Coursehero >> Texas >> Texas A&M >> ACCT 321

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

Visual Comput Sci DOI 10.1007/s00791-007-0078-5 REGULAR ARTICLE A multiscale Eulerian Lagrangian localized adjoint method for transient advection diffusion equations with oscillatory coef cients Hong Wang Yabin Ding Kaixin Wang Richard E. Ewing Yalchin R. Efendiev Received: 31 January 2005 / Accepted: 2 May 2006 Springer-Verlag 2007 Abstract We develop a multiscale Eulerian Lagrangian localized adjoint method for transient linear advection diffusion equations with oscillatory coef cients, which arise in mathematical models for describing ow and transport through heterogeneous porous media, composite material design, and other applications. Keywords Advection diffusion equations Eulerian Lagrangian method Multiscale method Reservoir simulation Porous medium ow 1 Introduction Transient advection diffusion partial differential equations arise in mathematical models for describing groundwater hydrology, environmental modeling and remediation, petroleum reservoir simulation and many other elds [2,6,9]. These problems admit solutions with moving steep fronts and complicated structures. Centered nite difference or nite element methods often generate numerical solutions with severe nonphysical oscillations. In industrial applications, Communicated by G. Wittum. H. Wang (B) Y. Ding Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA e-mail: hwang@math.sc.edu K. Wang School of Mathematics and System Sciences, Shandong University, Jinan, Shandong 250100, China R. E. Ewing Y. R. Efendiev Institute for Scienti c Computation, Texas A&M University, College Station, TX 77843-3404, USA upstream weighting techniques are commonly used to stabilize the numerical approximations to these systems in large-scale simulators. But upwind methods tend to produce excessive numerical diffusion, which smears out moving steep fronts of the solutions to the governing equations and introduces spurious grid orientation effect [6,17]. In addition, advection diffusion equations for describing porous medium ow through heterogeneous porous media usually contain solutions with multiple spatial and temporal scales. Consequently, a direct numerical solution strategy often requires extremely re ned space grids and time steps, due to the wide spectrum of spatial and temporal scales in the solutions and the advection diffusion feature of the governing equations. The Eulerian Lagrangian localized adjoint method (ELLAM) [4] provides a general framework for developing characteristic methods to solve transient advection diffusion equations with general boundary conditions in a massconservative manner. The ELLAM schemes have been successfully applied in the numerical modeling of petroleum reservoir simulation and groundwater contaminant transport and remediation. They generate accurate numerical solutions even if large time steps and spatial grids are used, and are very competitive with many numerical methods [3,8,14 16]. The multiscale nite element (or volume) method (MsFEM) [5,10,11] was introduced in recent years for solving partial differential equations with multiscale solutions. These methods aim at obtaining large scale solutions accurately and effectively without resolving small scale details. These ideas were carried out via the construction of basis functions that capture small scale information within each cell, so the effect of small scales on the large scale is captured correctly. We develop a multiscale Eulerian Lagrangian localized adjoint method (MsELLAM) for a transient linear 123 H. Wang et al. advection diffusion equation with oscillatory coef cients. The goal of this paper is to expose the fundamental idea in the development of such a scheme within the framework of the ELLAM formulation and the MsFEM, so we focus on a simple model problem in most of the paper. At the end of this paper we brie y discuss extensions of the MsELLAM scheme for coupled systems modeling porous medium ow. time at time step tn 1 and integrate the resulting equation to get tn b ct + V x, tn 1 a tn b x x c D x, cx x wd xdt (4) = tn 1 a f wd xdt. 2 An ELLAM formulation In this section we revisit the idea of the ELLAM framework [4] in the context of one-dimensional advection diffusion equations with oscillatory coef cients x x c D x, cx = f, ct + V x, x (1) x (a, b), t (0, T ], where V is velocity eld, D is the diffusion coef cient, f is a given source or sink term, and c(x, t) is a measure of concentration of a dissolved substance. Due to the heterogeneity of the porous medium, the velocity V and diffusion coef cient D may be random or highly oscillatory. Consequently, the solution could contain multiple scales. In the model problem (1), we assume the problem to have two scales: a scale of O(1) that represents the normal scale, and a scale of O( ) that represents a fast changing scale. Note that we do not impose a periodicity assumption on the problem. The ELLAM framework can treat advection diffusion equations with general in ow and out ow boundary conditions in a mass-conservative manner [4,14]. Since the goal of this paper is to explore how to utilize the idea of the multiscale nite element method [10] within the ELLAM framework to design a multiscale ELLAM scheme, we skip the treatment of boundary conditions by, e.g., assuming an initial condition with compact support or periodic boundary conditions. Equation (1) is also subject to the initial condition c(x, 0) = c0 (x), x [a, b]. 2.1 Weak formulation We begin by de ning a space time partition: b a I (3) T tn := n t, 0 n N , t := . N Due to various mathematical, numerical, and computational constraints, the size of spatial grids and time steps x and t often has to be chosen much larger than the ne scale in practical simulations. We multiply Eq. (1) by space time test functions w that vanish outside [a, b] (tn 1 , tn ] and are discontinuous in xi := a + i x, 0 i I, x := (2) Integration by parts in both space and time yields a weak formulation b tn b c(x, tn )w(x, tn )d x + a tn b tn 1 a D x, x cx wx d xdt tn 1 a b c wt + V x, x wx d xdt tn b = a + c(x, tn 1 )w(x, tn 1 )d x + tn 1 a f wd xdt. (5) + Here w(x, tn 1 ) := limt t + w(x, t) accounts for the disn 1 continuity of w(x, t) at time tn 1 . Based on the idea of the localized adjoint method or optimal test function method [4,12], the test functions w should be chosen from the solution space of the homogeneous adjoint equation of Eq. (1) wt V x, x x wx D x, wx x = 0. (6) 2.2 Operator splitting Because the solution space of ordinary differential equations is nite-dimensional, the development of optimal test function methods for the boundary-value problem of onedimensional ordinary differential equations is relatively simple [4,12]. In contrast, the solution space of the partial differential equation (6) is in nite-dimensional. Since the objective of a numerical procedure is to derive a nite-dimensional approximation, only a nite number of test functions should be chosen. Different choices of test functions lead to different classes of approximations. 2.2.1 An optimal test function operator-splitting In the context of the adjoint equation (6), one option is to split the adjoint operator into a spatial operator and a temporal operator. Namely, we require the test functions w(x, t) to satisfy the following system that consists of two subequations within each element (xi 1 , xi ) (tn 1 , tn ] 123 A multiscale ELLAM scheme (i = 1, 2, . . . , I ) x x wx D x, wx V x, wt = 0, x 3 An MsELLAM tracking ne-scale velocity The objective of this section is to derive an MsELLAM for problem (1), (2) by incorporating multiscale-handling capability into the ELLAM framework. 3.1 De nition of test functions One of the key issues in the development of MsELLAM is the choice of test functions in the ELLAM framework. In order for the MsELLAM to handle multiscale advection diffusion equations, we utilize the idea of multiscale nite element method (MsFEM) [10] in the construction of the test functions in the MsELLAM. 