A Dynamic, Adaptive, Locally Conservative, and Nonconforming …web.kaust.edu.sa/faculty/ShuyuSun/pub/DG_CEC2006.pdf · A Dynamic, Adaptive, Locally Conservative, and Nonconforming

A Dynamic, Adaptive, Locally Conservative, andNonconforming Solution Strategy for Transport

Phenomena in Chemical Engineering

SHUYU SUN AND MARY F. WHEELER

The Institute for Computational Engineering and Sciences (ICES),The University of Texas at Austin, Austin, Texas, USA

A family of discontinuous Galerkin (DG) methods are formulated and applied tochemical engineering problems. They are the four primal discontinuous Galerkinschemes for space discretization: symmetric interior penalty Galerkin, Oden-Babu�sska-Baumann DG formulation, nonsymmetric interior penalty Galerkin, andincomplete interior penalty Galerkin. Numerical examples of DG to solve typicalchemical engineering problems, including a diffusion-convection-reaction system ina catalytic particle, a problem of heat transfer in a fixed bed, and flow and contami-nant transport simulations in porous media, are presented. This article highlights thesubstantial advantages of DG on adaptive mesh modification over traditional meth-ods. In particular, we propose and investigate the dynamic mesh modification strat-egy for DG guided by mathematically sound a posteriori error estimators.

Keywords Discontinuous Galerkin methods; Dynamic mesh adaptation; Para-bolic partial differential equations; Transport phenomena

Introduction

Transport phenomena, including mass, momentum, and heat transfer and the coup-ling of transport and chemical reactions, play dominant roles in chemical engineeringprocesses (Bird et al., 1960). For example, a distillation process in a packed columninvolves fluid flow (momentum transfer) of liquid and vapor phases, mass transferwithin a homogeneous phase and on the liquid and vapor interface, and heat transferwithin and between phases (see, e.g., Sun et al., 1998). Transport phenomena are alsoimportant in biomedical engineering (see e.g., Sun et al., 2005). Mathematical mod-eling of transport phenomena has been widely employed in many areas of chemicalengineering. Analytical solutions for such modeling systems are usually not availabledue to the complexity of chemical engineering systems. Instead, numerical methodshave to be used extensively (see, e.g., Rubin and James, 1973; Rubin, 1983; Yeh andTripathi, 1991).

Finite difference methods (FDMs), in particular cell center finite differencemethods, are widely used for numerical simulation of transport phenomena arisingin chemical engineering due to their simplicity of implementation. However, FDMshave many limitations. It is difficult for them to handle complex geometry of the

Address correspondence to Shuyu Sun, The Institute for Computational Engineeringand Sciences, University of Texas at Austin C0200, 201 E. 24th St., ICES 5.316, Austin,TX 78712, USA. E-mail: [email protected]

Chem. Eng. Comm., 193:1527–1545, 2006Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440600584284

1527

domain. FDMs are generally low-order methods and cannot directly handle full ten-sor parameters (e.g., anisotropic diffusivity, conductivity, or permeability). More-over, FDMs do not provide the flexibility of local mesh adaptation without usingcomplicated arguments. Finite volume methods (FVMs) and finite element methods(FEMs) are two families of solution strategies that are flexible for the treatment ofcomplex geometry and local mesh refinement and are widely used in the field of com-putational fluid dynamics (CFD). The traditional FEMs (the continuous Galerkinmethods), however, are not locally conservative without post-processing. FVMs,by design, preserve conservation locally, but they are generally low-order methodsand are complicated to extend to high-order approximations. In addition, it is diffi-cult for FVMs to treat full tensor parameters.

Discontinuous Galerkin (DG) methods are specialized FEMs that utilize discon-tinuous spaces to approximate solutions, with boundary conditions and continuityon interelement boundaries weakly imposed through bilinear forms. DGmethods have recently gained popularity for their many attractive properties (seee.g., Cockburn et al., 2000, and references therein). First of all, the methods arelocally conservative while most classical FEMs are not. In addition, they have lessnumerical diffusion than most conventional algorithms, thus are likely to offer moreaccurate solutions, especially for convection-dominated transport problems. Theyhandle rough coefficient problems and capture discontinuities in solutions very wellby the nature of discontinuous function spaces. DG can naturally handle inhomoge-neous boundary conditions and curved boundaries. A DG solution may be extendedso that a continuous flux is defined over the entire domain; consequently, DG maybe easily coupled with conforming methods, in addition to coupling with noncon-forming schemes. Furthermore, with appropriate meshing, DG is capable of deliver-ing exponential rates of convergence. For time-dependent problems in particular, themass matrices of DG are block diagonal, which is not true for conforming FEMs.This provides DG a substantial computational advantage when explicit time integra-tions are used. A disadvantage of DG is a larger number of degrees of freedom thancontinuous FEMs due to the repeated unknowns on element interfaces. However, weshall show in this study that this weakness can be compensated, at least for high-order approximations in quadrilaterals, prisms, or hexahedra, by replacing thetensor-product polynomial spaces by complete polynomial spaces.

