Finite-Element Solution of

Reaction-Diffusion Equations with Advection

Biyue LiuMyron B. AllenHristo KojouharovBenito Chen

Department of Mathematics, University of Wyoming,Laramie, WY 82071, U.S.A., allen@uwyo.edu

Abstract

Reaction-diffusion equations arise in many chemical and biological settings.Solutions to these equations exhibit a wide variety of structures, includingpattern formation and traveling waves. In groundwater aquifers, reaction-diffusion equations govern kinetic adsorption and the growth and trans-port of biofilm-forming microbes, and the equations may contain advectiveterms. Accurate simulation in these cases is crucial to the developmentof contaminant remediation strategies. However, advection interferes withthe numerics, causing spurious oscillations or artificial diffusion near sharpfronts. These errors can trigger unrealistic reaction effects.

We examine finite-element techniques for solving reaction-diffusion prob-lems with advection. The techniques are based on the modified method ofcharacteristics, in some cases including streamline or velocity-weighted nu-merical diffusion to suppress numerical oscillations.

In contrast with linear reactions, nonlinear reactions may contain thresh-olds, and errors near these thresholds can affect the qualitative structure ofsolutions. Therefore, both the oscillations associated with poorly resolved,nondiffusive schemes and the smearing associated with diffusive schemescan distort the numerical solutions significantly.

1 Introduction

Reaction-diffusion equations govern an extraordinary variety of chemicaland biological phenomena [1]. In hydrology, equations of this type model thetransport and fate of adsorbing contaminants and microbe-nutrient systemsin groundwater. An interesting special case is the underground growth,transport, and kinetic attachment (adsorption) of biofilm-forming bacteriain porous media, since biofilms offer a promising mechanism for controllingaquifer contaminants with biobarriers [2].

We examine finite-element schemes for the two-dimensional, advectivereaction-diffusion equation,

φ∂c

∂t+ v · ∇c−∇ · (D∇c) = f(c), (1)

for concentration c(x, t). Here, φ is the porosity, v = (v1, v2) is the Darcyvelocity, and f(c) is a reaction term, simple models for which we specifylater. We assume that the hydrodynamic dispersion tensor D has the com-monly postulated form [5],

D = φdM

[1 00 1

]+dL|v|

[v2

1 v1v2

v2v1 v22

]+dT|v|

[v2

2 −v2v1

−v1v2 v21

]. (2)

As usual, dM is the molecular diffusion coefficient, while dL and dT are thelongitudinal and transverse dispersivities, respectively. In many settings,molecular diffusion is small, and dT < dL.

What makes equation (1) numerically challenging is the advective termv · ∇c. In problems where pumping induces field-scale pressure gradi-ents, this term typically dominates the transport, and the diffusion term−∇· (D∇c) exerts only modest smoothing effects on sharp plumes and con-centration fronts. One can measure the degree of advection dominance viathe dimensionless Peclet number P = ‖v‖∞L/D, where L is the diameterof the spatial domain and D = φdM + (dL + dT )‖v‖∞.

When P >> 1, sharp fronts and plumes remain sharp and cause nu-merical difficulties. Unless the spatial grid has a fine enough mesh sizeh, most high-order discrete methods produce spurious oscillations near thesteep fronts. Typically, the criterion for oscillation-free solutions requiresthat the grid Peclet number Ph = Ph/L = O(1) [6]. Grids this fine canbe expensive. A common alternative to fine gridding is to suppress theoscillations by adding numerical diffusion of various types [7].

Either type of error — oscillations or numerical diffusion — is bad enoughin passive transport problems or in problems involving first-order decay[3, 4]. But in problems involving nonlinear reactions they can lead to se-vere, qualitative distortions. Especially troublesome are reactions with mul-tiple equilibria and thresholds, since oscillations and numerical diffusion canartificially trigger or dampen threshold effects.

This paper explores these difficulties. In Section 2 we review a set of nu-merical methods for Equation (1), based on Eulerian-Lagrangian timestep-

ping and various types of numerical diffusion. In Section 3 we examineinstabilities excited by numerical oscillations, and in Section 4 we examinethe distortions inflicted by numerically diffusive schemes. Section 5 brieflydiscusses implications for realistic mathematical modeling.

2 A class of finite-element methods

The discretizations examined here are variants of a technique combiningthe modified method of characteristics (MMOC) [8] with the streamlinediffusion method (SDM) [9].

The idea behind the MMOC is to discretize the material derivativeDc/Dt = ∂c/∂t + v · ∇c along characteristic curves of the dispersion-freetransport equation. This tactic yields

Dc

Dt(x, tn) ' c(x, tn)− c(x∗, tn−1)

∆t,

where x∗ is an approximate solution at tn−1 to the problem dy/dt = v(y, t),y(tn) = x, say x∗ = x − v ∆t. Thus we treat the advective term in aLagrangian frame of reference, leaving the diffusion and reaction terms inan Eulerian frame. The MMOC tracks sharp fronts more accurately thanstandard timestepping schemes and symmetrizes the matrix equations tobe solved at each time level [8]. Still, the method does not automaticallyeliminate numerical oscillations near fronts.

