Eulerian-Lagrangian Localized Adjoint Method Research Articles

A two-grid Eulerian–Lagrangian localized adjoint method to miscible displacement problems with dispersion term

In this paper, we propose a novel two-grid method of Eulerian–Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solutions for nonlinear and coupled miscible displacement problems, which describe complex fluid flow processes in porous media. An ELLAM is used to solve the nonlinear convection–diffusion equation for concentration, and we present a MFEM to compute the pressure equation for the pressure and Darcy velocity. A two-grid scheme is investigated to linearize and decouple the mixed-method equations. It is illustrated that the coarse space can be selected as coarse as H=O(h1∕2) and the asymptotically optimal approximation as the nonlinear schemes can be still achieved. As a result, a large class of such nonlinear and coupled problems can be solved efficiently with large time steps.

An optimal-order error estimate of the lowest-order ELLAM-MFEM approximation to miscible displacement in three space dimensions

An optimal-order error estimate for the lowest-order ELLAM-MFEM time-stepping procedure for the coupled system of time-dependent nonlinear partial differential equations modeling miscible displacement that moves through the three-dimensional subsurface porous media is proved, under a practically reasonable time-step constraint of Δt=O(h). In the procedure, the Eulerian–Lagrangian localized adjoint method (ELLAM) and the mixed finite element method (MFEM) are used to solve the transport equation and the pressure equation, respectively. This mathematically justifies the numerical advantages of the ELLAM-MFEM solution procedure.

A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian–Lagrangian localized adjoint methods

In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/2) in this paper.

An Efficient ELLAM Implementation for Modeling Solute Transport in Fractured Porous Media

The goal of this study is to introduce an adaptation of the Eulerian-Lagrangian localized adjoint method (ELLAM) for the simulation of mass transport in fractured porous media, and to evaluate the performance of ELLAM in such a case. The fractures are represented explicitly using the discrete fracture model. The velocity field was calculated using the mixed hybrid finite element method. A sound ELLAM implementation is developed to address numerical artifacts (spurious oscillations and numerical dispersion) arising from the presence of fractures. The efficiency of the developed ELLAM implementation was further improved by taking advantage of the parallel computing on shared memory architecture for the tasks related to particles tracking and linear system resolving. The performance of ELLAM was tested by comparing its results with the Eulerian discontinuous Galerkin method based on several benchmark problems dealing with different fracture configurations. The results highlight the robustness and accuracy of ELLAM, as it allows the use of large time steps, and overcomes the Courant-Friedrichs-Lewy (CFL) restriction. The outcome also shows that ELLAM is more efficient when fracture density is increased.

A probabilistic collocation Eulerian–Lagrangian localized adjoint method on sparse grids for assessingCO2leakage through wells in randomly heterogeneous porous media

We develop a probabilistic collocation Eulerian–Lagrangian localized adjoint method on sparse grids for assessing CO 2 leakage through wells in randomly heterogeneous porous media , by utilizing the intrinsic mathematical, numerical, and physical properties of the mathematical model. We model the process in which CO 2 is injected into a deep aquifer, spreads within the aquifer and, upon reaching a leaky well, rises up to a shallower aquifer, to quantify the leakage rate, which depends on the pressure build-up in the aquifer due to injection and the buoyancy of CO 2 . The underlying Eulerian–Lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in CO 2 sequestration. The sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre-existing Eulerian–Lagrangian methods in a nonintrusive manner. It also provides a scalable framework to consider uncertainty in a straightforward parallel manner. Preliminary numerical experiments show the feasibility and potential of the method.

Modeling 2D Multispecies Reactive Transport in Saturated/Unsaturated Porous Media with the Eulerian–Lagrangian Localized Adjoint Method

In the present paper, the Eulerian–Lagrangian localized adjoint method (ELLAM) formulation developed by Younes et al. (Advances in Water Resources 29:1056–1074, 2006) is combined with the sequential noniterative approach to accurately simulate 2D multicomponent reactive transport in saturated/unsaturated porous media. The performance and accuracy of the developed model, named ELLAM_REACT, are compared against those of an existing numerical model based on a combination of discontinuous Galerkin and multipoint flux approximation methods (DGMPFA_REACT). Three studied test cases, dealing with reactive transport in saturated and unsaturated porous media and involving chemical reactions with only aqueous species or both fixed and aqueous species, show the superiority of the ELLAM_REACT model compared to the DGMPFA_REACT model.