The DG methods trace back to the work of Nitsche (1971). Other representativeearly works on DG include Reed and Hill (1973), Babu�sska (1973), Babu�sska andZlamal (1973), Wheeler (1978), Wheeler and Darlow (1980), and Arnold (1982).We refer the reader to the book of Cockburn et al. (2000) for a rich collection ofDG references in 1980s and 1990s. More recent work of DG to elliptic problemscan be found in Chen and Chen (2004), Karakashian and Pascal (2003), Larsonand Niklasson (2004), and Riviere et al. (1999, 2000, 2001). Recent developmentsof DG to hyperbolic conservation laws include Bey and Oden (1996), Bey et al.(1996a, b), and Johnson (1993). Developments of DG to solve parabolic equations,especially to simulate transport phenomena, are not as rich as the DG work on ellip-tic and hyperbolic problems; some representative works of the former includeDawson and Proft (2002a, b), Schotzau and Schwab (2001), Riviere and Wheeler(1999, 2002), Sun et al. (2001), Wheeler et al. (2003), and Sun and Wheeler(2004, 2005a–d, 2006a, b). A mixed form of DG, called local discontinuous Galerkin(LDG) methods, has received substantial attention (e.g., Castillo et al., 2000;

1528 S. Sun and M. F. Wheeler

Cockburn and Shu, 1998a, b; Dawson et al., 2000). We refer the reader to the workof Arnold et al. (2002) for a unified analysis, which applies to most DG variations.

In this article, we consider a family of DG methods applied to chemical engin-eering problems. They are the four primal discontinuous Galerkin schemes for spacediscretization: symmetric interior penalty Galerkin (SIPG) (Wheeler, 1978; Sun andWheeler, 2005c), Oden-Babu�sska-Baumann version of DG (OBB-DG) (Oden et al.,1998), nonsymmetric interior penalty Galerkin (NIPG) (Riviere et al., 2001) andincomplete interior penalty Galerkin (IIPG) methods (Sun, 2003; Dawson et al.,2004; Sun and Wheeler, 2005c). For time integration, we shall use both low-order(Euler) and high-order (Runge-Kutta) methods in the numerical examples. Formu-lation of these DG schemes will be first introduced to a general time-dependent para-bolic type equation, and we shall then apply them to simulations of transport andreaction arising from chemical engineering. Three representative chemical engineer-ing problems, namely, a diffusion-convection-reaction problem in a catalytic par-ticle, a problem of heat transfer in a fixed bed, and a contaminant transportproblem coupled with flow in porous media, will be presented. Efficient implemen-tation issues and advantages of DG will be addressed. In particular, dynamic meshadaptation strategies are proposed by which the local physical phenomena can beeffectively and efficiently captured.

Model Equation and Discontinuous Galerkin Algorithms

We let X be a polygonal and bounded domain in a d-dimensional space (d ¼ 1, 2, or3) and let T be the final simulation time. We consider the following general parabolicequation:

@/c

@tþr � ðuc�DrcÞ ¼ rðcÞ; ðx; tÞ 2 X� ð0;TÞ ð1Þ

where the dependent (unknown) variable is c. Here, /, a scalar, u, a vector, and D, asecond-order tensor, are given data and can be functions of time and space. Theright-hand side r(c) is a given function of the unknown variable c and can be func-tions of time and space as well: rðcÞ ¼ rðc; x; tÞ. Equation (1) can be used to modeltransport phenomena including heat and mass transfer processes, as well as fluidflow. For example, to model single-phase contaminant transport in porous media,the unknown variable c is considered as the concentration of a species (amountper volume); / is the porosity; u is the Darcy velocity; D is the dispersion=diffusionfusion tensor; r(c) is the term from well injection, extraction, and chemical reaction.More variations of the model equation will be presented in the numerical examplesections of this article.

We consider two types of general boundary conditions: the Dirichlet and theRobin types. We divide the domain boundary @X into the Dirichlet boundary CD

and the Robin boundary CR such that CD \ CR ¼ / and �CCD [ �CCR ¼ @X. The follow-ing boundary conditions are imposed for the problem (1):

cðx; tÞ ¼ cB; x 2 CD; t 2 ð0;TÞ ð2Þ

�Drc � n ¼ kBðc� cBÞ; x 2 CR; t 2 ð0;TÞ ð3Þ

Adaptive, Nonconforming Solution Strategy for Transport 1529

where cB and kB are given boundary data that can be constants or functions of timeand boundary space. We impose the following initial condition:

cðx; 0Þ ¼ c0 ð4Þ

where c0 is the given initial data that could be a function of space. We remark thatthe inflow, outflow, and no-flow boundary conditions employed in modelingcontaminant transport problems (see following sections of this article) are specialor limiting cases of the Robin boundary condition (3).

