Numerical Integration of Reaction–Diffusion Systems

Numerical Algorithms 31: 247–269, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Numerical integration of reaction–diffusion systems

Michelle Schatzman

MAPLY, Laboratoire de Mathématiques Appliquées de Lyon, CNRS et Université Claude Bernard,21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France

E-mail: [email protected]

Received 27 January 2001; revised 1 November 2001

Approximating numerically the solutions of a reaction–diffusion system in an efficientmanner requires the application of implicit methods, since the Courant–Friedrichs–Lewy con-dition on explicit methods imposes a time step of the order of the square of the space step. Inthis article, we review two types of strategies which are expected to yield reasonably precisesolutions within a reasonable computing time. The first examines methods for solving the lin-ear step necessary in any resolution procedure; estimates of CPU time in terms of the error aregiven in the non preconditioned and in the preconditioned case – provided that it is possible todefine an efficient preconditioner. The second strategy is based on splitting, with or withoutextrapolation. The respective faults and qualities of both strategies are examined; they lead toa list of difficult analytical and numerical problems with possible hints as to their solution.

Keywords: reaction–diffusion, alternate directions, preconditioning, high precision, stiffness

AMS subject classification: 35K57, 65M12, 65M20

1. Introduction

The typical application problem to keep in mind is the following reaction–diffusionsystem

∂u

∂t− M ⊗�u + f (u) = 0. (1)

Here u is a function from ⊂ Rd to R

m, and M is an m × m matrix, which does nothave to be diagonal; it satisfies the following assumption:

the spectrum of M is included in the complex half-plane {�z > 0}. (2)

If (2) did not hold, the linear problem would be ill-posed. The notation M⊗� designatesthe operator given by

248 M. Schatzman / Integration of reaction–diffusion systems

u = u1

...

up

, M ⊗ �u =

p∑j=1

�M1j uj

...p∑

j=1

�Mmjuj

.

Finally, the function f maps an open subset of Rm to R

m.We will say very little regarding boundary conditions: there are many difficult

questions when splitting is applied. Most of the present discussion on splitting is validin full space or with periodic boundary conditions.

This kind of problem appears typically in the modeling of chemical reactions,where usually the coefficients in the matrix M have very different orders of magnitude,and the function f may have very steep gradients, making the numerical approximationof (1) into a problem which is stiff for two different reasons:

1. we are discretizing a partial differential equation of second order in space, and weexpect the condition number of the discretization of −M ⊗ � to be at least equal tothe product of the condition number of M by δ−2, where δ is a measure of the spatialdiscretization step;

2. the reaction term f becomes after linearization v → f ′(u)v, which can also be quitestiff.

Moreover, this is usually also a very large problem, particularly if we are interested inspatial dimension d = 2 or d = 3.

It is important to note that the directions of stiffness of the two pieces of the prob-lem are radically different. To give an idea of what is going on, if we replace f ′(u) by amultiplication by a space dependent matrix potential V (x), then the spectrum of the op-erator u → V (x)u is simply the union of the matrix spectra of the matrices V (x) and thecorresponding “eigenfunctions” are Dirac masses at x multiplied by eigenvectors w ofV (x); on the other hand, the eigenvalues of −M⊗� are products of eigenvalues of � byeigenvalues of M, and the corresponding eigenfunctions are oscillating functions; there-fore, the eigenfunctions of the first operator are very well localized in physical space, andthe eigenfunctions of the second one are very well localized in frequency space. This isa prototype of a non-commutative situation, of which the simplest example would be thefollowing one-dimensional quantum mechanical situation

Au = i∂u

∂x, Bu = xu.

The domain of the momentum operator A and the position operator B can be defined insuch a way that they are self-adjoint in L2(R), and their commutator is

[A,B] = AB − BA = i.

M. Schatzman / Integration of reaction–diffusion systems 249

We will argue that any effort to improve the numerical resolution of (1) must takeinto account the fact that the stiffness directions of the two operators under considerationare foreign one to the other.

There are basically two possible approaches to the construction of efficient numer-ical schemes for reaction–diffusion systems: either apply any one of the good numericalschemes devised for integrating stiff systems, or split the operator into its linear and non-linear parts. In section 2, we count the number of operations needed for the numericaltime integration: first, we check that direct methods are not very applicable, then weestimate the numerical cost of the space time integration using an iterative strategy andthe relation of the CPU time to the error. In section 3, we examine the possibilities forpreconditioning, the expected benefits and the work to be performed in order to constructefficient preconditionings.

In section 4, we recall a number of results on exponential splitting, including theessential negative result of Q. Sheng, displaying the severe competition between orderand stability. We also describe a number of cases where estimates are precise enough toobtain good stability results for the extrapolation of exponential splitting.

In section 5, we turn to more practical results, examining what becomes ofPeaceman–Rachford formula when commutativity assumptions are discarded: it can beunstable, and sufficient conditions on stability involve assumptions on the commutators,which are satisfied in a number of particular cases.

In section 6 we construct extrapolations of the Peaceman–Rachford formula, andwe report on experimental numerics; the analysis seems to be still extremely difficult.

In section 7 we turn to non linear estimates, for the exponential the rational split-tings, and we state conditions under which the alternate direction methods work in thatsetting.

