# A preliminary investigation on an ELLAM scheme for linear transport equations

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<ul><li><p>A Preliminary Investigation on an ELLAM Schemefor Linear Transport EquationsMohamed Al-Lawatia,1 Hong Wang21Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36,Al-Khod Postal Code 123, Muscat, Sultanate of Oman</p><p>2Department of Mathematics, University of South Carolina, Columbia,South Carolina 29208</p><p>Received 7 August 2001; accepted 15 June 2002</p><p>DOI 10.1002/num.10042</p><p>We present an Eulerian-Lagrangian localized adjoint method (ELLAM) for linear advection-reaction partialdifferential equations in multiple space dimensions. We carry out numerical experiments to investigate theperformance of the ELLAM scheme with a range of well-perceived and widely used methods in uid dynamicsincluding the monotonic upstream-centered scheme for conservation laws (MUSCL), the minmod method, theux-corrected transport method (FCT), and the essentially non-oscillatory (ENO) schemes and weightedessentially non-oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is verycompetitive with these methods in the context of linear transport PDEs, and suggest/justify the development ofELLAM-based simulators for subsurface porous medium ows and other applications. 2002 Wiley Periodicals,Inc. Numer Methods Partial Differential Eq 19: 2243, 2003</p><p>Keywords: advection-reaction equations; characteristic methods; comparison of numerical methods;essentially nonoscillatory schemes; Eulerian-Lagrangian methods; transport equations</p><p>I. INTRODUCTION</p><p>Advection-dominated transport partial differential equations (PDEs) describe the displacementof oil by an injected uid in petroleum recovery, subsurface contaminant transport andremediation, and many other applications. The mathematical models used to describe thesecomplex ow processes are coupled systems of time-dependent nonlinear PDEs and constrain-ing equations. The numerical simulation of these systems presents severe numerical difculties[13]. One of the major difculties in the numerical simulation to these coupled systems is an</p><p>Correspondence to: Hong Wang, University of South Carolina, Department of Mathematics, Columbia, SC 29208(e-mail: hwang@math.sc.edu)Contract grant sponsors: Mobil Technology Company and ExxonMobil Upstream Research CompanyContract grant sponsor: South Carolina State Commission of Higher Education, South Carolina Research InitiativeGrant.Contract grant sponsor: National Science Foundation; contract grant number: DMS-0079549 2002 Wiley Periodicals, Inc.</p></li><li><p>accurate and efcient solution of the advection-dominated transport PDE for the concentration,which is virtually linear in terms of its primary unknown when the uids are fully miscible.Standard nite difference or nite element methods (FDMs, FEMs) tend to generate solutionswith severe nonphysical oscillations. Although classical upwind FDM could eliminate theseoscillations, it yields solutions with excessive smearing and potentially spurious effects relatedto the orientation of the grid. A huge variety of improved methods have been developed to solveadvection-dominated PDEs.</p><p>Most of these methods use xed spatial grids with some form of upstream weighting and thestandard temporal discretization. The optimal test function methods [46] attempt to minimizespatial errors and yield an upstream bias in the resulting numerical schemes. Hence, they aresusceptible to time truncation errors that introduce numerical dispersion and the restrictions onthe size of time steps. They tend to be ineffective for transient advection-dominated problems.Some other methods [79] attempt to reduce the local truncation errors by using nonzero spatialerrors to cancel temporal errors. The streamline diffusion nite element methods (SDFEMs) [10,11] add a numerical diffusion only in the direction of streamlines with no crosswind diffusionintroduced. Many high-resolution methods, such as the total variation diminishing methods(TVD) and the essentially nonoscillatory (ENO) methods [1216] are based on Godunovmethods [17] and are well suited for the solution of nonlinear hyperbolic conservation laws.They resolve shock discontinuities in the solutions of hyperbolic conservation laws withoutexcessive smearing or spurious oscillations. Moreover, they conserve mass; this property is ofessential importance in applications.