated by substituting the mixed-hybrid finite element solver used in the discretization of the dispersive (diffusive) term with a linear Galerkin finite element solver, which does not display such a strong ill conditioning. To illustrate the different mechanisms that come into play, we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh. A thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the Godunov mixed method is a structural feature of the mixed-hybrid method. In fact, the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion (diffusion) tensor influence the diagonal and off-diagonal terms of the final stiffness matrix. One and two dimensional test cases support our findings.

Three dimensional Godunov mixed methods on tetrahedra for the advection-dispersion equation*

Godunov Mixed Methods on triangular grids has been shown to be an effective tool for the solution of the two-dimensional advection-dispersion equation. The method is based on the discretization of the dispersive flux by means of the mixed hybrid finite element approach, while a high resolution Godunov-like finite volume scheme discretizes advection. The two techniques are combined together through a time-splitting algorithm that achieves formal second order accuracy if a corrective term is added in the finite volume stencil. In this paper we develop and study the extension of this approach to three dimensions employing tetrahedral elements and a fully 3D limiter. The numerical characteristics of the proposed method will be studied both theoretically and numerically using simple test problems.

Behavior of the mixed hybrid finite element method for the solution of diffusion equations on unstructured triangulations

In this paper we study the Mixed Hybrid Finite Element (MHFE) Method on unstructured triangular grids by considering two different basis functions for the space related to the approximation of the velocity. The final mixed hybrid formulation produces a system, with the Lagrange multipliers as unknowns, in which the matrix does not depend on the choice of the vector basis functions but only on the inner product of the outward normals to the edges of the triangulation. This is useful to study some properties of the final matrix as the propery M and the positivity of the inverse. It is known that the system matrix is an M-matrix if the angles of the triangulation are not bigger than π/2. An M-matrix has the characteristics that its inverse is nonnegative (all elements are nonnegative). Which implies the existence of a discrete maximum principle and thus stability and conservation properties of the discretization. We show that, when the triangulation is of Delaunay type, the inverse of the final matrix is always positive, even in presence of obtuse angles.

Comparison of 3D flow fields arising in mixed and standard unstructured finite elements

Computing 3D velocity fields is an important task in subsurface water flow analysis. While Finite Element (FE) simulations generally yields accurate estimates of the head, the numerical velocity may display errors and even unphysical behavior in some parts of the domain. Theoretically, the Mixed Hybrid FE (MHFE) approach provide a more accurate approximation to the velocities. In this communication we analyze a number of 3D-flow test cases, and compare the results obtained using FE and MHFE on tetrahedral meshes. Theoretical convergence estimates are numerically verified for a few simple problems. A more complex heterogeneous test case is used to show that, even for very fine meshes, the results obtained using the two discretization approaches may differ.