3.1.1 Spatial con guration of the test functions The second equation in the Eulerian Lagrangian operatorsplitting (8), which de nes spatial con guration of the test functions w in the ELLAM framework, is a self-adjoint elliptic equation. The original MsFEM was developed for such an equation [10]. We naturally follow the idea of MsFEM to de ne the spatial con guration of the test functions w at time step tn . Because the mesh size of the coarse grid x > , the test functions in a conventional nite element method, which is linear on [xi 1 , xi ], cannot handle ne-scale behavior of the solutions. In the MsFEM a subdivision x (9) xi, j := xi 1 + j x f , 0 j J, x f := J is introduced in [xi 1 , xi ], where x f = O( ) is chosen to account for the ne-scale behavior. Then two piecewiselinear basis functions wi(1) (x, tn ) and wi(2) (x, tn ) are determined by using a nite element method to solve the second equation in (8) in [xi 1 , xi ] with respect to the subdivision (9) and the boundary condition wi(1) (xi 1 , tn ) = 1, (2) wi (xi 1 , tn ) = 0, wi(1) (xi , tn ) = 0, (2) wi (xi , tn ) = 1. (1) (2) = 0. (7) The second equation in (7) is a singularly-perturbed, twoscale, steady-state advection diffusion equation. It de nes the spatial con guration of the test functions w. This type of splitting leads to a class of optimal test function methods involving exponential upstream weighting in space. The corresponding test functions w typically exhibit exponential boundary layers near the inter-element boundary, which require a ne grid for reasonable resolution. The rst equation in (7) de nes the temporal variation of the test functions w. Numerical methods resulting from these test functions tend to have large temporal truncation errors, numerical dispersion, and phase errors [4,14,15]. The combination of multiscale feature and advection dominance of the problems introduces extra numerical dif culties. 2.2.2 An Eulerian Lagrangian operator-splitting An alternative operator splitting of the adjoint equation (6) is to split the adjoint operator into a rst-order hyperbolic operator and a second-order diffusion operator. Namely, we require the test functions w(x, t) to satisfy the following system that consists of two sub-equations within each Eulerian Lagrangian element i , which is a space time strip emanating backward from (xi 1 , xi ) along the characteristics from time step tn to time step tn 1 x wx = 0, wt V x, x D x, wx = 0. x (8) In the ELLAM framework, the equations in (8) are imposed locally within the interior of each element i . As in the case of optimal test function splitting, the second equation in (8) de nes the spatial con guration of the test functions w. This equation is now an elliptic steady-state diffusion equation, in contrast to a steady-state advection diffusion equation. Consequently, the test functions w are not expected to exhibit boundary layers near the inter-element boundary. The rst equation in (8) de nes the temporal variation of the test functions w, which implies the Lagrangian feature of the test functions. We note that in the case of scale separation (e.g., problems with periodic coef cients) the solution of the local problems (7) and (8) can be approximated using auxiliary problems, which arise in homogenization. This approximation will reduce the computational cost of the proposed method. (10) The shape functions wi (x, tn ) and wi (x, tn ) are restricted to the element [xi 1 , xi ], and are extended outside of the element by 0. These shape functions for i = 0, 1, . . . , I form a basis for the MsFEM with the degrees of freedom being still located at coarse spatial grid nodes xi for i = 0, 1, . . . , I . In this way, the basis functions naturally build the multiscale behavior into their construction and re ect the multiscale behavior due to diffusion. 