Finally, we conclude with a list of open problems in section 8.

2. Operation counts for naïve computations

Denote by A the spatial discretization of −M ⊗ �, with boundary conditions;assume that you have used your favorite method for the spatial discretization, be it finitedifferences, finite elements, finite volumes, pseudo-spectral methods or a magic wand.

Given the state of the art, we can safely assume that we have obtained a very goodspatial discretization, which can be assumed to be of large order.

The spatial discretization scale will be called δ. It is then classical that the spectrumof A will be included inside the intersection of a cone included in the right-hand sidecomplex half-plane {z: �z > 0} and of a circle of radius C/δ2, C being an estimate ofthe eigenvalue of largest module of M.

For an explicit scheme, the number of operations per time-step is O(qmδ−d), q be-ing an estimate of the number of operations needed to advance one time step of the nu-merical method under examination, in the case of a scalar ordinary differential equation.


If the time integration is of precision order p and the space integration of precisionorder s, then, after 1/h time steps, the error at time O(1) will be of order

O(hp + δs

),

and if we choose

h = ε1/p and δ = ε1/s, (3)

the CFL (Courant–Friedrichs–Lewy) condition is satisfied provided that p � s/2; inthat case, the CPU up to time O(1), for a given error ε is equivalent to the number ofoperations, and it will be of order

Texplicit, automatic CFL(ε) = O(1)mqε−(1/p)−(d/s). (4)

On the other hand if p > s/2, we take δ as above and h = δ2; then the CPU time is ofthe order of

Texplicit, imposed CFL = O(1)mqε−(d+2)/s. (5)

Estimates (4) and (5) already show that a gain on the precision in time translates into again in CPU time; however, the condition p � s/2 says that the best choice is to take aspatial discretization which is twice more precise than the time discretization.

The computation times given by (4) and (5) can be considered as large. We shalllook now at implicit schemes: they are thought to be more efficient than explicit schemesfor stiff problems, but it may be that the situation is not quite so clear cut.

In any reasonably efficient implicit numerical scheme, the essential step will be tosolve a problem of the form

ui + h

q∑j=1

aij(Auj + f (uj )

) = data, 1 � j � q. (6)

This will be the case in an implicit Runge–Kutta method, in an implicit multistepmethod, and in any of the more modern methods, specifically those proposed by Butcherin [6–8].

The convergence of implicit Runge–Kutta schemes for parabolic problems hasbeen proved by Lubich and Ostermann in [35,36]. Here we will concentrate on Runge–Kutta methods.

A priori, one might think that it is best to take q = 1, and indeed if that is atall possible it should be done. However, we will argue that together with the use ofadequate preconditioned iterative methods, it is well possible that for small values ofq the computational cost remains acceptable. Therefore, instead of ruling out from thestart q > 1, we shall include it into our analysis, and it will enable the reader to judgethe feasibility on the basis of rational arguments.


Solving (6) will usually be done by a Newton’s method or one of its generaliza-tions; the main step will be

ui + h

q∑j=1

aij(Auj + Buj) = data, 1 � j � q. (7)

with B the multiplication by f ′(u).Let us start first by counting operations for direct methods, thinking of matrices

with a sparse structure. In the best possible case, −� can be dicretized as a matrix ofsize O(δ−d), with bandwidth O(δ−d+1). Therefore, −M ⊗ � wil be discretized as amatrix of size O(mδ−d) with bandwidth O(δ−d+1 + (m′ −1)δ−d), with m′ the bandwidthof M: if M happens to be diagonal, the band width of A is the same as the bandwidthof the discretization of −�. B is of bandwidth m, and therefore, adding it does notincrease the bandwidth. Because of the nonlinear term, the matrix decomposition mustbe performed anew at each time step, a very time consuming process.

The size of the matrix involved in (6) is O(qmδ−d) and its bandwidth is

b(δ, d, q,m,m′) = O(δ−d+1 + (m′ − 1)δ−d + (q − 1)mδ−d

).

Therefore, the cost of one step of a direct method is

O(1)δ−3d+2qm

(1 + m′ − 1

δ+ m(q − 1)

δ

)2

.

Therefore, if m′ > 1 or if q > 1, the resolution requires O(δ−2) more operations, whichexplain why fully implicit Runge–Kutta schemes are not usually chosen to treat this typeof problem.

In any case, the use of direct methods is usually out of the question for three di-mensional problems, since it requires storing an LU decomposition with bandwidthb(δ, d, q,m,m′), and size O(qmδ−d); with d = 3, q = 1, m′ = 1 and m = 100, whichis not unusual for reaction–diffusion problems in chemistry, this means O(100δ−5) num-bers, and for a spatial discretization step of δ = 10−2 this means 1012 numbers, a whollyimpractical proposition in the largest computers used these days, since it requires ac-cessing mass storage, a very slow process. On top of that, this grid is not even veryfine.