To discretize the space, we let Eh be a family of nondegenerate and possiblynonconforming partitions of the domain X composed of line segments for one spatialdimension, triangles and=or quadrilaterals for two spatial dimensions, or tetrahedra,prisms, and=or hexahedra for three spatial dimensions. The set of interior elementinterfaces is denoted by Ch. We define the average and jump for an element-wisesmooth function / on the interface c of two elements e1 and e2 as follows.

f/g ¼ 1

2/��e1

� �jc þ /

��e2

� �jc

� �½/� ¼ /

��e1

� �jc � /

��e2

� �jc

ð5Þ

The discontinuous finite element space is taken to be

DrðEhÞ ¼ f 2 L2ðXÞ : f je2 PrðeÞ; 8e 2 Eh

� �ð6Þ

where PrðeÞ denotes the complete polynomial space, i.e., the space of polynomials of(total) degree less than or equal to r on the element e. In this study, we takef1; x; y; x2; xy; y2g and f1; x; y; x2;xy; y2; x3; x2y; xy2; y3g as the basis functions intwo spatial dimensions for the function spaces P2ðeÞ and P3ðeÞ, respectively. Weremark that continuous FEMs work with the complete polynomial spaces for tri-angular and tetrahedral meshes, but require the tensor-product polynomial spacesfor quadrilaterals, prisms, and hexahedra. With no continuity constraints, DG meth-ods work well with the complete polynomial spaces for all types of elements. This pro-perty reduces the number of unknowns of DG for quadrilaterals, prisms, andhexahedra without a loss of accuracy.

We introduce the bilinear form Bðc;wÞ as follows:

Bðc;wÞ ¼Xe2Eh

Ze

Drc� cuð Þ � rw�Xc2Ch

ZcfDrc � ng½w� � sform

Xc2Ch

ZcfDrw � ng½c�

þXc2Ch

Zc

c�u � n½w� �Xc�CD

Zc

Drc � nw� sform

Xc�CD

Zc

Drw � nc

þX

c2CD[CR

Zc

cu � nwþXc�CR

Zc

kBcwþXc2Ch

r2rc

hc

Zc½c�½w� ð7Þ

where c� is the upwind value of c, and sform is taken to be �1 for NIPG and OBB-DG, 1 for SIPG, and 0 for IIPG. The penalty parameter rc is zero for OBB-DG,nonnegative for NIPG, and strictly positive for SIPG and IIPG. The linear func-tional Lðw; cÞ is defined as


Lðw; cÞ ¼Z

XrðcÞw� sform

Xc�CD

Zc

Drw � ncB þXc�CR

Zc

kBcBw ð8Þ

The continuous-in-time DG solution of the problem (1)–(4) is given by the functionCDGðx; tÞ that belongs to the space DrðEhÞ for any fixed time and satisfies the follow-ing equations:

@/CDG

@t;w

� �þ B CDG;w

� ¼ Lðw; CDGÞ; 8w 2 DrðEhÞ; 8t 2 ð0; tÞ;

ð/CDG;wÞ ¼ ð/c0;wÞ; 8w 2 DrðEhÞ; t ¼ 0 ð9Þ

A fully discretized DG algorithm can be obtained from the above formulation byfurther applying a standard time integration scheme such as the backward Eulermethod, the trapezoid rule or a Runge-Kutta method.

Four different primal DG schemes, i.e., OBB-DG, NIPG, SIPG, and IIPG, areunified in a single formulation (9), depending upon the choice of sform and the pen-alty term. The algorithm (9) has been known to possess local conservative property(Sun, 2003), i.e., the element-wise satisfaction of mass=heat=momentum balance.The existence of a unique solution of the equation system (9) has been mathemat-ically established (Sun and Wheeler, 2005c). It has been shown (Riviere and Wheeler,2002) that the error in the L2(H1) norm from OBB-DG converges optimally in h (themesh size) and nearly optimally in p (the degree of approximation):

ffiffiffiffi/

pCDG � c� ��

L1ðL2Þþ��D1=2r CDG � c

� ��L2ðL2Þ

� Khminðr;s�1Þ

rs�52

ð10Þ

where CDG is the DG solution, c the exact solution, r the order of approximation,h the maximum element length, and s the regularity of the solution. The positive con-stant K depends upon only the solution and is independent of the mesh size and theorder of approximation. Because of the penalty term, the convergence of NIPG,IIPG, and SIPG in the L2(H1) norm is not only optimally in h, but is also closerto optimality in p (Sun, 2003):

ffiffiffiffi/


L1ðL2Þþ��D1=2r CDG � c

� ��L2ðL2Þ

� Khminðr;s�1Þ

rs�32

ð11Þ

If the mesh is conforming and contains only line segments, triangles, or tetrahedra,the NIPG, IIPG, and SIPG methods achieve optimal L2(H1) convergence in both hand p (Sun, 2003).