3.1.2 Temporal variation of the test functions Once the spatial con guration of the test functions w(x, tn ) is determined at time step tn , the rst equation in the Eulerian 123 H. Wang et al. Lagrangian splitting (8) de nes the temporal variation of w(x, t) from time step tn to time step tn 1 by extending them backward along the characteristic curves y = r (t; x, tn ) de ned by y dy = V y, , dt y = x. t=tn (11) Because the test functions w satisfy the rst equation of (8) (at least approximately), the last term on the left-hand side of (5) vanishes (or at least is the same order as the global truncation error of the numerical scheme [7,13]. Hence, we drop this term in the numerical scheme. We then obtain an MsELLAM scheme by incorporating Eqs. (13) and (14) into Eq. (5). In the case of multiscale transient advection diffusion equations, the velocity eld exhibits multiscale or oscillatory behavior. Consequently, the evaluation of the test functions w, which is carried out via a characteristic tracking, requires an accurate resolution of problem (11) determined by the ne-scale velocity eld V . Since the size of the global time step t > , we introduce a local time-step partition tn,k := tn 1 + k t f , 0 k K , t f := t K (12) 4 An MsELLAM tracking coarse-scale velocity The MsELLAM scheme developed in the previous section requires a characteristic tracking of the ne-scale velocity eld. In this section we explore the possibility of developing an MsELLAM for problem (1), (2), which requires only tracking the coarse-grid velocity eld. 4.1 De nition of test functions One of the key issues in the development of MsELLAM is the choice of test functions in the ELLAM framework, which in turn is the result of an appropriate operator-splitting. 4.1.1 An alternative Eulerian Lagrangian operator-splitting To develop an MsELLAM scheme that requires only tracking the coarse-grid velocity eld, we decompose the ne-scale velocity eld V as V = V + V. (15) Here V and V represent the global, smooth part and the local, oscillatory part of the velocity eld V , respectively. We revise the Eulerian Lagrangian operator-splitting (8) as follows wt V wx = 0, (16) Vx (Dwx )x = 0. In the ELLAM framework, the equations in (8) are imposed locally within the interior of each element i . We note that a somewhat related modi ed Eulerian Lagrangian operatorsplitting was previously used in [7] for the purpose of improving the accuracy of the numerical approximation and simplifying the characteristic tracking algorithm. 4.1.2 Spatial con guration of the test functions We note that the second equation in the operator-splitting (16) is now a steady-state advection diffusion We equation. use the MsFEM idea to de ne the spatial con guration of the basis functions w at time step tn . In the current context, we solve the second equation in (16) by a standard nite element method with the ne-scale subdivision (9) on interval where the size of the local time step t f = O( ) is chosen to handle the multiscale behavior in time. 3.2 Numerical scheme With the test functions de ned in Sect. 3.1, we now derive an MsELLAM scheme. For clarity of exposition, we use the variable y to represent the spatial coordinate of any point in the domain. We use the change of variable y = r ( ; x, tn ) to evaluate the source term in Eq. (5) to obtain f (y, t)w(y, t)dydt i tn r (t;xi ,tn ) = f (y, t)w(y, t)dydt tn 1 r (t;xi 1 ,tn ) tn,k r (t;xi ,tn ) K (13) = k=1tn,k 1 r (t;x ,t ) i 1 n f (y, t)w(y, t)dydt. Then we can use a numerical quadrature to discretize the temporal integral on each local time interval [tn,k 1 , tn,k ] for k = 1, 2, . . . , K . We handle the diffusion term in a similar manner D y, i tn y c y (y, t)w y (y, t)dydt D y, y c y (y, t)w y (y, t)dydt y c y (y, t)w y (y, t)dydt. r (t;xi ,tn ) = (14) tn 1 r (t;xi 1 ,tn ) tn,k r (t;xi ,tn ) K = k=1tn,k 1 r (t;x ,t ) i 1 n D y, 123 A multiscale ELLAM scheme [xi 1 , xi ]. Because the ne-scale grid size x f = O( ) is chosen to be small enough so that the grid Peclet number is of O(1), we do not expect that the basis functions exhibit nonphysical oscillations [12]. 