If this were possible, CPU time can be estimated as follows:

O(1)h−1δ−3d+2qm

(1 + m′ − 1

δ+ m(q − 1)

δ

)2

. (8)

If q and m′ are equal to 1, this quantity is of the order of

Tdirect, small = mε−(3d+2)/s−1/p; (9)


this means indeed that we can make the most progress by increasing p. If q or m′ arenot equal to 1, then the quantity (8) is of the order of

Tdirect, large = O(1)qmε−3d/s−1/p(m′ − 1 + m(q − 1))2, (10)

and the conclusion is the same as above: increasing the order of the time integration isthe crucial point where we can gain numerical efficiency, and the question is so crucialthat a reasonable increase in q might well be an interesting choice, though it makes thecalculation significantly larger.

The comparison between (9), (10) and (4), (5) shows that explicit schemes are notnecessarily worse than implicit ones in terms of operation counts. What may govern thedecision is more the good or bad properties of either method, the control of dissipativitybeing a moot point; in any case, in dimension 2 and more, direct methods are very slow.

Let us try now an operation count for an iterative method, starting with a conjugategradient method, and assuming of course that the problem is self-adjoint: clearly, thisrules out q > 1.

We have first to estimate the cost of multiplying a vector u by the matrix A + B:this requires O(qmδ−d) operations. Then, we must estimate the number of steps neededto divide the residual by 2: let λmax and λmin be respectively the largest and the smallesteigenvalue of 1 + hA+ hB; then the factor of reduction of the residual is

ρ =√λmax + √

λmin√λmax − √

λmin. (11)

If we assume that B is not much stiffer than A, and that the eigenvalues of M are com-parable to 1, then

c1δ√h

� − ln ρ � c2δ√h,

which we will write also

− ln ρ ∼ δ√h.

This means that the number k of iteration steps needed to divide the residual by 2is comparable to

√h/δ, and therefore, we will consider that we need O(kmδ−d) =

O(m√hδ−d−1) operations to solve (6) for one time step. Assuming that Newton’s

method requires a finite number of iterations, the total number of operations up to afinal time O(1) is

O(mδ−d−1/

√h

)and with the choice (3), we find that the CPU time is comparable to

mε−(d+1)/s−1/2p, (12)

and, though this is a much better estimate than (9) or (10) the practical conclusion on thescientific programme remains: we must do better on time integration.


If we do not assume anymore that q = 1, we will have to use other iterative meth-ods, such as the biconjucate gradient; not much is known theoretically in general on theconvergence rates of these processes; nevertheless, the literature [2], regularly updatedon the Web, quotes a rate of convergence twice slower for the biconjugate gradient thanfor the conjugate gradient, with possible break-downs of the algorithm. See also therecent review article of Saad [44] with many interesting historical aspects.

Provided that the convergence rate is that one, the order of magnitude of the CPUtime remains the same, and instead of (12) the order of magnitude of the CPU timeshould be now

Titerative(ε) = qmε−(d+1)/s−1/2p. (13)

Therefore, experiments are called for: if, in practice, the CPU time is given by (13), itmakes much sense to apply even fully implicit Runge–Kutta methods to large reaction–diffusion systems, provided that appropriate iterative methods are used.

Observe also that for p = s/2, (13) and (4) are of comparable order; (13) is betteronly if p is larger than s/2.

3. Preconditioning

The issue of efficient preconditioning for reaction–diffusion systems is not yet wellunderstood. Let us recall that a preconditioner for a linear system

Ax = b (14)

is a nonsingular operator N which approximates the inverse of A and whose action iseasy to calculate; then the resolution of (14) is equivalent to the resolution of

NAx = Nb. (15)

If NA is close to the identity – i.e., if an appropriate condition number of NA is closeto 1 – iterative methods for (15) may be much faster than for (14). Of course, there isan exact preconditioner, viz. the inverse of A: but usually, its computation is not fast;therefore, I shall rule out this possibility.

Since alternate directions are a very successful approach, as we shall see in theforthcoming sections, one may wonder whether an alternate preconditioning strategycould be employed; this idea is indeed found in the article [5], without any theoreticalconsiderations.

I claim that this strategy meets with serious difficulties, even in the commutativecase. Assume indeed that A and B commute, that they are self-adjoint non-negativeand that the upper bound of their spectrum is a number X � 1. An alternate directionpreconditioning would be an expression of the form

N = R(hA, hB),


with R = P/Q a rational fraction in x and y, whose denominator will be of the form

Q(x, y) =k∏

i=1

(1 + aix)

l∏j=1

(1 + βjy)−1, (16)

with all the ai’s and bj ’s positive.The limitations of the method are stated as the following result:

Lemma 1. There exists no rational fraction R = P/Q, with Q of the form (16) forwhich the conditioning of NA is small relatively to hX, when hX � 1.

Proof. Let R be the rational fraction

R(x, y) = P (x, y)

Q(x, y), P (x, y) = (1 + x + y)P (x, y).

We shall prove that

if L � 1,max0�x,y�L |R(x, y)|min0�x,y�L |R(x, y)|

� 1. (17)

Let r be the degree of P ; by definition of P , r � 1. We write P as a sum of homogeneouspolynomials

P (x, y) =r∑

s=0

Ps(x, y),

where Ps(x, y) is homogeneous of degree s in (x, y). We have several cases to consider,according to the value of r relatively to k + l. The first case is

k + l > r.

We choose α > 0 and β > 0 such that

Pr(α, β) �= 0.