In the SDM, one adds numerical diffusion to the standard Galerkin finite-element approximations by perturbing the test functions. Instead of usingtrial-space basis functions ϕi(x) as test functions, one uses test functions ofthe form ϕi(x) + γv(x) ·∇ϕi(x). Here, γ is a nonnegative parameter of sizeO(h) that dictates the amount of numerical diffusion added. The advantageof the SDM is that it adds diffusion only in the local streamwise direction— the direction where it is needed for suppressing oscillations [9].

An effective way to combine the two schemes is to discretize the advectivepart of Equation (1) via the MMOC, retaining the numerically diffusive termthat arises from v · ∇c in the standard SDM but discarding all other termsof size O(γ). For a spatial domain Ω in the plane, we obtain

1

∆t

∫

Ω(cn+1h − c∗h)ϕ + γ

∫

Ω(v · ∇cn+1

h )(v · ∇ϕ)︸ ︷︷ ︸

(ND)

+∫

ΩD∇cn+1

h · ∇ϕ = f(cn+1h ).

(3)

Here, cn+1h (x) is the unknown trial function at time tn+1, and c∗h(x) = cnh(x∗).

We call this scheme MMOC-SDM. For a detailed analysis and convergenceproof in the absence of reactions, see [10]. We accommodate nonlinearreactions f(cn+1

h ) using a Picard iterative scheme at each time level.

The term labeled (ND) in Equation (3) adds numerical diffusion in thestreamwise direction. In the matrix equations to be solved at each timelevel, this term contributes matrix entries that mimic those associated withthe longitudinal part of the hydrodynamic dispersion tensor (2). The term(ND) thus effectively augments physical dispersion by the array

γ

[v2

1 v1v2

v2v1 v22

]. (4)

This numerical diffusion is positive semidefinite, having a zero eigenvalueassociated with eigenvectors orthogonal to the local velocity.

Alternatively, one can add velocity-weighted numerical diffusion alongcoordinate lines — analogous to standard upstream weighting methods —by dropping the off-diagonal entries in the array (4). We refer to this modi-fied version of MMOC-SDM as the diagonalized scheme. The numerical dif-fusion in this case is positive definite wherever the local velocity is nonzero.In what follows, we examine the effects that the MMOC, the MMOC-SDM,and the diagonalized scheme have on numerical solutions to Equation (1).

3 Effects of oscillations

To illustrate the effects of oscillations on problems with nonlinear reactions,we examine the model reaction term f(c) = rc(1 − c), r > 0. This termrepresents logistic growth; substitution into Equation (1) yields an advectiveversion of Fishers’ equation [11]. In the absence of diffusion, the Lagrangianequation Dc/Dt = rc(1−c) has a stable equilibrium at c = 1 (the “carryingcapacity”) and an unstable one at c = 0. When the initial function c(x, 0)has values restricted to the interval (0, 1), the exact solution values remainin (0, 1). However, the unstable equilibrium is numerically problematic,since small negative excursions in c, representing nonsensical concentrationvalues, grow in magnitude without bound.

When diffusion is weak (that is, grid Peclet number Ph is too large),numerical oscillations near sharp fronts can produce precisely such negativeexcursions. Figure 1 shows what happens when this occurs. The figure isa perspective plot of the MMOC solution to a rotating plume problem onΩ = (0, 1) × (0, 1), in which the velocity field is v(x, y) = ( 1

2− y, x − 1

2),

dM = 10−4, dL = 2×10−3, dT = 6×10−4, and r = 22. The initial conditionis a “Gauss hill,”

c(x, y, 0) = exp−σ

[(x− 1

2)2 + (y − 3

8)2],

centered at (12, 3

8), with σ = 20. We truncate the hill to satisfy the bound-

ary condition c = 0 on ∂Ω (although the code handles other boundaryconditions). The spatial grid is uniform, with mesh size h = 2−6, and thetime step is ∆t = h. The plot shows the graph of ch(x, 5∆t), the solutioncorresponding to the last time step before machine blowup.

00.2

0.40.6

0.81

0

0.5

1−1.5

−1

−0.5

0

0.5

1

1.5

Figure 1. Perspective plot of a numerical solution, using MMOC,to a rotating plume problem with a logistic reaction term. Theview is from the third quadrant in (x, y), at an elevation of π/15,with contours shown underneath. The numerical solution is on theverge of machine blowup because of the reaction-driven negativeexcursion.

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

1.2

Figure 2. Perspective plot of the initial condition for a rotatingplume problem with a cubic reaction term. The view is from thethird quadrant in (x, y), at elevation π/15. In the exact solution,the plumes are advected, while their peaks grow and their lowerslopes decay as a consequence of the reaction.

In this problem, incorrect negative solution values grow as they prop-agate, eventually overwhelming any resemblance between the exact andnumerical solutions. The MMOC discretization is stable for the linear casewhen f(c) = ±rc, and it is possible to prove that this scheme converges ash,∆t → 0. Nevertheless, in the nonlinear case the method is unstable forthe spatial mesh size used.

