## 1. Introduction

[2] The numerical simulation of advection processes is a crucial issue for many groundwater modeling applications. This is because (1) many transport problems can be reduced to their advective component, which is affected by the only uncertainty of the flow field, and (2) the more general advection-diffusion equation is often solved by splitting techniques, in which the solution of the diffusive component is usually the easiest [*Abbott*, 1979; *Holly and Preissmann*, 1977].

[3] In spite of the importance of the advection problem, computational difficulties remain for its numerical solution, mainly for 2-D and 3-D cases, where nonstructured grids and irregular elements are used.

[4] The available methods can be classified as Eulerian and Lagrangian. Eulerian methods compute the unknown function at the nodes or at the elements of a computational mesh fixed in space, after spatial and time discretization of the PDE. It is well known that the classical Eulerian finite difference or finite element methods provide numerical solutions affected by numerical diffusion or oscillations [*Bella and Grenney*, 1970; *Gray and Pinder*, 1983; *Venezian*, 1984]. A reduction of numerical diffusion can be obtained by evaluating the spatial derivatives starting from the function values at distant grid points, as in the QUICK and QUICKEST procedure [*Leonard*, 1979], that can be coupled to a limiting algorithm like ULTIMATE [*Leonard*, 1991], aiming at eliminating spurious oscillations. A popular Eulerian approach is the so-called Runge-Kutta discontinuous Galerkin (RKDG) method [*Cockburn and Shu*, 1998]. The RKDG combines a piecewise linear discontinuous finite element spatial approximation of the unknown function with a time discretization that guarantees the so-called “total variation diminishing (TVD)” property, that is an increasing (in time) spatial regularity of the solution. After a spatial discretization of the equation that guarantees approximate Riemann fluxes along the element discontinuities, the proposed time discretization provides, with simple low-order matrix operations, a system of ODEs that can be solved, for given temporal step, using a Runge-Kutta high-order accurate scheme. Other methods are the streamline upwind Petrov-Galerkin [*Brooks and Hughes*, 1982], the Taylor-Galerkin [*Donea*, 1984] and the Galerkin-least squares schemes [*Hughes et al.*, 1989]. If a fully implicit time discretization is used in Eulerian methods, no limitation exists for the choice of the time step; however, this technique requires the solution of large nonsymmetric algebraic systems for each time step, with a fast growth of the numerical effort with the number of elements. Most of the recently adopted Eulerian methods use explicit time discretization, are second-order accurate, but have limitations on the size of the Courant number, that must always be taken smaller than one, corresponding to the well-known Courant-Friederichs-Lewy (CFL) condition. An exception is an algorithm for the finite difference solution of the 1-D case, proposed by *Ponce et al.* [1979]. The CFL condition fulfillment does not produce a large increment of the numerical effort in structured meshes, because it is not affected by the number of elements. Nevertheless, it can limit the efficiency of the algorithm in nonstructured meshes, obtained by automatic mesh generators. In this case, the existence of even a single small element can require the use of a small time step for all the elements, with a strong increment of the computational effort and a potential loss of accuracy due to the use of very small Courant numbers in some parts of the domain. The Courant number can be held within the stability limit using different grid refinement techniques in computational elements with changing velocity values [*Sobey*, 1984], but this lead to very large time-consuming procedures, specially in the 2-D and 3-D cases.

[5] In the Lagrangian approaches, the computational grid is not fixed in space, moving along the characteristic lines with the same velocity of the flow field. Then, it is necessary to accurately track the fluid particles and to evaluate their trajectories during their motion. The particle tracking technique is often the critical point of the procedure [*Oliveira and Baptista*, 1998]. Both forward and backward tracking techniques have been proposed in the past, but the search for an accurate method is still in progress [*Bensabat et al.*, 2000; *Pokrajac and Lazic*, 2002]. An example of Lagrangian approach is the second moment method, originally developed for the case of air pollution [*Egan and Mahoney*, 1972] and subsequently applied also to shallow flow [*Nassiri and Babarutsi*, 1997]. The disadvantage of the Lagrangian approaches is that they are difficult to be applied to 2-D and 3-D problems with irregular boundaries and heterogeneous domains. Moreover, many Lagrangian techniques do not guarantee mass conservation.

[6] In the semi-Lagrangian approaches for the solution of the advection-diffusion problem, the use of Lagrangian techniques is restricted to the advective component and limited to the time step used for the solution of the next diffusive problem. The method of characteristics (MOC) solves the advective problem by locating, at each time step, the foot of the characteristic line ending in each node of a fixed grid. This means that a potentially dissipative interpolation of the known values of the surrounding grid nodes must be performed. Many examples of similar methods can be found in the literature, like the two-point fourth-order interpolation [*Holly and Preissmann*, 1977], the finite element characteristic [*Wang et al.*, 1988], the minimax characteristics [*Li*, 1990] methods, as well as the cubic-spline interpolation [*Schohl and Holly*, 1991], or the characteristic Galerkin scheme, recently developed for the 3-D case also [*Kaazempur-Mofrad and Ethier*, 2002]. The MOC based techniques have the advantage of allowing the use of large time steps and preserving the solution from oscillations and numerical diffusion, but they seldom guarantee local and global mass balance [*Chilakapati*, 1999]. Volume tracking techniques [*Van Leer*, 1977; *Rider and Kothe*, 1998] follow the volume evolution of the mass initially present inside each element of the mesh and guarantee its conservation. The semi-Lagrangian approaches, also known as Eulerian-Lagrangian approaches, are nowadays probably the most popular tools for the solution of the advection problem equations [*Celia et al.*, 1990; *Healy and Russell*, 1993].

[7] Recently, some grid-free methods have been proposed, based on the interpolation of scattered data [*Behrens and Iske*, 2002], but they suffer of the same limitations of the Lagrangian methods.

[8] In the following section an Eulerian procedure is proposed, that embraces several advantages of the Eulerian and Lagrangian approaches. It is explicit, mass conservative, but also unconditionally stable with respect to the Courant number. It adopts a second-order approximation of the unknown advected function in each computational element and shows an average convergence order equal to 2 in 1-D numerical tests with smooth initial concentration. The procedure is restricted to the case of irrotational flow fields, as it is more extensively discussed in section 4. In section 2 the proposed algorithm is presented for the 1-D case, in section 3 it is extended to the 2-D case. Extension from the 2-D to the 3-D case is straightforward. Some benchmark problems are solved in each section, in order to show the features of the method.