4.1.3 Temporal variation of the test functions Once the spatial con guration of the test functions w(x, tn ) is determined at time step tn , the rst equation in the modi ed Eulerian Lagrangian splitting (16) de nes the temporal variation of w(x, t) from time step tn to time step tn 1 by extending them backward along the characteristic curves y = r (t; x, tn ) de ned by y dy = V y, , dt y = x. t=tn by c0 (x) = exp (x x0 )2 , 2 2 (20) where the center x0 = 0.5 and the spread = 0.1 that determines the steepness of the Gaussian pulse. A two-scale oscillatory velocity led V (x/ ) and diffusion coef cient D(x/ ) is chosen as follows x x D V 2 x , 0.01 = , 2 x 1 + 0.8 sin = 1 + cos (21) (17) where = 0.2 which determines the magnitude of the oscillatory velocity eld. We use = 0.02, 0.01 and 0.005 to investigate the convergence rate. The source term f is chosen to be zero. 5.1 Results with ELLAM 4.2 Numerical scheme The diffusion term and source term in the reference equation (5) can be handled in a similar manner to Eqs. (13) and (14). The rst equation in the modi ed Eulerian Lagrangian operator-splitting (16) implies tn b c(wt + V wx )d xdt tn 1 a (18) vanishes, or at least, is within the global truncation error of the numerical scheme [7]. The oscillatory part V of the velocity eld on the left-hand side of the reference equation (5) remains, which can be approximated as follows V y, i tn y c(y, t)w y (y, t)dydt V y, y c(y, t)w y (y, t)dydt We use an ELLAM scheme with extremely re ned spatial grid and time step to generate a reference solution, which is presented in Fig. 1. In Table 1 and Fig. 1 we present numerical results generated by an ELLAM scheme with an accurate characteristics tracking using a second-order Runge Kutta formula. These numerical results show that the ELLAM scheme converges only when the meshsize x < . This is fully understandable and can be explained as follows. Although ELLAM schemes have been successfully applied to simulate transient advection diffusion equations in different applications [3,8,14 16], they are not designed to solve advection diffusion equations with multiple scales. This is similar to the case why conventional nite element methods do not work well for multiscale elliptic equations. r (t;xi ,tn ) = (19) 0.5 tn 1 r (t;xi 1 ,tn ) tn,k r (t;xi ,tn ) K = k=1tn,k 1 r (t;x ,t ) i 1 n y c(y, t)w y (y, t)dydt. V y, Reference Solution dx=1/5 dx=1/10 dx=2/25 dx=1/20 dx=1/25 0.4 We handle the temporal integral in this term as we did with the source term in Eq. (13) and with the diffusion term in (14) in Sect. 3. 0.3 0.2 5 Numerical experiments We simulate the transport of a one-dimensional Gaussian pulse over the spatial domain [0, 4]. In the example runs the time interval is [0, T ] = [0, 1]. The initial condition is given 0.1 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Fig. 1 Results by an ELLAM scheme with different grids 123 H. Wang et al. Table 1 Results with an ELLAM ( = x 1 5 4 25 1 10 2 25 1 20 1 25 1 50 1 100 2 250 1 200 1 250 1 500 1 100 , t = 1 1000 ) L 1 error 0.0809 0.0888 0.0651 0.0518 0.0499 0.1269 0.0346 0.0581 0.0484 0.0260 0.0153 0.0045 L 2 error 0.0784 0.0908 0.0753 0.0602 0.0564 0.1390 0.0390 0.0644 0.0551 0.0297 0.0182 0.0054 L error 0.1758 0.1981 0.1432 0.1202 0.1225 0.2263 0.0729 0.1376 0.1229 0.0609 0.0484 0.0150 0.5 Reference Solution dx=1/5 dx=1/10 dx=2/25 dx=1/20 dx=1/25 0.4 0.3 0.2 0.1 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Fig. 2 Results of MsELLAM tracking ne-scale velocity This is the reason why we use the idea of MsFEM to develop an MsELLAM scheme. 5.3 Results of the MsELLAM tracking coarse-scale velocity We present the numerical results generated by the MsELLAM scheme that just track the coarse-scale velocity in Table 3 and Fig. 3. We see that this scheme generates simiTable 3 Results of the MsELLAM tracking coarse-scale velocity 1 ( = 100 ) x 1 5 1 10 1 20 1 25 1 50 1 100 5.2 Results of the MsELLAM tracking ne-scale velocity In Table 2 we present the numerical results obtained by the MsELLAM scheme that requires tracking characteristic curves de ned by the ne-scale velocity. In these numerical example runs the size x of the coarse spatial grid varies 1 from 1 to 100 . The size of the ne spatial grid x f is chosen 5 1 to be xed x f = 1000 . We choose the size t of the coarse temporal step is chosen to be t = x. The size t f of the ne-scale temporal step is chosen to be t f = x f . The numerical results in Table 2 and Fig. 2 show that the MsELLAM converges for x > . As x is close to , the MsELLAM does not converge due to resonance effect [10]. In other words, the MsELLAM exhibits the same convergence behavior