It should be pointed out that SIPG is the only primal DG method that possessesoptimal L2(L2) convergence in h (Sun and Wheeler, 2005c):

ffiffiffiffi/


L2ðL2Þ� K

hminðrþ1;sÞ

rs�32

ð12Þ

Because of this optimality in the L2(L2) norm, the SIPG method is particularlyimportant, especially if the scalar unknown variable c is of primary interest ratherthan its flux. SIPG is also interesting from a computational point of view, since it


is the only DG method that gives a symmetric algebraic system for diffusion-reactiontype equations, which could be exploited to construct fast linear solvers. On theother hand, the rest of primal DG methods are useful in certain other situations.In particular, OBB-DG and NIPG methods are capable of handling problemsinvolving highly varying coefficients, and IIPG is the only fully compatible methodamong the four for coupled flow and transport problems (Dawson et al., 2004).

Numerical Examples without Mesh Adaptation

A Diffusion-Convection-Reaction System in a Catalytic Particle

We first apply DG to the classic chemical engineering problem concerning diffusion-convection and reaction in a catalytic particle (e.g., Levenspiel, 1998). The modelequation is given by

@c

@t¼ @2c

@x2� km

@c

@x� U2c; 0 < x < 2; t > 0 ð13Þ

The variable c, a normalized concentration, is a function of time t and one-dimensional space x. We note that, even though physical phenomena occur in athree-dimensional space, the model equation reduces to a single spatial dimension bytaking advantage of the symmetry of the problem. The variable t is the time normalizedby the diffusion time constant. Similarly, the space variable x has been normalized bythe half-thickness of the particle. km is the intraparticle Peclet number and U is theThiele module. We impose the normalized boundary and initial conditions:

cðx; 0Þ ¼ 0; 0 < x < 2

cð0; tÞ ¼ cð2; tÞ ¼ 1; t > 0ð14Þ

Simulations are conducted for different values of parameters km and U in order to testthe performance of the algorithm. To demonstrate that DG works well even withoutmesh adaptation, we use a uniform grid with 100 elements. Although the other threeDG schemes could be used as well, OBB-DG is chosen for this numerical example.OBB-DG is more convenient than the other three in the sense that it does not containa penalty term, thus eliminating the choice of penalty parameters. The complete quad-ratic basis functions (i.e., P2) are used for each element. The backward Euler methodwith a uniform time step Dt ¼ 0:01 is used for time integration. We note that, in allDG computations throughout this article, no slope limiter is used. Even though slopelimiters are useful in controlling the total variation of the solution and in reducing sol-ution oscillations, the employment of slope limiters often results in additional numeri-cal diffusion and a reduction of numerical accuracy. It was found (Sun and Wheeler,2006a, b) that slope limiters can be generally avoided if a proper spatial discretizationis chosen. In particular, adaptive mesh modification eliminates most oscillations in DGsolutions.

Concentration profiles are plotted in Figures 1 and 2 at various times, fromt ¼ 0.1 to t ¼ 1, with various combination of model parameters. We have verifiedthe convergence of DG solutions by comparing with DG solutions in more refinedmeshes (not shown for brevity) and with published solutions in the literature(Coimbra et al., 2004). Figure 1 compares the concentration profiles for differentintraparticle Peclet numbers. We remark that Figure 1(a) displays the behavior of


the diffusion-reaction system in the absence of convection. Figure 2 illustrates theinfluence of the Thiele modules on the concentration profile. Clearly, the behaviorsof the diffusion-convection-reaction system are thoroughly reproduced by the DGscheme in a wide range of system parameters. In addition, it is shown that DG iseffective for both the convection-free diffusion-reaction system [Figure 1(a)] andthe convection-dominated diffusion-convection-reaction problem [Figure 2(a)], aswell as the reaction-dominated system [Figure 2(f)].

Figure 1. Effect of the intraparticle Peclet number on the concentration profile (U ¼ 1): (a)km ¼ 0, (b) km ¼ 1, (c) km ¼ 5.

Figure 2. Effect of the Thiele module on the concentration profile (km ¼ 10): (a) U ¼ 0, (b)U ¼ 1, (c) U ¼ 2, (d) U ¼ 3, (e) U ¼ 5, (f) U ¼ 10.


A Heat Transfer Process in a Fixed Bed

The study of heat transfer in fixed beds is of considerable importance in various unitoperations in chemical engineering. In particular, simulations of this process are cru-cial to the analysis and design of separation units in which a fluid passing throughthe bed exchanges mass and heat with the particles in the presence of chemical reac-tions (e.g., exothermic catalytic reaction) or in the absence of chemical reactions(e.g., non-isothermal adsorption).