Then, for L � 1,

P (αL, βL) ∼ Pr(α, β)Lr,

and

Q(αL, βL) ∼ (αL)k(βL)lk∏

i=1

ai

l∏j=1

bj ,

and therefore there exists a number C such that for all large enough L

max0�x,y�L

∣∣R(x, y)∣∣ � CLk+l−r .


Since R takes bounded values in the neighborhood of 0, then (17) holds.The second case is k + l � r. Observe that we cannot have k = l = r, since

r is at least equal to 1. If l and k vanish, then it is plain that (17) holds, since in thiscase R(x, y) is a polynomial of degree at least equal to 1. Assume that only one of theintegers l and k vanishes, for instance l. Since P is the product of P with 1 + x + y, wemay write

Pr (x, y) =t∑

n=s

cnxr−nyn, (18)

with 1 � s � t − 1 � r − 2, and cs �= 0, ct �= 0. But, for x = L and y = 1, we have

P (L, 1) ∼ csLr−s,

so that

R(L, 1) ∼ csL−s∏k

i=1 ai

and this implies that

R(L, 1) = O(L−s

),

proving (17). Assume now that neither k nor l vanishes. We may still write (18), and weobserve that

R(L, 1) ∼ csLr−s−k∏k

i=1 ai∏l

j=1(1 + bj )

and

R(1, L) ∼ ctLt−l∏

i=1k (1 + ai)∏k

j=1 bj.

We cannot have at the same time

r = s + k and t = l,

since, together with the inequality s � t − 1 it would imply

r � l + k − 1

and contradict the assumption k + l � r. Therefore, one of the exponents r − s − k ort − l is different of 0 and this proves (17) also in this case. To conclude the proof, itsuffices to take L = hX. �

Remark 2. Observe that the condition hX � 1 means that the problem is stiff: in par-ticular, thinking of its parabolic part, this condition means that the Courant–Friedrichs–Lewy number is large, i.e. the interesting case for implicit schemes. Of course, if we arenot willing to take hX large, we can as well work with explicit schemes.


If V is not stiff, we can very well precondition only the part 1 − hM ⊗ �, usingfor instance the generalization of Bornemann’s preconditioner [3,4] to systems. Then, ifthis generalization works – we are presently checking all the details with Magali Ribot– the condition number of NA is O((ln(1/δ))2) and then the number of operations ish−1qmδ−d � (1/δ). Under assumptions (3), the CPU time for integrating up to a timeO(1) would be

Tpreconditioned iterative = O(1)qm ln(1/ε)ε−(d/s)−(1/p),

a definite improvement over (13) if p > s/2, and we find once again the necessity ofimproving the precision of the time integration. If V is stiff, we need preconditionerswhich would efficiently take V into account.

4. Splitting: known results

We shall limit ourselves to the linearized problem

ut − M�u + V u = 0, x ∈ Rd, t > 0,

where V is a matrix valued potential satisfying the condition

∀x ∈ Rd, ∀ξ ∈ R

d, ξTV (x)ξ � −λ|ξ |2,and λ is a real number. Under this assumption, V and −M ⊗ � generate a semi-groupof operators in L2(Rd)m, which are denoted by an operator exponential. The classicalLie and Strang [50] splittings can be written as follows

L(t) = exp(−tB) exp(−tA), S(t) = exp(−tB/2) exp(−tA) exp(−tB/2),

or the alternate forms obtained by exchanging A and B. Both splittings have the niceproperty that they lead to automatically stable approximations; L is an approximation oforder 1 and S is of order 2.

However, there is a limit on the order of automatically stable split approximations:given a finite sequence of non-negative numbers αj , βj , 1 � j � k, write

P(α, β, t)= exp(−tα1A) exp(−tβ1B) exp(−tα2A) exp(−tβ2B) · · ·× exp(−tαkA) exp(−tβkB); (19)

then Sheng has proved in [46] the following result: any approximation to the exponentialexp(−t (A + B)) by a linear combination with non-negative coefficients of products ofthe form P(α, β, t) is at most of order 2, unless A and B satisfy some commutationrelations; the result should have been stated as a result in the free Lie algebra with twogenerators A and B, i.e. in an algebra where A and B satisfy no commutation propertywhatsoever; but Sheng’s proof does exactly that, though the author does not say so.

In an unpublished article, I proved that the same result holds if the exponentialsare replaced by rational fractions which approximate them. Therefore, if we requireautomatic stability, we cannot go beyond Strang’s formula, i.e. order 2.


Thus, higher order formulas can be obtained only if we relax the requirement ofautomatic stability. If we can establish an estimate of the form∥∥etAe−2tBe−tA − e−2t (A+B)

∥∥L(L2)

� Ct, (20)

then formulas obtained by extrapolation from Strang’s formula will be stable; in partic-ular, the first extrapolation given by

4S(t/2)2 − S(t)

3

is of order 4 – at least formally – and will be stable if estimate (20) holds.Let us review some positive results obtained on exponential splitting, extending

somewhat the class of operators of interest.In [16,18] we have considered the case

A = − ∂

∂xa(x, y)

∂

∂x, B = − ∂

∂yb(x, y)

∂

∂y(21)

where a and b are strictly positive smooth functions of the periodic variables a and b.We proved that (20) holds by Dunford integral arguments. The same type of argumentshave been used in [19] and in the more general article [11] to treat the case

A = −�, B = V,

where � operates in Rd , and V is a potential in R

d . Let us describe the result obtainedin [14]: we denote by greek letters α, β, . . . multi-indices belonging to N

d ; |α| = α1 +· · · + αd is the length of α. ∂α is the derivative ∂α1

1 . . . ∂αdd .