Efficiency of the Eulerian–Lagrangian localized adjoint method for solving advection–dispersion equations on highly heterogeneous media

SUMMARYThe Eulerian–Lagrangian Localized Adjoint Method (ELLAM) is well adapted for advection‐dominated transport problems. The method can use large time steps because the advection term is accurately approximated without any CFL restriction.A new formulation of ELLAM was developed by Younes et al. (Adv. Water Resour. 2006; 29:1056–1074) for unstructured triangular meshes. This formulation uses only strategic numerical integration points and thus requires a very limited number of integration points (usually 1 per element). To avoid numerical dispersion due to interpolation when several time steps are used, the method continuously tracks characteristics, and only changes due to dispersion are interpolated at each time step.In this work, we study the applicability and efficiency of this formulation for highly heterogeneous domains. The results show that (i) this formulation remains efficient even for highly heterogeneous domains, (ii) special care must be taken for the approximation of the dispersive term when large time steps are used, (iii) interpolation at intermediate times can be used to reduce memory requirement for problems with a large number of elements and small time steps, and (iv) the method can be used successfully for heterogeneous problems including injection and pumping wells, and remains much more efficient than the discontinuous Galerkin finite element method even when small time steps are required. Copyright © 2011 John Wiley & Sons, Ltd.

An Optimal-Order Error Estimate to ELLAM Schemes for Transient Advection-Diffusion Equations on Unstructured Meshes

The Eulerian–Lagrangian localized adjoint method (ELLAM) provides a general characteristic procedure for solving transient advection-diffusion equations with general boundary conditions in a mass-conservative manner. In this paper we prove an optimal-order error estimate in the $L^2$ norm and a superconvergence estimate in the energy norm for the ELLAM scheme to d-dimensional transient advection-diffusion equations with general flux boundary conditions on unstructured meshes.

On the use of large time steps with ELLAM for transport with kinetic reactions over heterogeneous domains

AbstractAn Eulerian Lagrangian localized adjoint method (ELLAM) is considered for the resolution of advection‐dominated transport problems in porous media. Contrary to standard Eulerian methods, ELLAM can use large time steps because the advection term is approximated accurately without any CFL restriction. However, it is shown in this article that special care must be taken for the approximation of the dispersive and reactive terms when large time steps are used over heterogeneous domains. An alternative procedure is proposed. It is based on an equivalent dispersion coefficient or an equivalent reaction rate when different zones are encountered during the tracking. Numerical experiments are performed with variable dispersion or variable reaction rates over space (including nonlinearity). When classical ELLAM require numerous time steps to handle heterogeneity, the alternative procedure is shown to perform with the same accuracy in a single time step. © 2009 American Institute of Chemical Engineers AIChE J, 2009

Multidimensional Finite-Element Particle Tracking Method for Solving Complex Transient Flow Problems

This paper proposes a particle tracking technique for applying the Eulerian–Lagrangian localized adjoint method or the Eulerian–Lagrangian method to solve transport equations in a rapidly changing unsteady flow. The proposed particle tracking technique uses a linear temporal interpolation of velocity to avoid the error introduced when using a stepwise temporal approximation of the velocity, which has been adopted in many existing particle tracking techniques. Numerical experiments were carried out on several example problems to demonstrate the superior accuracy and efficiency of the proposed particle tracking technique over Cheng’s and Pollock’s methods. The numerical results show that the use of a linear temporal interpolation of velocity can allow an increase in time step size without causing a significant decrease in accuracy, particularly under a complicated transient flow or a rapidly changing velocity field with time.

A finite-volume ELLAM for non-linear flux convection–diffusion problems

In this paper, a modified finite-volume Eulerian–Lagrangian localized adjoint method (FVELLAM) extended for convection–diffusion problems with a non-linear flux function is introduced. Tracking schemes are discussed using viscous Burgers’ equation. It is shown that in order to have smooth results, only the new time level values should be used in tracking process. Then, the proposed method is employed to study immiscible incompressible two-phase flows in porous media. Various one- and two-dimensional test cases involving internal sources and sinks are solved and accuracy of solution and performance of the method are investigated by comparing the results obtained using FVELLAM with those of fine grid solutions. Finally, it is concluded that although proposed FVELLAM produces satisfactory results even on coarse grids and allows fairly large time-steps, its major advantage is in solving convection-dominated problems in heterogeneous media.

An Optimal-Order Error Estimate for a Family of ELLAM-MFEM Approximations to Porous Medium Flow

Mathematical models used to describe porous medium flow lead to coupled systems of time-dependent nonlinear partial differential equations, which present serious mathematical and numerical difficulties. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid-orientation effect. The ELLAM-MFEM time-stepping procedure, in which an Eulerian-Lagrangian localized adjoint method (ELLAM) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this paper we prove an optimal-order error estimate for a family of ELLAM-MFEM approximations.

A multiscale Eulerian–Lagrangian localized adjoint method for transient advection–diffusion equations with oscillatory coefficients

We develop a multiscale Eulerian–Lagrangian localized adjoint method for transient linear advection– diffusion equations with oscillatory coefficients, which arise in mathematical models for describing flow and transport through heterogeneous porous media, composite material design, and other applications.