</p><p>Because of the hyperbolic nature of advective transport, characteristic methods have beensuccessfully applied to solve linear transport PDEs [1822]. Characteristic methods carry outthe temporal discretization by following the movement of particles along the streamlines.Because the solutions of transport PDEs are much smoother along the characteristics than theyare in the time direction, characteristic methods generate accurate solutions even if very largetime steps are used. However, characteristic methods raise many implementational and analyt-ical issues that need to be addressed. Traditional particle tracking methods advance the gridsfollowing the characteristics. They greatly reduce temporal errors and, thus, generate fairlyaccurate solutions. However, they often severely distort the evolving grids and greatly compli-cate the solution procedures. The modied method of characteristics (MMOC) [19] follows theow direction by tracking the characteristics backward from a xed grid at the current time stepand hence, avoids the grid distortion problems present in forward tracking methods. The MMOCsymmetrizes and stabilizes the governing PDEs, and greatly reduces temporal errors. It allowsfor large time steps in a simulation without loss of accuracy and eliminates the excessivenumerical dispersion and grid orientation effects. The major drawbacks of many previouscharacteristic methods are that they fail to conserve mass and have difculties in treating generalboundary conditions.</p><p>The Eulerian-Lagrangian localized adjoint method (ELLAM) was introduced by Celia et al.in solving (one-dimensional constant-coefcient) advection-diffusion PDEs [23] and was thengeneralized to solve linear advection-reaction transport PDEs [24, 25]. The ELLAM method-ology provides a general characteristic solution procedure and a consistent framework fortreating general boundary conditions and conserving mass. Thus, it overcomes the two principalshortcomings of the previous characteristic methods while maintaining their numerical advan-tages. In this article we present an ELLAM for multidimensional linear advection-reactiontransport PDEs. We then carry out numerical experiments to compare the performance of theELLAM scheme with high-resolution methods in the context of linear advection-reactiontransport PDEs, including the monotonic upstream-centered scheme for conservation laws</p><p>AN INVESTIGATION OF AN ELLAM SCHEME 23</p></li><li><p>(MUSCL), the minmod scheme, the ux-corrected transport method (FCT), and the essentiallynonoscillatory (ENO) and the weighted ENO (WENO) schemes [1214, 16, 2629].</p><p>The rest of the article is organized as follows: In Section 2 we present an ELLAM scheme.In Section 3 we briey recall the MUSCL, minmod, FCT, ENO, and WENO schemes. InSection 4 we conduct numerical experiments to investigate the performance of the ELLAM andthese methods. Section 5 contains summary and discussions.</p><p>II. AN ELLAM SCHEMEA. Denition of Test Functions</p><p>We consider a multidimensional linear advection-reaction PDE</p><p>ct vcx, t Kx, tc Fx, t, x, t 0, T,</p><p>cx, t gx, t, x, t I 0, T (2.1)</p><p>where d is a bounded domain with a Lipschitz continuous boundary . A boundarycondition is specied at the inow boundary (I) identied by (I) {xx , v n 0}. Inaddition, an initial condition c(x, 0) c0(x) is specied to close the problem (2.1).