Let ρ be a nonnegative number, write q = 2 max(�1/ρ�, 2). We assume that V isof class Cq , that it is nonnegative, and that there exists a number λ such that

for all α ∈ Nd such that |α| � q,

∣∣∂αV (x)∣∣ � λ(V (x)

)(1−ρ|α|)+. (22)

Then, the following estimate holds:

∀t ∈ [0, 1], ∥∥S(t) − exp(−t (A+ B)

)∥∥L(L2)

= O(t1+2 min(ρ,1/2)), (23)

Analogous results have been proved by Ichinose and Tamura [25], with estimatesin trace norm using analytical techniques; Takanobu has used probabilistic methodsin [52] to obtain estimates of the same type; in [51], Takanobu obtained estimates ofthe form (23), also by probabilistic techniques, using a Feynman–Kac formula and re-fining thus [23], which overlaps with [20]. Previously, Ichinose and Tamura had obtainedestimates on propagators of parabolic equations, allowing for a constant A and a timedependent B(t), with B(t) dominated by Aα , α < 1 [26]; they obtained also estimateson operator norm in the time independent case [24].

Other results on exponential splitting include the article of Jahnke and Lubich [27]:they prove estimates when [A,B] is dominated by Aα , α � 1, [A, [A,B]] is domi-nated by A, and A generates a holomorphic semigroup. These estimates are applied to


a Schrödinger equation with d-dimensional periodic variable and smooth potential; theyalso prove estimates for a pseudospectral discretization of this problem.

In a case which is closer to the present preoccupations, we have also proved thefollowing result [14]: let M satisfy assumption (2) and let V be a mapping of class C4

from Rd to the space of m × m matrices. It is assumed that V is bounded as well as all

its derivatives of order at most 4. Then, taking

A = −M ⊗ �, B = V,

we have proved that for all r ∈ (1,∞) the following estimate holds:∥∥S(t) − exp(−t (A + B)

)∥∥L(Lr)

= O(t).

We conclude then, using the regularizing property of the semi-group e−tA that if u be-longs to W 4,p(Rd)m, 1 < p < ∞, then

∀t ∈ [0, T ], ∣∣S(t)nu− exp(−t (A+ B)

)u∣∣Lp � Cnt3eαt |u|W 4,p ;

the constant α and C are independent of u, t > 0 and the integer n.The exponential splitting has mainly a theoretical interest, since we almost never

calculate exponentials of matrices in practice.Therefore, we move on to more practical splittings.

5. Some practical splittings and their analysis

The alternate direction methods are old methods which were initially introducedbecause the machines of the fifities and sixties did not have enough room to store the LUor Cholesky decompositions of the matrix of the Laplace operator in a two-dimensionaldomain [43]: the Peaceman–Rachford method, also called ADI (alternate directions im-plicit) was explicitly introduced with this purpose in mind: the Laplace operator, orrather its centered difference approximation on a uniform mesh was split into its x andy centered difference approximations: they are linear and they commute, so that theanalysis of the problem was quite straightforward.

We may write the approximation scheme as

un+1/4 = un − δth

2Aun,

un+1/2 = un+1/4 − h

2Bun+1/2,

un+3/4 = un+1/2 − h

2Bun+1/2,

un+1 = un+3/4 − h

2Aun+1.

(24)

In practice, up to special initial and final steps, the ADI method amounts to the successiveapplication of Crank–Nicolson schemes for A and B, with time step h/2.


The corresponding operator is

P(t) = (1 + tA/2)−1(1 − tB/2)(1 + tB/2)−1(1 − tB/2)−1. (25)

Under this form, it is clear that for all t > 0,

P(t)P (−t) = 1. (26)

If A and B are matrices, relation (26) makes sense for all small enough t ; if A and B areoperators in infinite dimensional space, it is not even clear that the operations involvedin the definition (25) of P can be performed, and it may be a nontrivial problem to showthat P makes sense on an appropriate space, and that its powers are well defined, at leastfor small enough values of t .

Of course, when A and B commute, the analysis is very easy, and the reader isreferred, for instance, to [38–41], for a review of a large number of alternate directionsmethods. See also the articles by Verwer and his coauthors [34,53] who try to identifysystematically examples where commutativity hypotheses can be satisfied.

In [49], Sportisse and Verwer include a stiff linear term in a linear ODE situation,and they observe an order reduction phenomena when integrating by splitting. In [28],Kozlov and Owren analyse the order reduction phenomenon for split operator formulas,in the presence of a stiff linear term.

Removing the commutativity assumption leads to some unexpected phenomena,such as the following instability result, proved in [45]: assume that a is strictly positive,that the absolute value of the real number b is less than a and that r is also strictlypositive; define matrices

MA =(a b

b a

), MB =

(r−2 00 1

),

and operators

A = −MA ⊗ �, B = −MB ⊗ �.

Then it is possible to choose the values of a, b and r in such a way that

∀t > 0,∥∥P(t)∥∥L(L2(Rd)2)

� 2.