for multiscale transient advection diffusion equations as the MsFEM for multiscale elliptic equations. L 1 error 0.0801 0.0158 0.0095 0.0093 0.0092 0.0092 L 2 error 0.0739 0.0202 0.0123 0.0118 0.0114 0.0114 L 0.1584 0.0616 0.0358 0.0330 0.0305 0.0301 0.5 Reference Solution dx=1/5 dx=1/10 dx=2/25 dx=1/20 dx=1/25 0.4 Table 2 Results with the MsELLAM scheme tracking ne-scale 1 velocity ( = 100 ) x 1 5 1 10 1 20 1 25 1 50 1 100 0.3 L 1 error 0.0803 0.0163 0.0103 0.0101 0.0099 0.0099 L 2 error 0.0743 0.0208 0.0130 0.0125 0.0122 0.0121 L error 0.1590 0.0624 0.0365 0.0345 0.0319 0.0315 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.1 0.2 Fig. 3 MsELLAM velocity splitting scheme with different meshsize 123 A multiscale ELLAM scheme lar results as the MsELLAM scheme requiring tracking the ne-scale velocity, even though only the coarse-scale velocity is tracked. pressure equation (K p) = 0 (23) 6 Discussion and future work We developed a multiscale Eulerian Lagrangian localized adjoint method (MsELLAM) for time-dependent advection diffusion equations with highly oscillatory coef cients. Preliminary numerical experiments have shown the potential of these MsELLAM schemes. Although we restrict the development of the MsELLAM to a simple one-dimensional advection diffusion equation, our goal is to develop an MsELLAM for coupled systems in porous medium ow. We take the system of single-phase ow as an example to demonstrate the idea. Let c(x, t) be the concentration of an invading uid and let p(x, t) and u(x, t) be the pressure and Darcy velocity of the uid mixture. The mass conservation equation for the uid mixture incorporated with the incompressibility condition, Darcy s law, and the mass conservation equation for the invading uid lead to the following coupled system of PDEs [2,6] K p, (c) ct + (uc D(x, u) c) = cq, x , t [0, T ]. u = q, u= on each spatial cell with a ne-scale subdivision. The construction of the MsFEM basis functions is usually an expensive process. Note that the viscosity = (c) changes very little in majority of the physical domain , where the concentration c is smooth. changes rapidly within the neighborhood of the moving steep front region. Hence, in most of the domain where c is smooth 1 K p (K p). (24) (c) (c) In other words, the MsFEM basis functions constructed at time t = 0 can still be used at later time steps. One needs only to update the MsFEM basis functions in the region with moving steep fronts. We notice that similar arguments work for the transport equation in the system (22). So the spatial con gurations of the MsELLAM basis functions constructed at time t = 0 can be used at later time steps, except within the region with moving steep fronts where the MsELLAM basis functions need to be updated. The temporal variation of the test functions is evaluated via a characteristic tracking, which is needed in the evaluation of other terms in the MsELLAM schemes anyway. Moreover, the characteristic tracking is local and is fully parallelizable. We will pursue our research in this direction in the near future. Finally, we address some implementational issues in the context of the MsELLAM for multiscale advection diffusion equations in multiple space dimensions. In the MsELLAM, the spatial integrals on each coarse cell are evaluated via a numerical quadrature on every ne cell in the coarse cell. A forward tracking algorithm seems to be a feasible choice in this case. The option of tracking a coarse velocity eld seems more attractive, since it greatly simpli es the tracking procedure and avoids the inter-element boundary layer of the basis functions. Numerically, the coarse velocity eld is often chosen as the mean velocity eld on the coarse mesh. As pointed out in [10], an oversampling technique could be used to compute the shape functions on a slightly larger domain to avoid boundary layers in multiple space dimensions. Acknowledgements This research is partially supported by NSF CMG/DMS grant 0621113, DOE grant DE-FG02-06ER25727, National Natural Science Foundation of China No. 10771124, and the Research Fund for Doctoral Program of High Education by