We now consider a two-dimensional pseudo-homogeneous model described bythe following governing dimensionless equation (e.g., Smith, 1956; Ferreira et al.,2002):

ð1þ nhÞ@T

@t¼ 1

Peh

@2T

@x2þ L

Per

@2T

@r2� @T

@xþ L

rPer

@T

@r;

0 < x < 1; 0 < r < 1; t > 0 ð15Þwhere T is the dimensionless temperature, x the normalized axial variable along thefixed bed, r the normalized radial variable, and t the dimensionless time. There arefour dimensionless parameters in the differential equation: nh, the thermal capacityfactor, Peh, the axial thermal Peclet number, Per, the radial thermal Peclet number,and L, the ratio of the bed length and radius. There is one more dimensionless num-ber, Bi, the thermal Biot number, in the following imposed boundary conditions:

Tð0; r; tÞ ¼ 1; 0 � r < 1; t > 0

@T

@xð1; r; tÞ ¼ 0; 0 � r < 1; t > 0

@T

@rðx; 0; tÞ ¼ 0; 0 < x < 1; t > 0

@T

@rðx; 1; tÞ ¼ �BiTðx; 1; tÞ; 0 < x < 1; t > 0

8>>>>>>>>>><>>>>>>>>>>:

ð16Þ

The initial condition is given by

Tðx; r; 0Þ ¼ T0; 0 < x < 1; 0 � r < 1 ð17Þ

We carry out the simulation with the following parameters: nh ¼ 1:3, Peh ¼ 100,Per ¼ 500, L ¼ 20, Bi ¼ 8, and T0 ¼ 0. We use OBB-DG for this numericalexample and remark that the other three DG schemes could be applied as well. Auniform 32-by-32 mesh and the complete quadratic basis functions for eachelement are used for spatial discretization. The backward Euler method with auniformtime step Dt ¼ 0:01 is used for time integration. Temperature profiles arepresented in Figure 3 at various times, from t ¼ 0.05 to t ¼ 10. The convergenceof DG solutions may be confirmed by comparing with solutions in refined discreti-zations and with published solutions in the literature (Coimbra et al., 2004; Ferreiraet al., 2002). It is easy to see that the enthalpy (heat energy) is transferred into thebed mainly by convection at the entrance of the bed and is transferred away fromthe bed by both the conduction through the wall and the convection at the bed exit.Inside the bed, heat transfer is dominated by convection in early times and isbalanced by convection and conduction at later times. The temperature profile ulti-mately approaches a steady state, where heat is transferred in the radial direction


only by conduction and in the axial direction mainly by convection. This numericalexample demonstrates that DG is effective for both the conduction-dominated andthe convection-dominated problems.

Flow and Contaminant Transport in Porous Media

The aim of this example is to show the flexibility of DG in treating complex geome-tries with unstructured mesh and the ability of DG to handle variable (jumping)material properties. In addition, we shall demonstrate that the spatial discretizationof DG works well with high-order time integration. We consider the following nor-malized single-phase flow and contaminant transport system in geological media:

�r � ðKrpÞ ¼ r � u ¼ 0; x 2 X

u � n ¼ 0; x 2 CN

p ¼ pB; x 2 CD

8><>: ð18Þ

Figure 3. Normalized temperature profiles at various times: (a) t ¼ 0.05, (b) t ¼ 0.1, (c) t ¼ 0.2,(d) t ¼ 0.5, (e) t ¼ 1, (f) t ¼ 2, (g) t ¼ 3, (h) t ¼ 4, (i) t ¼ 10.


@/ec

@tþr � ðuc�DrcÞ ¼ 0; ðx; tÞ 2 X� ð0;TÞ

ðuc�DrcÞ � n ¼ 0; ðx; tÞ 2 Cin � ð0;TÞ

ð�DrcÞ � n ¼ 0; ðx; tÞ 2 Cout � ð0;TÞ

cðx; 0Þ ¼ c0ðxÞ; x 2 X

8>>>>>><>>>>>>:

ð19Þ

The computational domain X [shown in Figure 4(a)] has two different rock types.In the circular area XC centered at (1, 0.5) with a radius 0.1, the conductivity K is 0.01,while it is 0.1 elsewhere. The pressure P is specified as 1 on the left edge of the domain(i.e., at x ¼ 0) and as 0 on the right edge (i.e., at x ¼ 2). On other boundaries of thedomain, the no-flow boundary condition is imposed. For the transport problem, wedenote by Cin the inflow boundary and by Cout the outflow=no-flow boundary. Aconstant diffusion-dispersion diagonal tensor D is assumed with Dii ¼ 0.01. The effec-tive porosity /e is 0.3 in XC and 0.1 elsewhere, due to different adsorption behaviors inthe two types of rock. The initial concentration is 1 in XC and zero elsewhere.

OBB-DG is employed to solve for both the flow and transport equations. Wedid not explicitly write down the formulation of DG methods for time-independentflow problems in previous sections, but the DG formulation for flow equations issimilar to that for transport except the time accumulation term is dropped out.The simulation time interval is (0, 2), and we use the explicit fourth-order Runge-Kutta method for time integration with a uniform time step Dt ¼ 1� 10�5. The com-plete quadratic basis function (i.e., the P2 space) is used in each element for both flowand transport. The employed unstructured mesh is shown in Figure 4(a), where therock XC is plotted in red.