The problem lies with the order of the commutator of A and B, given by

[A,B] =(

0(1 − r−2

)b(

r−2 − 1)b 0

)⊗ �2,

i.e. it is of order 4 unless r = 1. The norm of P(t) can easily be found in Fouriervariables, since the multiplication by the operator P(t) is in Fourier variable the multi-plication by the matrix

q(τ) = (12 + τMA)−1(2(12 + τMB)

−1 − 12)(12 − τMA), τ = t|ξ |2

2.

It is then an exercise in matrix algebra to prove the result of instability.


It is possible to prove the stability of the Peaceman–Rachford case in the non-commutative case, under appropriate assumptions on A and B. Assuming that A and B

are self-adjoint in a Hilbert space, and bounded from below, one may assume withoutloss of generality that A and B are non negative. Then, they have positive square rootswhich will be denoted

√A and

√B. The condition under which stability is proved in [45]

says essentially that the commutator of√A and

√B is dominated by

√A + √

B. Theexact assumption is considerably more technical, and will not be given here. If we thinkof A and B as pseudodifferential operators, the assumption says basically that A andB behave as operators of order 2 in some appropriate scale of asymptotics. However,the use of asymptotics may not be the most pleasant solution, since it requires muchregularity: in fact, only finite regularity is required to perform the proof, which usesrational fractions instead of asymptotic expansions.

Just to give an idea of what is involved, I shall reproduce here one step of the proof:let us show that (

1 + t√A

)−1(1 + t

√B

)−1√A = O

(1/

√t).

Indeed, formally[(1 + t

√B

)−1,√A

] = (1 + t

√B

)−1[√A,

√B

](1 + t

√B

)−1. (27)

Let us write the assumption on the commutator of√A and

√B under the form[√

A,√B

] = √Am1 + √

Bm2 + m3, (28)

where m1, m2 and m3 are bounded operators. Let now

g(t) = (1 + t

√A

)−1(1 + t

√B

)−1√A and g0(t) = (

1 + t√A

)−1√A

(1 + t

√B

)−1.

It is obvious that ∥∥g0(t)∥∥ = O

(1/

√t).

We infer from (27) and (28) the identity

g(t) = g0(t) + tg(t)m1(1 + t

√B

)−1 + (1 + t

√A

)−1h(t), (29)

with

h(t) = t(1 + t

√B

)−1(√Bm2 + m3

)(1 + t

√B

)−1,

whose norm is bounded as follows:∥∥h(t)∥∥ = O(√

t).

We infer from (29) that for t small enough:

g(t)(1 − tm1

(1 + t

√B

)−1) = g0(t) + (1 + t

√A

)−1h(t),

and therefore ∥∥g(t)∥∥ = O(1/

√t).


Similar but algebraically much more complicated arguments lead to the following esti-mate:

if j � 2 and k � 2,∥∥(1 + tA)−1(1 + tB)−1Aj/2Bk/2

∥∥ = O(t (j+k)/2).

This result can be reached only under appropriate assumptions on the commutators ofthe mj ’s,

√A and

√B: only a finite number of hypotheses are necessary. We write then

P(t) = q(t) + r(t), q(t) = (1 + tA/2)−1(1 − tA/2)(1 + tB/2)−1(1 − tB/2),

and we are able to prove that ∥∥r(t)∥∥L(H)= O

(√t), (30)

which does not suffice to conclude to stability. In order to reach the result, we calculatethe square of the operator norm of P(t):∥∥P(t)∗P(t)∥∥L(H)

= ∥∥q(t)∗q(t) + q(t)∗r(t) + r(t)∗q(t) + r(t)∗r(t)∥∥L(H)

.

We know that the operator norm of q(t) is at most equal to 1, and estimate (30) lets usdeal with the term r(t)r (t). There remains to estimate

q(t)∗r(t) + r(t)∗q(t),

which seems a priori of size O(√t ). However, both terms contain the bracket [A,B]

but with opposite signs. By means of a long, mainly algebraic proof, chasing a crowd ofcommutators out of their nooks and crannies, it is possible to show that this feature letsus gain one half-power of t and show stability.

6. Extrapolation of Peaceman–Rachford

The essential algebraic feature of the Peaceman–Rachford formula is its reversibil-ity (26), a consequence of which is the formal series expansion

lnP(t) =∞∑j=1

P2j+1t2j+1;

therefore, we expect the extrapolation of a reversible approximation to the exponentialto be more precise than the extrapolation of an arbitrary approximation.

However, stability can be a bad problem when trying to extrapolate the simplestinstance of a Peaceman–Rachford extrapolation, i.e. the case B = 0, which is simply aCrank–Nicolson formula:

c(t) = (1 + tA/2)−1(1 − tA/2).

Indeed in the self-adjoint case, the spectrum of

4c(t)2 − c(t)

3


is the image I of the spectrum of A by the mapping

x → 1

3

(4(1 − tx/4)2

(1 + tx/4)2− 1 − tx/2

1 + tx/2

),

and if A is unbounded, sup I � 5/3. Therefore, Richardson’s extrapolation is a badsolution for this problem.