</p><p>We dene a quasi-uniform temporal partition on [0, T] by 0 t0 t1 t2 . . . tN1 tN T. Multiplying Eq. (2.1) by space-time test functions w(x, t) that are continuous andpiecewise smooth, vanish outside the space-time strip [tn1, tn], and are discontinuous intime at time tn1, we obtain a space-time weak formulation</p><p>cx, tnwx, tndx tn1</p><p>tn </p><p>v ncx, twx, tdsdt</p><p> tn1</p><p>tn </p><p>cx, twt v w Kwx, tdxdt</p><p>cx, tn1wx, tn1 dx </p><p>tn1</p><p>tn </p><p>Fx, twx, tdxdt, (2.2)</p><p>where w(x, tn1 ) limt3tn1 w(x, t), which takes into account the fact that w(x, t) is discontin-uous in time at time tn1.</p><p>In the ELLAM framework [23], the test functions w are chosen to satisfy the adjoint equationof Eq. (2.1)</p><p>wt v w Kw 0.</p><p>This equation can be rewritten as the following differential equation</p><p>dd wr; x , t</p><p>, Kr; x , t, wr; x , t, 0,</p><p>wr; x , t, t wx , t,</p><p>24 AL-LAWATIA AND WANG</p></li><li><p>along the characteristic y r(; x , t) dened by</p><p>dyd vy, , with yt x .</p><p>Solving the adjoint equation along the characteristic r(; x , t) yields the following expression forthe test function w(x, t)</p><p>wr; x , t, wx , te</p><p>t Kr;x ,t,d</p><p>.</p><p>Therefore, the test functions w in Eq. (2.2) should vary exponentially along the characteristicsr(; x , t). Once w(x , t) is specied, w(r(; x , t), ) is determined completely along thecharacteristic r(; x , t). Thus, to dene the test functions w in the space-time strip [tn1,tn], we only need to dene w on at the time tn and on the space-time outow boundary (O) [tn1, tn] with (O) {x v n 0}.B. Derivation of a Reference Equation</p><p>To avoid confusion, we replace the dummy variables x and t in the second term on theright-hand side of Eq. (2.2) by y and and reserve x and t for the points in at time tn or at [tn1, tn]. Let () be the set of the points that will ow out of the domain duringthe time period [, tn]. For any y (), there exists an x such that y r(; x, tn).Likewise, for any (y, ) (), there exists a pair (x, t) (O) [tn1, tn] such that y r(;x, t). Therefore,</p><p>tn1</p><p>tn </p><p>Fy, wy, dyd tn1</p><p>tn </p><p>Fr; x, tn, wr; x, tn, drd</p><p> tn1</p><p>tn </p><p>Fr; x, t, wr; x, t, drd. (2.3)</p><p>The rst term on the right-hand side of Eq. (2.3) is evaluated by applying the Euler formulaat time tn, leading to</p><p>tn1</p><p>tn </p><p>Fr; x, tn, wr; x, tn, drd</p><p>t*x</p><p>tn</p><p>Fr; x, tn, wr; x, tn, r; x, tnx ddx</p><p>Fx, tnwx, tnt*x</p><p>tn</p><p>eKx,tntnddx E1f, w </p><p>1x, tnFx, tnwx, tndx E1F, w.</p><p>AN INVESTIGATION OF AN ELLAM SCHEME 25</p></li><li><p>Here the space-dependent time step t(I)(x) tn t*(x), where t*(x) tn1 if thecharacteristic r(; x, tn) does not backtrack to the boundary during the time period [tn1, tn],or t*(x) [tn1, tn] is the time when r(; x, tn) intersects the boundary otherwise (cf. Fig. 1).(1)(x, tn) (1 eK(x,tn)t</p><p>(I)(x))/K(x, tn) if K(x, tn) 0, or t(I)(x) otherwise. E1(F, w) is thelocal truncation error.</p><p>The second term on the right-hand side of Eq. (2.3) is treated similarly. We obtain</p><p>tn1</p><p>tn </p><p>Fr; x, t, wr; x, t, drd</p><p> tn1</p><p>tn O</p><p>v n2x, tFx, twx, tdsdt E2F, w.</p><p>Here t(O)(x, t) tn t*(x, t) for (x, t) (O) [tn1, tn], where t*(x, t) tn1 ifr(; x, t) does not backtrack to the boundary during the time period [tn1, t], or t*(x, t) [tn1, t] is the time when r(; x, t) intersects the boundary otherwise. (2)(x, t) (1 </p><p>eK(x,t)t</p><p>(O)(x, t))/K(x, t) if K(x, t) 0 or t(O)(x, t) otherwise. E2(F, w) is the local truncationerror.</p><p>Incorporating the two equations above into Eq. (2.3) and the inow boundary condition inproblem (2.1) into Eq. (2.2), we obtain the following reference equation:</p><p>cx, tnwx, tndx tn1</p><p>tn O</p><p>vx, t nxcx, twx, tdsdt</p><p>cx, tn1wx, tn1 dx </p><p>1x, tnFx, tnwx, tndx</p><p> tn1</p><p>tn O</p><p>2x, tvx, t nxFx, twx, tdsdt</p><p> tn1</p><p>tn I</p><p>vx, t nxgx, twx, tdsdt Ew, (2.4)</p><p>where</p><p>FIG. 1. Illustration of characteristic tracking.