The simulated pressure field superimposed with velocity is shown in Figure 4(b).Clearly, the velocity is relatively small in the circular rock XC due to the low conduc-tivity there. This velocity solution is used to compute the advection term in the trans-port system. Figure 5 shows the simulated concentration profiles at various times.It is observed that diffusion dominates during early times (roughly before t ¼ 0.1)and advection starts to play a dominant role at later times (roughly after t ¼ 1).To demonstrate the convergence of solutions in both time integration and spatialdiscretization, we rerun the simulation using smaller time steps and a refined spatialdiscretization with all other parameters being the same as before. Figure 6(a)is the concentration profile at t ¼ 0.5 with a uniform time step Dt ¼ 1� 10�6

Figure 4. Flow and transport example: (a) unstructured mesh on a domain consisting of tworock types; (b) pressure contour superimposed with velocity simulated by OBB-DG.


and the P2 space for all elements, and the concentration result at t ¼ 0.5 plotted inFigure 6(b) is obtained from OBB-DG with a uniform time step Dt ¼ 1� 10�6 andthe P3 space (the complete cubic basis function) for all elements. Consistency ofFigures 5(c), 6(a), and 6(b) confirms that the solution we obtained is indeed a con-vergent one. As shown in this and previous examples, the explicit time integrationrequires substantially smaller time steps than the implicit approach due to the tight

Figure 6. Flow and transport example, convergence study: (a) Dt ¼ 1� 10�6 and the P2 space,(b) Dt ¼ 1� 10�6 and the P3 space.

Figure 5. Flow and transport example, normalized concentration profiles at various times: (a)t ¼ 0.1, (b) t ¼ 0.2, (c) t ¼ 0.5, (d) t ¼ 1.0, (e) t ¼ 1.5, (f) t ¼ 2.


stability condition (the CFL condition) for explicit methods. However, due to theirsimplicity and low computational cost, explicit implementations are often used forhigh-order time integration except for highly stiff systems, where implicit or mixedimplicit-explicit methods have to be employed. Of course, the small time steprequirement in this example also resulted from the nonuniformity of the mesh struc-ture and velocity field. It should be noted that explicit DG methods need to solveonly local algebraic equation systems (arising from the mass matrix) element-wise,or do not need to solve any algebraic equation system at all if orthogonal basis func-tions are used. In contrast, implicit DG methods require solving a global algebraicequation system (arising from the stiffness and mass matrices) each time step. Thisjustifies the preference for high-order explicit time integration for systems with alarge number of degrees of freedom, even though explicit methods demand smallertime steps.

Mesh Adaptation Strategies

For any adaptive mesh modification, we need to know which set of elements have tobe refined or coarsened to improve our solution. This goal is often achieved byemploying an a posteriori error estimate (or error indicator), which provides valuableerror information and can be used to guide adaptive modifications of the mesh (see,e.g., Remacle et al., 2003; Biswasa et al., 1994; Adjerid et al., 2002; Karakashian andPascal, 2003). DG possesses substantial advantages over classic FEMs in term ofmesh adaptivity. The approximation spaces for DG are localized in each element,which provides a capability allowing for general nonconforming meshes with vari-able degrees of approximation. In particular, nonmatching grids and hanging nodesare treated naturally in DG. This results in substantially more flexible implemen-tation of adaptivity for DG than for conventional FEM approaches. This flexibilityalso increases the efficiency of adaptivity because the unnecessary areas do not needto be refined in order merely to maintain conformity of the mesh.

Using a duality argument, we have developed an explicit L2(L2) a posteriorierror estimate for SIPG applied to time-dependent problems (Sun and Wheeler,2006a):

ffiffiffiffi/


L2ðL2Þ� K

Xe2Eh

g2e

!1=2

ð20Þ

where the constant K is independent of the mesh size h, and the local error indicatorge is defined by

g2e ¼ h4

e RIk k2L2ðL2ðeÞÞþ

1

2

Xc�@en@X

hc RB0k k2L2ðL2ðcÞÞ þ

Xc�@e\@X

hc RB0k k2L2ðL2ðcÞÞ

þ 1

2

Xc�@en@X

h3c RB1k k2

L2ðL2ðcÞÞ þX

c�@e\@Xh3

c RB1k k2L2ðL2ðcÞÞ ð21Þ

Here, the residual quantities are defined below:

RI ¼ r CDG�

� /@CDG

@t�r � CDGu�DrCDG

� ð22Þ


RB0 ¼CDG�

; x 2 c; c 2 Ch

cB � CDG; x 2 c; c � CD

0; x 2 c; c � CR

8<: ð23Þ

RB1 ¼DrCDG � n�

; x 2 c; c 2 Ch

kB CDG � cB

� þDrCDG � n; x 2 c; c � CR

0; x 2 c; c � CD

8<: ð24Þ

This a posteriori error estimate is especially valuable in cases where the primal scalarunknown rather than the flux is of interest. It is efficient to compute and also easy toimplement. Numerical experiments show that it is good in capturing the areas withlarge errors and could guide effective mesh modifications to achieve efficient adap-tivities (Sun and Wheeler, 2005d, 2006b). In particular, they are effective in capturingconcentration fronts in reactive transport problems. As for the other primal DGschemes, we have established a unified a posteriori error estimate approach in theL2(H1) norm (Sun and Wheeler, 2005b):

ffiffiffiffi/


L1ðL2Þþ��D1=2ðuÞr CDG � c

� ��L2ðL2Þ

� KXe2Eh

g2e

!1=2

ð25Þ

where the constant K is independent of the mesh size h, and the local error indicatorge is defined by

g2e ¼ h2

e RIk k2L2ðL2ðeÞÞþ

Xc�@e\@X


Xc�@e\@X

1

hcRB0k k2

L2ðL2ðcÞÞ

þ 1

2

Xc�@en@X


1

2

Xc�@en@X

1

hcRB0k k2

L2ðL2ðcÞÞ

þ 1

2

Xc�@en@X


1

2

Xc�@en@X

hc @RB0=@tk k2L2ðL2ðcÞÞ ð26Þ

This approach is flexible and applies to all four versions of DG, namely, OBB-DG,SIPG, NIPG, and IIPG. The a posteriori error estimate in L2(H1) is explicit andresidual based and thus is also computationally efficient. General boundary con-ditions can be easily incorporated into the L2(H1) error estimate. In addition, noregularity assumption for the dual problem is required for this estimate. Numericalperformance of the a posteriori error estimators under two major categories, namelyestimators in the L2(L2) and L2(H1) norms, has been investigated and compared.Results indicate that L2(H1) type error indicators are more flexible, but L2(L2) typeerror indicators are more effective for many reactive transport cases.

With a dynamic adaptive approach, one often starts with a fine mesh and modi-fies the mesh dynamically, performing both refinement and coarsening according toan error indicator. Dynamic mesh modifications are particularly effective for transi-ent problems involving a long period of simulation time, because the location ofstrong physics usually moves with time. For such problems, it is recommended tocompute the error indicator for only a short time interval involving the current time.We divide the entire simulation time into a collection of time slices, each of whichmay in turn contain a certain number of time steps. Due to the computational costassociated with mesh modifications, we modify the mesh at the end of each time slice


rather than doing it for each time step. For each time slice (Tn�1, Tn), we first com-pute the solution at Tn-1 using either the initial condition (for the first time slice, i.e.,n ¼ 1) or the solution at the end of the last time slice (if n > 1) by a local projection.We then compute the DG approximation of the partial differential equation (PDE)for the time slice (Tn�1, Tn) and compute an error indicator for each element. Basedon the computed error indicator, we refine a certain number of elements and coarsenthe same number of other elements, maintaining the number of total elementunchanged. We denote by a modification factor a 2 ð0; 1Þ the ratio of the numberof the refined=coarsened elements to the number of the total elements. We repeatthe above procedure Mn times (a preset iteration number), each time using anupdated mesh before moving to the next time slice.

A Numerical Example with Dynamic Mesh Adaptation

We consider the following normalized contaminant transport problem in a singlephase in porous media:

@/ec

@tþr � ðuc�DrcÞ ¼ s; ðx; tÞ 2 X� ð0;TÞ

ðuc�DrcÞ � n ¼ 0; ðx; tÞ 2 Cin � ð0;TÞ

ð�DrcÞ � n ¼ 0; ðx; tÞ 2 Cout � ð0;TÞ

cðx; 0Þ ¼ 0; x 2 X

8>>>>>>><>>>>>>>:

ð27Þ

where the domain X ¼ ð0; 10Þ2 has an inflow boundary Cin and an outflow=no-flowboundary Cout. The diffusion-dispersion D is a constant diagonal tensor withDii ¼ 0.01; and the velocity is u ¼ (0.2, 0) uniformly across the domain. The domainX is divided into two parts, i.e., the lower half Xl ¼ (0,10)� (0,5) and the upper halfXu ¼ (0,10)� (5,10). The effective porosity /e in Xu is 0.1. Adsorption occurs only inthe lower part of the domain, which results in an effective porosity /e ¼ 0.2 in Xl .The contaminant release rate s is 1.0 inside the square centered at (5,5) with the sizeof 0.3125� 0.3125 and is 0.0 elsewhere.

SIPG is employed to solve this problem because the scalar variable (concen-tration) is of primal interest in this problem and SIPG is the only one among the fourprimal DG methods that owns optimal a priori and a posteriori error estimators inL2(L2). The penalty parameter is chosen according to the recommendation by Sunand Wheeler (2006). The simulation time interval is (0, 2) and we use the backwardEuler method for time integration with a uniform time step Dt ¼ 0.01. The completequadratic basis function is used for each element. The initial mesh is a 32� 32uniform rectangular grid. Dynamic mesh adaptation is incorporated into the DGalgorithm, guided by the a posteriori error indicator in L2(L2). The time derivativeterm in the interior residual, which is needed in the computation of the error indi-cator, is approximated by a finite difference. The modification factor for thedynamic mesh adaptation is chosen to be a ¼ 0.05. We partition the simulation timeinterval (0, 2) into 20 time slices uniformly. The iteration number is chosen to be fivefor the initial time slice and two for the remaining slices.