We proposed in [17] to construct a stable extrapolation of the form

n∑j=1

ajP (t/j)j

by choosing the coefficients aj in a way that removes the above-mentioned difficulty; thecommutative case can guide us to understand the requirements: if A and B commute, itsuffices to look at the symbol of P(t) which is

P (t, λ, µ) = (1 − tλ/2)(1 − tµ/2)

(1 + tλ/2)(1 + tµ/2).

In the simplest case, we look for coefficients a1, a2 and a3 such that the following orderconditions are satisfied

a1 + a2 + a3 = 1,a1

9+ a2

4+ a3 = 0

and the condition ensuring the stability for λ → ∞ and µ → 0

−a1 + a2 − a3 = 0.

The unique solution of this system of equations is

a3 = 45

64, a2 = 1

2, a1 = −13

64,

which leads to the approximation formula

P4(t) = 45

64P

(t

3

)3

+ 1

2P

(t

2

)2

− 13

64P(t). (31)

We have given higher order formulas, namely

P6(t) = 197

7360f

(t

2

)− 407

690f

(t

4

)2

+ 3483

7360f

(t

6

)3

+ 376

345f

(t

8

)4

, (32)

and

P8(t)= − 11281

6174720f

(t

2

)+ 9137

42210f

(t

4

)2

− 12167739

9605120f

(t

6

)3

+ 5984

21105f

(t

8

)4

+ 30578125

17289216f

(t

10

)5

. (33)


These formulas have been tested numerically by Dia in his thesis [15], on the exam-ple (21), with different types of spatial discretizations: finite differences and finite ele-ments of different kinds.

Formally, Pj is of order j , but the numerical experiments show that the formalorder is not reached: the experimental precision is smaller than the theoretical precision,and the difference is smaller than 1; we have not been able to explain theoretically thisfinding.

7. Nonlinear splitting

Before reviewing some recent results on splitting, it is important to mention threeother strategies, based on implicit–explicit multistep schemes: one is the strategy of Ma-day et al. [37] which is specifically geared towards the resolution of problem involvinga convective term, and splits into the convective part and the nonconvective part; then,a change of unknown functions leads to an integral formulation along characteristics,and the standard technique for obtaining multistep approximations is used to get a dif-ference scheme. The other strategy is that of Akrivis et al. [1]: the authors couple twomultistep schemes, an implicit scheme for the parabolic part and an explicit scheme forthe nonlinear part, and they prove convergence, provided that the linear operator comingfrom the parabolic part is A(0)-stable: this holds if it is self-adjoint. Moreover, sincethe nonlinear part is treated by an explicit method, it means that it must not be stiff. Thelast strategy is explained at length in Lang’s book [31], and in the articles [29,30,32,33];there the authors discretizes in time and then in space, the so called Rothe method; thetime discretization is performed by an implicit Rosenbrock method, and an adaptivestrategy is developed for the spatial approximation. In [22], Gerisch and Verwer applysplitting techniques with low order Rosenbrock methods to integrate reaction–diffusionequations from biology.

It should be noted that convergence has not yet been proved for split schemes rel-ative to a reaction–diffusion problem with stiff reaction term; however, the question isimportant for practical applications, see, for instance, the work of Sportisse [47,48].

In [13], we implemented the Richardson extrapolation of an exponential splittingmethod for a reaction–diffusion system, with periodic boundary conditions. More pre-cisely, we had a one-dimensional quintic complex-valued Ginzburg–Landau with peri-odic boundary conditions, that can be written

ut = (m + iαµ0)uxx − (m + iαµ1)u

+ (1 + iαµ2)|u|2u− (1 + iαµ3)|u|4u, x ∈ R/LZ, (34)

with initial condition

u(x, 0) = u0(x).

We denote by Zt the flow of (34):

u(x, t) = (Ztu0

)(x).


We associated with the differential part the linear semigroup

Xt = e(m+iαµ0)∂xx

and to the other remaining terms the flow Y t of the differential system; in other words,define a function f from C to C by

f (u) = −(m + iαµ1)u+ (1 + iαµ2)|u|2u− (1 + iαµ3)|u|4uand let Y t be the solution of

∂Y t

∂t(x) = f

(Y t(x)

), X0(x) = x.

Then a Strang approximation of the flow Zt of the partial differential equation (34) canbe written

St = Xt/2 ◦ Y t ◦ Xt/2,

and its Richarson extrapolation is

T t = 1

3

(4St/2 ◦ St/2 − St

). (35)

In this special case, the periodic boundary condition enable us to use an exact formulafor Xt , provided that we use an appropriate spectral basis; we replaced Y t by a Runge–Kutta approximation of order 4, since the problem was not very stiff, and we ran simu-lations to see how efficient the idea was. Since we needed 5 fast Fourier transforms forupdating by one time step, the operation count for one time step was O(δ ln(1/δ)). Wemade an honest comparison by running a Strang formula for the same problem, approx-imating the nonlinear flow Y t this time by a modified Euler scheme, i.e. a method oforder 2. The numerical experiments showed the scheme defined by (35) to be of order 4in time. In terms of computational time versus error size, the experiments gave a veryclear advantadge to our scheme above the scheme of order 2.