4 Effects of numerical diffusion

One can eliminate the troublesome oscillations by using a sufficiently finespatial grid, but this strategy is expensive in complicated problems, whichoften involved coupled reaction-diffusion equations in three space dimen-sions. The strategy may be impossible if dissipative effects are negligible.A more common approach is to “stabilize” the discretization by addingnumerical diffusion. The SDM is one of the more delicate approaches fordoing this. In the logistic growth problem just analyzed, the addition ofsome numerical diffusion can suppress the negative excursions and hencethe instability.

Nevertheless, numerical diffusion can also cause severe errors when non-linear reactions are present. To demonstrate this fact, we examine the cubicreaction term f(c) = rc(c− 1

2)(1− c), with r = 22. The diffusion-free equa-

tion Dc/Dt = f(c) now has an unstable equilibrium at c = 12

and stableequilibria at c = 0, 1. The equilibrium c = 1

2is a threshold: Solution

values initially greater than 12

tend to 1, while those initially less than 12

tend to 0. Numerical diffusion can artificially push solution values belowthe threshold, resulting in numerical solutions that differ qualitatively fromexact solutions.

Figures 2 through 5 illustrate this possibility. All figures are perspectiveplots of numerical solutions to a rotating plume problem with v and D asabove. This time, c(x, 0) is a superposition of three Gauss hills, centered at(11

16, 7

32), (1

2, 3

8), and ( 5

16, 17

32) and having width parameter σ = 110. Figure 2

shows this initial condition. All three discretizations use h = 2−7, ∆t = 2h.Figure 3 shows the solution at t = 100∆t, computed using the MMOC

without any numerical diffusion. Here, the spatial grid is fine enough toyield oscillation-free solutions, and the initial ridge of Gauss hills evolvesinto three distinct, cylindrical peaks, rotated with the velocity field as ex-pected. With the exception of the rotation, this numerical solution differsonly slightly from that computed using v = 0.

Figures 4 and 5 show corresponding solutions at the same time level, com-puted using the MMOC-SDM and the diagonalized scheme, respectively. Inboth cases γ = 0.05. In Figure 4, the peak undergoing the strongest advec-tion (furthest from ( 1

2, 1

2)) suffers significant streamwise distortion, since ch

undergoes artificially diffusive transport in that direction. Diffusive damp-ing causes some concentrations to drop below the threshold value 1

2; hence

the outermost peak shows unrealistic decay. In Figure 5, the artificial dif-

0

0.5

1 00.2

0.40.6

0.81

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 3. Perspective plot of a numerical solution, using MMOC,to a rotating plume problem with a cubic reaction term. The viewis from the fourth quadrant in (x, y), at elevation π/15, with con-tours shown underneath. The numerical solution agrees qualita-tively with the expected exact solution.

fusion is not just longitudinal but acts along coordinate lines. Here, theoutermost peak undergoes enough damping to trigger its virtual disappear-ance, a stark contrast from Figure 3.

5 Conclusions

The most straightforward implication of these numerical experiments is thatneither spurious oscillations nor artificial diffusion is tolerable in numeri-cal methods for advection-reaction-diffusion equations. When the reactionterms are simple and the transport equations are uncoupled, it is possibleto recognize unrealistic numerical artifacts. But in more complex problems,where numerical models are essential, discrepancies between numerical be-havior and exact solutions may go undetected.

A more subtle implication concerns the mathematical model used for hy-drodynamic dispersion in the presence of field-scale heterogeneities. Thereis not yet universal agreement on the form that this model should take,but consensus is growing that the standard model (2) may require scaled

0

0.5

1 00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4. Perspective plot of a numerical solution, using MMOC-SDM, to a rotating plume problem with a cubic reaction term. Theview is from the same point as in Figure 3. The numerical solutionundergoes unrealistic damping, apparent in the peak undergoingstrongest advection.

0

0.5

1 00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5. Perspective plot of a numerical solution, using the diag-onalized scheme, to a rotating plume problem with a cubic reactionterm. The view is from the same point as in Figure 3. Numericaldiffusion has altogether suppressed the peak undergoing strongestadvection.

dispersivities or may even be qualitatively inappropriate (see, for example,[12]). Numerics suggest that advection-diffusion-reaction equations can bedistressingly sensitive to the form and strength of dispersion, whether nu-merical or physical. Any change in dispersion models is apt to have profoundeffects on numerical predictions of reactive transport.

Acknowledgments

The Wyoming Water Research Center partially supported this work throughfunding from the U.S. Geological Survey. We also received support fromNSF grant OSR/9550477 and AFOSR grant F49620-94-1-0194. We aregrateful to Professors Al Cunningham and Martin Hamilton at the Centerfor Biofilm Engineering, Montana State University, for valuable guidance.

Key words

Reaction-diffusion equations, Eulerian-Lagrangian methods, advective flows.

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