Figure 7 illustrates the DG solutions powered by dynamic mesh modification atvarious times. Due to retardation effects arising from adsorption, the contaminanttransport is slower in the lower part of the domain. A continuous concentration


Figure 7. DG solutions (a, c, e, and g at t ¼ 0:5, t ¼ 1, t ¼ 1:5, and t ¼ 2, respectively) and themesh structures (b, d, f, and h at t ¼ 0:5, t ¼ 1, t ¼ 1:5, and t ¼ 2, respectively) using dymanicmesh adaptation.


profile is observed because of diffusion-dispersion. It should be observed that themesh is densely refined around the moving contaminated region and at the plumefront. This is driven by physics, as convection is large at the high concentration area(plume center) and diffusion is significant at the high concentration gradient area(plume edge). The mesh follows the movement of the contaminant plume andspreads in a way similar to the behavior of the plume itself. Clearly, the dynamicmesh modification guided by the L2(L2) error indicator sharply captures the time-dependent local behavior of the advection-diffusion-adsorption process. In addition,one sees that the length of an element can be at least as small as 1=26 of that of thelargest element in a mesh. Obviously, this adaptation in spatial resolution substan-tially reduces the computational time and storage requirement without a loss ofaccuracy.

Due to the discontinuous space used in DG, the projections of concentrationduring mesh modifications involve only local computations and are locally massconservative, which ensures both the efficiency and accuracy of DG during dynamicmesh modifications. We emphasize that the simultaneous locality in both thecomputation and conservation is a unique advantage of DG that is not satisfiedby FVMs and classical FEMs. We also observe that the iteration number in eachtime slice can be as small as two, which further assists the computational efficiencyof adaptive DG.

Discussion and Conclusions

In this article, the four primal DG methods are formulated to construct efficient solu-tions of parabolic partial differential equations. Several representative example pro-blems in chemical engineering are solved using DG. The numerical resultsdemonstrate the ability of DG to solve a set of rather diverse time-dependentproblems in chemical engineering. In addition, for mass transfer simulation, DG treatsboth the convection-dominated and the diffusion-dominated systems very well. Forheat transfer simulation, DG is effective for both the convection-dominated and theconduction-dominated problems. DG has less degree of numerical diffusion comparedwith other classic algorithms. Due to the discontinuous nature of approximation spaces,DG preserves the steep moving fronts and allows highly varying and jumping problemcoefficients (see also Sun and Wheeler, 2006b).

A highlight of this article is to show that DG possesses substantial advantageson adaptive mesh modification over finite difference methods, finite volume meth-ods, and classic finite element methods. Effective adaptivity allows the attainmentof accurate solutions with substantially less computational effort. In this article, adynamic mesh modification strategy is formulated and studied for DG methodsguided by mathematics-based and physics-driven a posteriori error estimators.Numerical results illustrate the advantage of adaptive approaches. In particular,we see that the flexibility of DG in allowing nonmatching meshes substantially sim-plifies the implementation of the mesh adaptation as the local element refinement isindependent of neighborhood elements. In addition, this flexibility increases theefficiency of adaptivity because the unnecessary areas do not need to be refined justfor maintaining the conformity of the mesh. Moreover, DG errors are localized; inother words, there is less pollution of errors. This leads to a more effective adaptivityfor DG than for conforming methods. Because of this, we see that DG withadaptivity sharply captures local physical phenomena. It is also observed that the


proposed dynamic strategy performs very well for transient problems with a longperiod of simulation time in aspects of both the numerical accuracy and computa-tional cost.

Some transient problems in chemical engineering involve moving interfaces,examples of which include drying, solidification, and crystallization processes. Effec-tive simulations of such processes require dynamically moving grids. DG methodsare good candidates for such problems because of the flexibility of DG in allowinggeneral nonmatching unstructured meshes and the locality of DG projections fromthe old mesh to the updated mesh (Li and Tang, 2006). Consequently, numericalinvestigation of DG to such moving interface problems in chemical engineering is,an appealing future work. Another interesting future research topic is p- and hp-adaptive DG applied to chemical engineering problems. This is particularlyintriguing for nonconforming hexahedral meshes because DG allows the replace-ment of tensor-product polynomial spaces by complete polynomial spaces, leadingto a reduction in the number of degrees of freedom in the system approximatedby high-degree polynomials. For simplicity, all simulations in this study were per-formed by serial computations. However, due to its locality in data communications,DG is expected to have excellent parallel efficiency. Investigation of efficient paralleladaptive DG with automatic load balancing is a topic we are currently pursuing.

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