The theory is less advanced than the experiments: we have been able to prove inthe case of a scalar reaction–diffusion in R an estimate of the form∥∥(

T h)nu0 − Znhu0

∥∥L∞(R)

� C√h

as h tends to 0 and nh tends to τ ; here u0 is a bounded continuous function on R.Descombes has proved in [10] for a system of the form (1) the following estimate

in L2 ∣∣T hu0 − Zhu0

∣∣L2(Rd)m

� C(u0)t|u0|L2(Rd)m,

and if u0 belongs to H 2(Rd)m, the exponent of t becomes 2.There is even more: Descombes proves the convergence and gives an error esti-

mate, using the regularizing character of the operator Xt , and obtained the estimate:∥∥(T h

)nu0 − Znhu0

∥∥L2(Rd)m

� Ch| ln h|‖u0‖L2(Rd)m. (36)


This result holds if u0 is essentially bounded, square integrable, and four times contin-uously differentiable. Descombes had also to assume that the exact solution belongs toC0([0, T ];L2(Rd)m ∩ L∞(Rd)m). Estimate (36) is obtained by a combination of thelinear estimates as in [11], Taylor expansion and the regularizing property of partiallydiffusive semigroups.

All these results indicate that the stability, the order estimates and the convergencecan probably be established for the fully discretized system in time and space, with theflow Y t is replaced by a Runge–Kutta explicit scheme of order 4; however, even if itis true, it is probably an extremely technical result, and it is definitely not yet proved.Another observation is that the case with periodic boundary conditions is probably noharder than the case in R

d ; however, other boundary conditions will probably causemany difficulties, which we cannot manage presently.

Descombes and Ribot have studied theoretically and numerically the convergenceof the Peaceman–Rachfdord scheme for a system of reaction–diffusion, and they reporttheir results in [12]. More precisely, with the notations (1), they consider

P(t) =(

1 + tF

2

)−1(1 + tM�

2

)(1 − tM�

2

)−1(1 − tF

2

).

They prove that the scheme converges; more precisely, if f is of class C7 and u0 issmooth enough, they show the estimate∣∣P(h)mu0 − Znhu0

∣∣L2(Rd)m

� Ch2|u0|L2(Rd)m.

They implemented a computation in the case of the leopard spots system given by Mur-ray [42, p. 438]:

∂u

∂t− ν�u + f (u, v)= 0,

∂u

∂t− dν�u + g(u, v)= 0.

(37)

The functions f and g are defined as follows:

l(u, v) = ρuv

1 + u + kuv, f (u, v) = a− u− l(u, v), g(u, v) = α(b−v)− l(u, v).

This has been proposed as a model of a Turing system where spatial structures are cre-ated by interaction of non linear phenomena and spectral properties. The numericalparameters are

a = 92, b = 64, α = 1.5, ρ = 18.5,

k = 0.1, d = 10, γ = 1, ν = 0.001.(38)

In practice, they replaced the nonlinear implicit step by an approximation of it of order 2:

(1 + tF/2)−1 = 1 − (t/2)F (1 − tF/2).

Some of the numerical results are given in figure 1.


Figure 1. The evolution of the solutions of (37), starting from initial data with one peak. Computation ofDescombes and Ribot.

8. Open questions

Here is a list of open questions most of which are probably quite difficult.

1. Find efficient preconditioners for (7); the first level is to do that for systems when thereaction term is not stiff, and we probably know how to do that but need some moretime to check the preconditioner even in non symmetric cases and run experiments;the second level is to tackle the case of equations, with q = 1 and a stiff reactionterm; the third level is to find something for systems, with q > 1 and a stiff reactionterm.

2. Define an adaptative strategy, so as to localize the regions where stiffness makes theresolution difficult.

3. Domain decomposition: problem 2 leads to a domain decomposition strategy, and itis very tempting to try to include this strategy in a splitting formulation. However,splitting may involve a projection step, as explained in the seminal work of Chorin etal. [9], where it is applied to fluid mechanical problems: in that case, the projection isperformed on a space of divergence free functions. As soon as a projection is includedin a splitting strategy, the reversibility (26) fails, and there is a significant loss inprecision of the time integration. The domain decomposition strategy developed forinstance in the article of Gaiffe et al. [21] includes a projection so as to satisfy thetransmission condition at the domain boundaries. It remains therefore a challengingquestion to use an operator splitting approach in a domain decomposition setting,while getting a high precision for the time integration.

4. Give sufficient conditions of stability for the extrapolations (31), (32) and (33) ofPeaceman–Rachford formulas. A preliminary study has shown that in the case (21),a version of Weyl pseudodifferential calculus with parameters should give the answer.But this is a very heavy theory and I backed down before the enormous number ofpages that would be needed to prove a statement which concerns a rather particularcase.

5. Prove convergence of fully discretized approximations, be it in the case of exponen-tial splittings or of Peaceman–Rachford formulas.


Acknowledgements

I would like to thank Stéphane Descombes and Magali Ribot for letting me read anadvance copy of their results on Peaceman–Rachford approximation and for giving mepermission to illustrate this article with their numerical results. Many thanks are due toJohn Butcher and the organizing committee of the ANODE 2001 conference, who gaveme the opportunity to speak with the voice of partial differential equations in an ordinarydifferential equations meeting, and a very exciting one.

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Numerical Integration of Reaction–Diffusion Systems