ELLAM for resolving the kinematics of two-dimensional resistive magnetohydrodynamic flows

We combine the finite element method with the Eulerian–Lagrangian Localized Adjoint Method (ELLAM) to solve the convection–diffusion equations that describe the kinematics of magnetohydrodynamic flows, i.e., the advection and diffusion of a magnetic field. Simulations of three two-dimensional test problems are presented and in each case we analyze the energy of the magnetic field as it evolves towards its equilibrium state. Our numerical results highlight the accuracy and efficiency of the ELLAM approach for convection-dominated problems.

Direct and split operator approaches with ELLAM for reactive transport equations

AbstractNumerical models for solving reactive solute transport in groundwater are often based on standard Eulerian methods, such as finite elements (FE). In this work, we combine the moving mesh Eulerian Lagrangian Localized Adjoint Method (ELLAM) with the direct substitution approach (DSA) and the sequential non‐iterative approach (SNIA) to accurately solve advection dominated problems including chemical reactions. Performances of ELLAM are evaluated in comparing SNIA_ELLAM and DSA_ELLAM to SNIA_FE and DSA_FE for (1) kinetic reactions with first‐order decay, Monod biodegradation, and interphase mass transfer problems and (2) equilibrium reactions with linear and nonlinear sorption problems. DSA_ELLAM (respectively SNIA_ELLAM) is shown to be more efficient and accurate than DSA_FE (respectively SNIA_FE). For the test cases with kinetics, SNIA_ELLAM is highly accurate and efficient. However, for equilibrium reactions, it induces numerical diffusion and DSA_ELLAM reveals the more efficient and accurate method. © 2007 American Institute of Chemical Engineers AIChE J, 2007

A uniform estimate for the ELLAM scheme for transport equations

AbstractWe prove a priori error estimate in a weighted energy norm for the Eulerian‐Lagrangian localized adjoint method (ELLAM) for the transport equations, without any special refinement or numerical stabilization introduced. The estimate holds uniformly with respect to ϵ. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

An efficient algorithm for characteristic tracking on two-dimensional triangular meshes

Tracking characteristics on unstructured meshes is an important part of many numerical methods in computational fluid mechanics. In this paper, we propose an efficient algorithm for characteristic tracking on two-dimensional unstructured triangular meshes. Numerical experiments, including an example for applying this algorithm with the Eulerian-Lagrangian localized adjoint method (ELLAM) to solve a convection-dominated convection-diffusion problem, are presented to demonstrate the efficiency of this algorithm.

A new approach to avoid excessive numerical diffusion in Eulerian–Lagrangian methods

AbstractLumping is often used to avoid non‐physical oscillations for advection–dispersion equations but is known to add numerical diffusion. A new approach is detailed in order to avoid excessive numerical diffusion in Eulerian–Lagrangian methods when several time steps are used. The basic idea of this approach is to keep the same characteristics during all time steps and to interpolate only the concentration variations due to the dispersion process. In this way, numerical diffusion due to the lumping is removed at the end of each time step. The method is combined with the Eulerian–Lagrangian localized adjoint method (ELLAM) which is a mass conservative characteristic method for solving the advection–dispersion equation.Two test problems are modelled to compare the proposed method to the consistent, the full and the selective lumping approaches for linear and non‐linear transport equations. Copyright © 2007 John Wiley & Sons, Ltd.

A fractional step ELLAM approach to high-dimensional convection–diffusion problems with forward particle tracking

In this paper, a fractional step method combined with an Eulerian–Lagrangian localized adjoint method (ELLAM) is proposed to solve high-dimensional convection–diffusion problems. The method reduces high-dimensional problems to a series of uncoupled one-dimensional problems in each time step interval, in which one-dimensional ELLAM is used to solve the one-dimensional splitting equations. The approach takes the attractive advantages of the ELLAM method and the fractional step technique. It reduces computational complexities, large memory requirements, and long computation durations due to the application of the splitting technique. It reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation. It effectively eliminates non-physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way. Numerical experiments show the efficient performance of the approach.

Characteristic‐mixed covolume methods for advection‐dominated diffusion problems

AbstractCharacteristic‐mixed covolume methods for time‐dependent advection‐dominated diffusion problems are developed and studied. The diffusion term in these problems is discretized using covolume methods applied to the mixed formulation of the problems on quadrilaterals, and the temporal differentiation and advection terms are treated by characteristic tracking schemes. Three characteristic tracking schemes are studied in the context of mixed covolume methods: the modified method of characteristics, the modified method of characteristics with adjusted advection, and the Eulerian–Lagrangian localized adjoint method. The proposed methods preserve the conceptual and computational merits of both characteristics‐based schemes and the mixed covolume methods. Existence and uniqueness of a solution to the discrete problem arising from the methods is shown. Stability and convergence properties of these methods are also obtained; unconditionally stable results and error estimates of optimal order are established. Copyright © 2006 John Wiley & Sons, Ltd.