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In this paper a time-splitting technique for the two dimensional advectiondispersion equation is proposed. A high resolution in space Godunov method for advection is combined with the RT0 Mixed Finite Element for the discretization of the dispersion term. Numerical tests on an analytical one dimensional example ascertain the convergence properties of the scheme. At di erent Peclet numbers, the choice of optimal time step size used for the two equations is discussed, showing that with accurate selection of the time step sizes, the overall CPU time required by the simulations can be drastically reduced. Results on a realistic test case of groundwater contaminant transport con rm that the proposed scheme does not su er from Peclet limitations, and always displays only small amounts of numerical di usion across the entire range of Peclet numbers.

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The meshless local Petrov-Galerkin (MLPG) method is a meshfree 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.

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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

SUMMARYA meshless model, based on the meshless local Petrov-Galerkin (MLPG) approach, is developed and implemented for the solution of axi-symmetric poroelastic problems. The solution accuracy and the code performance are investigated on a realistic application concerning the prediction of land subsidence above a deep compacting reservoir. The analysis addresses several numerical issues, including the parametric selection of the optimal size of the local sub-domains for the weak form and the nodal supports, the appropriate integration rule, and the linear system solver. The results show that MLPG can be more accurate than the standard finite element (FE) method on coarse discretizations, with its superiority decreasing as the nodal resolution increases. This is due to both a slower convergence rate and a progressively higher computational cost compared to FE. These drawbacks can be partially mitigated by improving the efficiency of the numerical integration and the system solver with the aid of projection techniques based on Krylov subspace methods. The outcome of the present analysis supports the development of coupled methods where a limited number of MLPG nodes are used to locally improve a FE solution.

The density dependent flow and transport problem in groundwater\ud on three dimensional triangulations is solved numerically by\ud means of a Mixed Hybrid Finite Element scheme for the flow\ud equation combined\ud with a Mixed Hybrid Finite Element-Finite Volume (MHFE-FV)\ud time-splitting based technique\ud for the transport equation. This procedure is analyzed and shown to be an\ud effective tool in particular when the process is advection dominated\ud or when density variations induce the formation of instabilities\ud in the flow field.\ud From a computational point of view, the most effective strategy\ud turns out to be a combination of the MHFE and a\ud spatially variable time splitting technique in which the FV scheme\ud is given by a second order linear reconstruction based on the least\ud square minimization and the Barth-Jespersen limiter.\ud The recent saltpool problem introduced as a benchmark test for\ud density dependent solvers is used to\ud verify the accuracy and reliability of this approach