</p><p>26 AL-LAWATIA AND WANG</p></li><li><p>Ew tn1</p><p>tn </p><p>cx, twtx, t vx, t wx, t Kx, twx, tdxdt E1F, w E2F, w.</p><p>C. A Numerical Scheme</p><p>In the ELLAM scheme, the trial space h consists of piecewise linear (or d-linear) functions in at time tn and on (O) [tn1, tn]. Because of the boundary condition (2.1), no degrees offreedom should be introduced at (I) at time tn. Similarly, because the solutions are known at thetime step tn1, no degrees of freedom should be introduced at (O) at time tn1. However, toconserve mass, all the test functions should sum exactly to one [23]. Therefore, the test space h is obtained by modifying the trial space h: For a basis function associated with a node at(I) at time tn, we add it to the basis function associated with its adjacent node that is inside .The basis functions associated with the nodes at (O) at time tn1 are treated similarly. AnELLAM scheme is formulated as follows: nd c(x, t) h, which satises the boundarycondition (2.1), such that @w(x, t) h</p><p>cx, tnwx, tndx tn1</p><p>tn O</p><p>vx, t nxcx, twx, tdsdt</p><p>cx, tn1wx, tn1 dx </p><p>1x, tnFx, tnwx, tndx</p><p> tn1</p><p>tn O</p><p>2x, tvx, t nxFx, twx, tdsdt</p><p> tn1</p><p>tn I</p><p>vx, t nxgx, twx, tdsdt. (2.5)</p><p>By using a Lagrangian coordinate, the ELLAM scheme (2.5) signicantly reduces the temporaltruncation errors and generates accurate solutions even if very large time steps are used.Moreover, it symmetrizes the governing PDE (2.1) and generates a well-conditioned, symmetricand positive-denite coefcient matrix. Thus, the discrete system can be solved efciently by,for example, the conjugate gradient method in an optimal order without any preconditioningneeded. Furthermore, this scheme naturally incorporates the boundary condition (2.1) into itsformulation and conserves mass. Finally, most terms in the scheme are standard in the FEM andcan be solved in a straightforward manner. In the rst term on the right-hand side, the trial andtest functions are actually dened at different time steps. Hence, its evaluation could be verychallenging and raises serious numerical difculties. We use a forward Euler or second-orderRunge-Kutta tracking algorithm [30] to evaluate this term.</p><p>III. NUMERICAL EXPERIMENTS</p><p>Many high-resolution methods are based on Godunov schemes and are well suited for thesolution of nonlinear hyperbolic conservation laws and resolve shock discontinuities in the</p><p>AN INVESTIGATION OF AN ELLAM SCHEME 27</p></li><li><p>solutions without excessive smearing or spurious oscillations. In this section we carry outnumerical experiments to investigate the performance of the ELLAM scheme and several highresolution methods for linear advection-reaction Eq. (2.1). This list includes the MUSCLscheme, the minmod scheme, the FCT method, and the third- and fourth-order ENO and WENOschemes [1214, 16, 2629]. These methods are developed as an improvement over traditionalxed-stencil high-order FD interpolations, which are known to be oscillatory in nature espe-cially near discontinuities of the solutions. The resulting oscillations do not decay as the meshis rened and lead to further instabilities in the solution.</p><p>A. Test Problem</p><p>The test problem is an incompressible ow in a two-dimensional homogeneous medium, wherethe analytical solution is known. This example entails the rotation of the initial conguration andis a widely used to test for a variety of numerical artifacts including deformation, nonphysicaloscillations, numerical diffusion, numerical stability, and phase error. We refer readers to [31,32] for the performance of the ELLAM scheme for problems with discontinuities.</p><p>In the numerical experiments, the spatial domain is (0.5, 0.5) (0.5, 0.5). Arotating velocity eld of v(x, y) (4y, 4x) is used, which gives one complete rotation in thetime interval of [0, T] [0,...</p></li></ul>