Station Education Ministry of China No. 20060422006. (22) System (22) models miscible displacement of one incompressible uid by another or groundwater contaminant transport through a two-dimensional horizontal porous medium reservoir over a time period of [0, T ]. In this system K(x) is the 2 2 permeability tensor of the medium, = (c) is the concentration-dependent viscosity of the uid mixture, q(x, t) is the external source and sink term, (x) is the porosity of the medium, D(x, u) is the diffusion-dispersion tensor. c(x, t) is either the speci ed concentration of the injected uid at sources or c(x, t) = c(x, t) is the resident concentra tion at sinks. The rst two equations in system (22) form a secondorder elliptic equation for the pressure p. For heterogeneous porous media, K(x) exhibit multiscale behavior. The multiscale nite element or volume method developed in [10, 11] can be applied to solve the pressure equation. In fact, the MsFEM framework can be improved to take the full advantage of the physical and mathematical properties of the porous medium ow. The basic idea can be summarized as follows [1]: At time t = 0, the MsFEM basis functions are constructed by using a nite element method to solve the homogeneous References 1. Aarnes, J., Espedal, M.S.: A new approach to upscaling for twophase ow in heterogeneous porous media. Fluid ow and transport in porous media: mathematical and numerical treatment, 1 11, 123 H. Wang et al. Contemp. Math., 295, American Mathenatical Society, Providence, RI (2002) Bear, J.: Dynamics of Fluids in Porous Materials. Elsevier, New York (1972) Binning, P.J. Celia M.A.: A nite volume Eulerian Lagrangian localized adjoint method for solution of the contaminant transport equations in two-dimensional multi-phase ow systems. Water Resour. Res. 32, 103 114 (1996) Celia, M.A., Russell, T.F., Herrera, I., Ewing, R.E.: An Eulerian Lagrangian localized adjoint method for the advection diffusion equation. Adv. Water Resour. 13, 187 206 (1990) Efendiev, Y.R., Hou, T.Y., Wu, X.H.: Convergence of a nonconforming multiscale nite element method. SIAM J. Numer. Anal. 37, 888 910 (2000) Ewing, R.E. (eds.): The Mathematics of Reservoir Simulation. Research Frontiers in Applied Mathematics, 1, SIAM, Philadelphia (1984) Ewing, R.E., Wang, H.: An optimal-order error estimate to Eulerian Lagrangian localized adjoint method for variablecoef cient advection reaction problems. SIAM Numer. Anal. 33, 318 348 (1996) Healy, R.W., Russell, T.F.: Solution of the advection-dispersion equation in two dimensions by a nite-volume Eulerian Lagrangian localized adjoint method. Adv. Water Res. 21, 11 26 (1998) Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997) 10. Hou, T.Y., Wu, X.H.: A multiscale nite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169 189 (1997) 11. Jenny, P., Lee, S.H., Tchelepi, H.A.: Multiscale nite-volume method for elliptic problems in subsurface ow simulation. J. Comput. Phys. 187, 47 67 (2003) 12. Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (1996) 13. Wang, H.: A family of ELLAM schemes for advection diffusion reaction equations and their convergence analyses. Numer. Methods for PDEs 14, 739 780 (1998) 14. Wang, H., Dahle, H.K., Ewing, R.E., Espedal, M.S., Sharpley, R.C., Man, S.: An ELLAM Scheme for advection diffusion equations in two-dimensions. SIAM J. Sci. Comput. 20, 2160 2194 (1999) 15. Wang, H., Ewing, R.E., Qin, G., Lyons, S.L., Al-Lawatia, M., Man, S.: A family of Eulerian Lagrangian localized adjoint methods for multi-dimensional advection reaction equations. J. Comput. Phys. 152, 120 163 (1999) 16. Wang, H., Liang, D., Ewing, R.E., Lyons, S.L., Gin, G.: An approximation to miscible uid ows in porous media with point sources and sinks by an Eulerian Lagrangian localized adjoint method and mixed nite element methods. SIAM J. Sci. Comput. 22, 561 581 (2000) 17. Yanosik, J., McCracken, T.: A nine-point, nite difference reservoir simulator for realistic prediction of adverse mobility ratio displacements. Soc. Pet. Eng. J. 19, 253 262 (1978) 2. 3. 4. 5. 6. 7. 8. 9. 123

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: