a monotonically-damping second-order-accurate unconditionally-stable numerical scheme for diffusion

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 133: 1559–1573 (2007)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.116

A monotonically-damping second-order-accurateunconditionally-stable numerical scheme for diffusion

Nigel Wood,* Michail Diamantakis and Andrew StaniforthMet Office, Exeter, UK

ABSTRACT: We present a new two-step temporal discretization of the diffusion equation, which is formally second-order-accurate and unconditionally stable. A novel aspect of the scheme is that it is monotonically damping: the dampingrate is a monotonically-increasing function of the diffusion coefficient, independent of the size of the time step, whenthe diffusion coefficient is independent of the variable being diffused. Furthermore, the damping rate increases withoutbound as the diffusion coefficient similarly increases. We discuss the nonlinear behaviour of the scheme when the diffusioncoefficient is a function of the diffused variable. The scheme is designed to maintain any steady-state solution. We presentexamples of the performance of the scheme. Crown Copyright 2007. Reproduced with the permission of the Controllerof HMSO. Published by John Wiley & Sons, Ltd.

KEY WORDS boundary-layer scheme; damping; two-time-level

Received 8 January 2007; Revised 17 April 2007; Accepted 14 May 2007

1. Introduction

Diffusion operators typically arise in atmospheric modelsfor either of two reasons:

• To parametrize the influence that unresolved scales ofmotion have on resolved scales. A first example isthe parametrization of planetary-boundary-layer turbu-lence, which in large-scale models usually acts only inthe vertical direction. Such schemes may additionallybe applied above the boundary layer to represent theunresolved mixing due to, for example, gravity-wavebreaking. A second example is the parametrizationof horizontal turbulence to maintain realistic spectralslopes for atmospheric variables.

• To control problems of numerical origin: for exam-ple, undesirable noise due to poor initialization,poorly-resolved small-scale forcing from parametriza-tion schemes, and weak computational instability.

However, the numerical implementation of such diffusionoperators presents a significant challenge in numericalweather and climate prediction models. This is becausethe time steps used in such models are usually largecompared with the typical time-scale of the diffusionoperator. The system is then stiff. This has severalimportant consequences in relation to the followingcriteria:

1. Unconditional stability. Explicit schemes are unlikelyto be viable, since the time step will violate the

* Correspondence to: Nigel Wood, Met Office, FitzRoy Road, ExeterEX1 3PB, UK. E-mail: [email protected]

condition for stability. Thus an implicit scheme isgenerally required. (Otherwise, some form of timesubstepping is required in order to guarantee thatthe effective time step satisfies the condition forstability.) This can adversely affect the efficiency ofthe scheme. For diffusion in one direction only (suchas the vertical, for boundary-layer schemes), this isgenerally not an issue. However, when the diffusionoperator acts in two or more directions (as horizontaldiffusion does), solving the implicit operator can leadto a significant computational overhead. This may beovercome, for example, by approximating the multi-dimensional operator as a product of several one-dimensional operators.

2. Accuracy. Different degrees of accuracy may bedesirable in different situations. Wherever the time-scale of diffusion is long compared with the timestep, the diffusion process will be well representedby the model, and second-order (spatial and temporal)accuracy is desirable. However, if the diffusion time-scale is shorter than the time step, the diffusionprocess will not be well represented, and in the contextof a full atmospheric model, second-order-accurateschemes can generate considerable noise (Dubal et al.,2004) unless the model is perfectly initialized. Insuch circumstances it is desirable to use a first-orderscheme, which effectively damps small-scale noise.The problem is that the two regimes may coexist,either in different areas of the model’s domain or atdifferent times in the evolution of the model’s forecast.

3. Monotonic damping. Higher-order-accurate schemes(such as the Crank–Nicolson scheme) can exhibitunphysical behaviour, in that the damping rate (theamount by which the magnitude of a quantity is

Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO.Published by John Wiley & Sons, Ltd.

1560 N. WOOD ET AL.

reduced in a time step) decreases when the diffusioncoefficient is increased. Only the fully-implicit andover-weighted implicit schemes (all first-order) do notexhibit such behaviour for any value of time step whenthe diffusion coefficient is independent of the variablebeing diffused. However, the over-weighted schemeshave the undesirable property that their damping ratetends – albeit monotonically – to a constant, finitevalue as the diffusion coefficient increases.

4. Maintenance of steady-state solutions. Some existingschemes do not properly maintain steady-state solu-tions, but instead wrongly distort them (Caya et al.,1998; Murthy and Nanjundiah, 2000; Dubal et al.,2006).

The problem, then, is that second-order-accurateschemes can exhibit unphysical behaviour and canalso lead to noisy solutions. But first-order-accurateschemes may not be accurate enough when the diffusioncoefficient is such that the process is well resolved bythe model’s time step. Here, we develop a scheme that isoptimal in the sense that for small time steps it is second-order-accurate. The scheme adapts to an increasing timestep, dynamically lowering the scheme’s accuracy. Itthereby damps spurious noise when the diffusion processis poorly resolved in time, while maintaining monotonicdamping whatever the size of the time step. In Section2 below, we motivate, describe and analyse the schemefor linear diffusion, where the diffusion coefficients areindependent of the diffused variable.

A further consequence of the stiffness of the problem(Kalnay and Kanamitsu, 1988; Girard and Delage, 1990;Benard et al., 2000) arises when the diffusion coefficientis a function of the variables on which the diffusionoperator is itself acting – as is the case for planetary-boundary-layer schemes. The resulting nonlinearity canthen lead to numerical instability. It also reduces thethreshold value of the time step for which unphysicalbehaviour occurs. In Section 3 we motivate and describean extension of the scheme to handle nonlinear diffusion.

In Section 4 we present results, using the full nonlinearscheme, the Met Office’s operational parametrizationschemes to determine the diffusion coefficients, and realdata. Conclusions are presented in Section 5.

2. Derivation of the proposed scheme: the linearcase

2.1. Preliminaries

Consider the one-dimensional diffusion equation:

∂F

∂t= ∂

∂x

(K

∂F

∂x

). (1)

Here F is the scalar variable being diffused, t is time, x

is an independent spatial variable, and K is the diffusioncoefficient, which for the moment we assume to beconstant (so that the equation is linear in F ). Only

the temporal aspects of two-time-level discrete forms ofEquation (1) are of interest here. It is therefore assumedthat F can be Fourier-decomposed as

F(x, t) =∑

k

Fk(t)eikx, (2)

where k is a wave number and Fk is the amplitude of thecorresponding component. Then Equation (1) reduces tothe first-order damping equation:

dF

dt= −βF, (3)

where the damping coefficient is β ≡ k2K , and subscriptsk have been dropped for convenience. (The same proce-dure can be formally followed for spatial discretizationsof Equation (1), provided β is modified to include theresponse function of the discretization.) Integrating Equa-tion (3) between time t and time t + �t , where �t is thetime step, gives the exact result:

F t+�t = e−β�tF t . (4)

Consider a general two-time-level scheme of the form:

F t+�t − F t

�t= −β

(1 + ε

2F t+�t + 1 − ε

2F t

). (5)

(The parameter ε is related to the weighting parameterγ – as used, for example, in scheme (g) of Kalnay andKanamitsu (1988) – by (1 + ε)/2 = γ and (1 − ε)/2 =1 − γ .) Setting ε = −1 gives the explicit scheme. TheCrank–Nicolson scheme is obtained with ε = 0, and thefully off-centred scheme arises with ε = 1. Here, weretain ε as an arbitrary function, assumed only to beindependent of F and time. The response function forthe scheme (5) is:

E ≡ F t+�t

F t = 1 − 12 (1 − ε)β�t

1 + 12 (1 + ε)β�t

. (6)

2.2. Requirements and their satisfaction

As discussed in Section 1, there are four requirementsfor a good scheme. The first three of these will be usedto determine the optimum form for ε, and thence theproposed new scheme. (The fourth is applicable to theforced nonlinear problem examined in Section 3.)

1. Unconditional stability. This requires |E| ≤ 1 for all�t . This holds provided that both β�t ≥ 0 (i.e. thephysical system is stable) and ε ≥ 0 (corresponding tothe usual requirement of off-centring weights ≥ 1/2).

2. Accuracy. Second-order accuracy for small �t

requiresE = Eexact + O(�t3),

where, from Equation (4),

Eexact = e−β�t .

Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 133: 1559–1573 (2007)Published by John Wiley & Sons, Ltd. DOI: 10.1002/qj

A MONOTONICALLY - DAMPING NUMERICAL SCHEME FOR DIFFUSION 1561

Therefore, expanding Eexact and E from Equation (6)under the assumption that both β�t and εβ�t are small,second-order accuracy requires:

1 − β�t + 1 + ε

2(β�t)2 + O(�t3)

= 1 − β�t + 1

2(β�t)2 + O(�t3). (7)

From this it follows that second-order accuracyrequires ε = O(�t), or, noting that ε is dimension-less and that β is the only inverse time-scale in theproblem, ε = O(β�t). (This is trivially satisfied byε = 0, consistent with the second-order accuracy ofthe Crank–Nicolson scheme.)

3. Monotonic damping. This requires that the dampingrate, which is inversely proportional to |E|, increasesmonotonically as β increases, i.e. that ∂|E|2/∂β < 0,thereby mimicking the damping effect of positive β

in the analytical solution. From Equation (6), andassuming that both β�t ≥ 0 and ε ≥ 0, as required forunconditional stability, monotonic damping thereforerequires that:

{1 − β�t

2(1 − ε)

} {1 − (β�t)2

2

∂ε

∂(β�t)

}> 0.

(8)

In other words, the two factors enclosed in braces inEquation (8) must have the same sign.

It can be shown that Equation (8) is satisfied (bothfactors being positive) for all β�t with the choice

ε = nβ�t

1 + nβ�t, (9)

where n > 1/2. Note that ε then varies monotonicallyas a function of β�t between 0 and 1, and also thatε = O(β�t) for small β�t , as required for second-orderaccuracy.

Thus the scheme

F t+�t − F t

�t= −β

{1

2

(1 + nβ�t

1 + nβ�t

)F t+�t

+1

2

(1 − nβ�t

1 + nβ�t

)F t

}, (10)

meets the three conditions of unconditional stability,second-order accuracy for small β�t , and monotonicdamping. This scheme works essentially because itdynamically keeps the off-centring parameter close tozero for small damping coefficients but asymptoticallyapproaches fully-implicit off-centring as the dampingincreases.

Note, though, that for a general implementation, whereβ is a spatial operator, it may be problematic thatthe off-centring weights are functions of the damping

coefficients. This motivates the following developmentof the scheme to address this drawback.

2.3. Development of the proposed scheme

The response function for the scheme (10) is:

E = 1 + (n − 12 )β�t

1 + (n + 12 )β�t + n(β�t)2 . (11)

Provided that n ≥ √2 + 3/2 (which guarantees n > 1/2),

the denominator can be factorized in real space, andEquation (11) can be rewritten as:

E = 1 − (1 − a − b)β�t

(1 + aβ�t)(1 + bβ�t), (12)

where a and b are the two roots of

y2 −(n + 1

2

)y + n = 0. (13)

This is the response that would be obtained by acombination of one explicit step and two implicit steps,in any order, in the form of a symmetrized split-implicit scheme (Staniforth et al., 2002a, 2002b). If wechoose the most symmetric form (implicit step – explicitstep – implicit step), then the scheme (10) is equivalentto:

F ∗ − F t

�t= −aβF ∗

F ∗∗ − F ∗�t

= −β(1 − a − b)F ∗

F t+�t − F ∗∗�t

= −bβF t+�t

. (14)

Varying n, subject to n ≥ √2 + 3/2, gives a continuum

of schemes. As n increases for fixed β�t , the off-centringof the scheme increases. Therefore, to ensure the optimallevel of second-order accuracy, n should be chosen assmall as permitted. Thus it is chosen to equal the limitingvalue for real factorization, n = √

2 + 3/2. This thenmeans that a = b = 1 + 1/

√2, and the symmetry of the

scheme is optimized. The proposed scheme is therefore:

F ∗ − F t

�t= −

(1 + 1√

2

)βF ∗

F ∗∗ − F ∗�t

=(

1 + √2)

βF ∗

F t+�t − F ∗∗�t

= −(

1 + 1√2

)βF t+�t

. (15)

This represents an explicit ‘anti-diffusion’ step sand-wiched between two implicit ‘over-diffusive’ steps. Theresponse function for this scheme is:

E =1 +

(1 + √

2)

β�t(1 +

(1 + 1√

2

)β�t

)2 , (16)

1562 N. WOOD ET AL.

and it can be verified that this has all the requiredproperties. In fact, it can be shown that (15) is theonly symmetric scheme of this form (one explicit andtwo implicit steps) that has the three desired propertiesof unconditional stability, second-order accuracy, andmonotonic damping.

The scheme has the additional desirable propertythat the response function (16) (which is inverselyproportional to the damping rate) tends to zero as β�t →∞, as does the exact response function (see Equation (4)).This is in contrast to all schemes with constant ε, exceptfor the off-centred scheme with ε = 1 (see Equation (6)).

In Figure 1, the response function of the proposedscheme is compared to the analytic response function,and to the response functions of the explicit (ε =−1) scheme, the Crank–Nicolson scheme (implicit withε = 0), and the fully-off-centred scheme (implicit withε = 1). The explicit scheme becomes non-monotonicand oscillatory (E < 0) when β�t > 1, and unstable(|E| > 1) for β�t > 2. The Crank–Nicolson scheme hasgood accuracy for small β�t but becomes oscillatoryfor β�t > 2. The proposed scheme and the off-centredscheme have similar behaviour, remaining stable andmonotonic for all β�t . However, the proposed schemehas better accuracy for all values of β�t , and for smallβ�t it is formally second-order, whereas the off-centredscheme is first-order.

Since the proposed scheme requires the solution oftwo implicit equations, a fairer test – in terms of com-putational expense (in this linear case for which thediffusion coefficient is independent of time) – is to com-pare the proposed scheme with the implicit schemes(ε = 0 and ε = 1) substepped twice. This means thateach scheme is applied twice per time step, each appli-cation using a half-length time step. This comparisonis shown in Figure 2. The improvement in accuracy ofboth the implicit schemes is evident, especially for the

0 1 2 3 4 5

β∆t

−1.0

−0.5

0.0

0.5

1.0

E

AnalyticExplicitImplicit ε=0Implicit ε=1New

Figure 1. Response functions of the analytic solution (solid), the explicitscheme (dotted), the implicit scheme with ε = 0 (dash-dotted), theimplicit scheme with ε = 1 (dash-dot-dot-dotted), and the new scheme(dashed), plotted against the dimensionless damping coefficient β�t .

AnalyticImplicit ε=0, 2 substepsImplicit ε=1, 2 substepsNew

0 1 2 3 4 5

β∆t

−0.5

−1.0

0.0

0.5

1.0

E

Figure 2. As Figure 1 (without the explicit scheme), but with theimplicit schemes now applied with two substeps.

Crank–Nicolson case (ε = 0). However, though it isbarely evident in the plot, this scheme is non-monotonicfor β�t > 4. The advantage of the proposed schemeover the substepped fully-off-centred scheme remains itshigher-order accuracy for small β�t . Furthermore, itbehaves better in the nonlinear case, as discussed in thenext section.

3. The nonlinear case

3.1. Preliminaries

In large-scale models, vertical diffusion arises as aparametrization of boundary-layer mixing. Then the dif-fusion coefficient β is a function of various prognosticvariables. In particular, it can be a function of F , andhence of time. The problem becomes nonlinear. In mostsuch models, the diffusion coefficient is estimated fromthe values of the fields (here F ) at the present time step:i.e. it is explicitly evaluated. This can lead to unphysi-cal behaviour. Such behaviour was analysed by Kalnayand Kanamitsu (1988) in the framework of the simplifiedscheme:

dF

dt= −(KF P )F + S. (17)

(Since the equation is now nonlinear, it can only beviewed as an exemplary model of the temporal behaviourof the more general diffusion equation (1) when K =K(F).) Here S is a constant forcing, or source, term,and KF P is the (redefined and nonlinear) diffusioncoefficient, with K constant. P is assumed to be positive.The steady-state solution of Equation (17) is

F0 =(

S

K

) 1P + 1

. (18)

A linear problem is obtained for analysis purposesby assuming that departures of F from F0 are small,

i.e. that F ′ ≡ F − F0 satisfies |F ′| |F0|. Then lineariz-ing Equation (17) about the steady-state solution gives:

dF ′

dt= −(KF P

0 )(F ′ + PF ′). (19)

This can be integrated between time t and time t + �t

to give the exact result:

(F ′)t+�t = e−β(1+P)�t (F ′)t , (20)

where β ≡ KF P0 .

Consider now application of the approach given inSection 2 to this slightly revised problem. We consideronly two-time-level schemes in which the diffusioncoefficient is evaluated explicitly. This means that theterm PF ′ is evaluated explicitly as P(F ′)t . (Henceforth,the primes are dropped for clarity, so that F indicatesperturbations from F0.) Discretization (5) applied toEquation (19) gives:

F t+�t − F t

�t= −β

{1 + ε

2(F t+�t + PF t)

+1 − ε

2(F t + PF t)

}. (21)

The response function for this scheme is:

E = 1 − β�t( 1−ε2 + P)

1 + β�t( 1+ε2 )

. (22)

3.2. Requirements and their satisfaction

As discussed in Section 1, there are four requirements fora good scheme:

1. Unconditional stability. This requires |E| ≤ 1, whichin turn, from Equation (22), requires:

(P − ε)β�t ≤ 2. (23)

(With ε replaced by 2γ − 1, this is the stabilityrequirement for scheme (g) of Kalnay and Kanamitsu(1988) (see their table 1).) Therefore, unconditional(on β�t) stability is obtained provided

ε ≥ P. (24)

Typically P ≥ 1, and so the condition (24) motivatesthe over-weighting of the implicit scheme used in theoperational boundary-layer schemes of, for example,ECMWF and the Met Office’s Unified Model (UM).

2. Accuracy. Expanding Equation (22) in β�t for smallβ�t , and setting the resulting expression equal to that

obtained from the exact result derived from Equation(20), gives the requirement that

1 − β�t(1 + P)

+ (β�t)2

2(1 + ε)(1 + P) + O((β�t)3)

= 1 − β�t(1 + P)

+ (β�t)2

2(1 + P)2 + O((β�t)3). (25)

Therefore second-order accuracy requires that

ε = P + O(β�t). (26)

3. Monotonic damping. We require that

∂|E|2∂β

< 0.

Note that the derivative is evaluated keeping P

constant, i.e. increased diffusion is considered to bedue to increased values of either K or F0, but not P .This leads, similarly to before, to the requirement:

{1 − β�t

2(1 − ε + 2P)

}

×{

1 + P + β�tP (1 + ε)

− (β�t)2

2(1 + P)

∂ε

∂(β�t)

}> 0. (27)

In other words, the two factors enclosed in braces inEquation (27) must have the same sign.

4. Maintenance of steady-state solutions. The exactsteady-state solution (18) should be maintained.

Inspired by the previous results, we find that the first threerequirements – (24), (26), and (27) (with both factorspositive) – are met if we choose

ε = P + (1 + P)nβ�t

1 + nβ�t, (28)

with

n >1 + P

2. (29)

Thus the scheme

F t+�t − F t

�t

= −β(1 + P)

{1

2

(1 + nβ�t

1 + nβ�t

)F t+�t

+1

2

(1 − nβ�t

1 + nβ�t

)F t

}, (30)

1564 N. WOOD ET AL.

with the condition n > (1 + P)/2, meets the three con-ditions of unconditional stability, second-order accuracyfor small β�t , and monotonic damping. (This reducesto the scheme (10) when P = 0, i.e. when the nonlinearcoefficient K reduces to a constant, and the problem thenbecomes linear.)

Again, though, for a general implementation whereβ is a spatial operator, it may be problematic that theoff-centring weights are functions of the damping coeffi-cients. This motivates the following further development.

3.3. Development of the proposed scheme

The response function for the scheme (30) is:

E = 1 + (n − P+1

2

)β�t

1 + (n + P+1

2

)β�t + n(P + 1)(β�t)2 . (31)

The problem of the off-centring weights being a functionof β was circumvented above by choosing n so thatEquation (11) factorized into a product and quotientof three terms of the form 1 + αβ�t , and hence thescheme was written as a three-step scheme. Here, though,this will not work, as can be seen by considering thefull, underlying problem (19) and noting that the PF

term (recall that the prime has been dropped) is alwaysevaluated explicitly. This shows that the operation of β onany discrete representation of F has to be associated witha corresponding term PF t , as in Equation (21). Thus,

βF −−−→ β(F ∗ + PF t), (32)

where the arrow indicates the operation of discretization,and F ∗ is any discrete representation of F . So a viablescheme will only be obtained if, instead of factorizing E,the alternative response function

E∗ ≡ F t+�t + PF t

F t + PF t = E + P

1 + P, (33)

(obtained by replacing F ∗ by F ∗ + PF t in the definition(6) of E) can be factorized into a product and quotientof terms of the form 1 + αβ�t . From Equation (31),

E∗ = 1 + (n + P−1

2

)β�t + nP (β�t)2

1 + (n + P+1

2

)β�t + n(P + 1)(β�t)2 . (34)

For the scheme to be viable, both the quadratics (of thenumerator and denominator) must factorize in real space,so that E∗ can be written as

E∗ = (1 + E1β�t)(1 + E2β�t)

(1 + I1β�t)(1 + I2β�t), (35)

with E1, E2, I1 and I2 real. This requires that(n + P − 1

2

)2 ≥ 4nP(n + P + 1

2

)2 ≥ 4n(P + 1)

. (36)

For this it suffices that

n ≥(√

2 + 3

2

)(P + 1) >

P + 1

2. (37)

If we choose the smallest viable value for n,

n =(√

2 + 3

2

)(P + 1),

then Equation (34) factorizes to the form (35), with

E1 =(

1+ 1√2

) (P + 1√

2±

√P(

√2 − 1)+ 1

2

), (38)

E2 =(

1+ 1√2

) (P + 1√

2∓

√P(

√2−1)+ 1

2

), (39)

and

I1 = I2 =(

1 + 1√2

)(1 + P). (40)

Equation (35) represents a combination of two explicitand two implicit steps, in any order. The scheme maytherefore be written as follows:

F ∗ − F t

�t= −I1β(F ∗ + PF t)

F ∗∗ − F ∗�t

= E1β(F ∗ + PF t)

F ∗∗∗ − F ∗∗�t

= E2β(F ∗∗ + PF t)

F t+�t − F ∗∗∗�t

= −I2β(F t+�t + PF t)

. (41)

Note that when P ≡ 0, whichever of E1 and E2 has thenegative root vanishes, and the corresponding step (eitherof the middle two steps) in the scheme (41) becomes anull step, so that either F ∗∗ ≡ F ∗ or F ∗∗∗ ≡ F ∗∗, and thescheme reduces to (15).

As for the linear case, the proposed scheme has thedesirable property that its response function (31) tendsto zero as β�t → ∞, as does the analytic responsefunction (see Equation (20)). Furthermore, this propertyis preserved independently of the choice made for P indetermining E1, E2, I1 and I2 from Equations (38)–(40).This is in contrast to all schemes with constant ε, exceptfor the case ε = 1 + 2P (see Equation (22)), which isdependent on the value of P .

In order to completely specify the scheme, we haveto assume a value for P . For boundary-layer-diffusionapplications, a typical range for P is found empiricallyto be about 1/2 ≤ P ≤ 2. Single-column model results(see Section 4) indicate that a choice of P = 3/2 inEquations (38)–(40) gives reasonable results. Figure 3shows the effect of assuming this value when the actualvalue of P varies between 1/2 and 2, and compares theproposed scheme with the over-weighted implicit schemewith ε = 2, both with and without application of twosubsteps. In all these cases, the diffusion coefficient iscomputed solely in terms of F t . The substepped schemeis accurate for small β�t , but becomes non-monotonicfor increasingly small values of β�t as the actual P



Actual P = 1/2

0 1 2 3 4 5

β∆t

0 1 2 3 4 5

β∆t

0 1 2 3 4 5

β∆t

0 1 2 3 4 5

β∆t

−1.0

−0.5

0.0

0.5

1.0

E

−1.0

−0.5

0.0

0.5

1.0

E

−1.0

−0.5

0.0

0.5

1.0

E

−1.0

−0.5

0.0

0.5

1.0

E

AnalyticImplicit, ε=2Implicit, ε=2, 2 substepsNew scheme with P=3/2




Actual P = 1

Actual P = 3/2 Actual P = 2

(a)

(c) (d)

(b)

Figure 3. Comparison of the analytical solution (solid) with various schemes when the actual value of P is: (a) 1/2, (b) 1, (c) 3/2, and (d) 2.The dash-dotted line represents the new scheme assuming P = 3/2; the dotted line represents the over-weighted implicit scheme with ε = 2,

without substepping; and the dashed line represents the over-weighted implicit scheme with ε = 2, with substepping.

increases, and eventually it becomes unstable. The non-substepped implicit scheme becomes non-monotonic forsome value of β�t when the actual P exceeds 1. Theproposed scheme is well behaved for all values of theactual P , except for P = 2, when it becomes non-monotonic (though even then the resulting oscillatoryresponse is strongly damped). However, its response isnot very accurate for small values of the actual P . Inchoosing the value of P to assume, there is clearly acompromise to be made between loss of accuracy forsmall values of the actual P and good behaviour for largevalues of P . This motivates a choice of P dependenton the boundary-layer stability (see Section 4 for furtherdiscussion).

3.4. Full nonlinear form

The full nonlinear equivalent of the scheme (41), corre-sponding to the proposed discretization of Equation (17),is:

F ∗ − F t

�t= −I1

(K(F t)P F ∗ − S

)F ∗∗ − F ∗

�t= E1

(K(F t)P F ∗ − S

)F ∗∗∗ − F ∗∗

�t= E2

(K(F t)P F ∗∗ − S

)F t+�t − F ∗∗∗

�t= −I2

(K(F t)P F t+�t − S

)

. (42)

Here, the F now denote the full fields. Including thesource term S in this way ensures that the scheme obtainsthe exact steady-state solution (18), and thereby satisfiesthe fourth requirement – maintenance of the steady-statesolution – whose importance was demonstrated by Cayaet al (1998). Eliminating F ∗, F ∗∗ and F ∗∗∗, we obtain:

F t+�t

= {(1 + E1K(F t)P �t

) (1 + E2K(F t)P �t

)F t

+ (1 + (I1I2 − E1E2)K(F t)P �t

)S�t

}× {(

1 + I1K(F t)P �t) (

1 + I2K(F t)P �t)}−1

,(43)

where we have used the fact that

I1 + I2 − E1 − E2 = 1,

(as required for consistency of the discrete equation, andsatisfied by all the schemes considered here). SettingF t+�t = F t = F0 in Equation (43) recovers the steady-state solution (18).

Returning briefly to Equation (35), in going from theresponse function E∗ to the proposed formulation, anypair consisting of one explicit and one implicit equationcan be combined so that the scheme is reduced to two


1566 N. WOOD ET AL.

steps, each resembling a semi-implicit step. In particular,the two-step scheme

F ∗ − F t

�t= −I1β(F ∗ + PF t)

+E1β(F t + PF t)

F t+�t − F ∗�t

= −I2β(F t+�t + PF t)

+E2β(F ∗ + PF t)

, (44)

is an equally viable candidate, having exactly the samelinear response as (41). The nonlinear form of (44) is:

F ∗ − F t

�t= −I1K(F t)P F ∗

+E1K(F t)P F t

+(I1 − E1)S

F t+�t − F ∗�t

= −I2K(F t)P F t+�t

+E2K(F t)P F ∗+(I2 − E2)S

, (45)

where the particular form for the source terms ensuresthat the exact steady-state solution is obtained. It can beverified that this nonlinear form of the two-step schemehas the same response (43) as the four-step nonlinearscheme of (42), so that the two schemes (42) and (45)are equivalent.

4. Application of the new scheme to boundary-layer vertical diffusion

In this section, the scheme (45) is applied to theboundary-layer vertical diffusion problem:

∂X

∂t= ∂

∂z

(K∂X

∂z

)+ S, (46)

where z is the height above the surface of the Earth,t is time, X may be a wind component, temperature,moisture, or any tracer, and K is the exchange coefficient,which (in addition to other variables) usually depends onX, i.e. K ≡ K(X), implying that the partial differentialequation is nonlinear. S is a forcing term from thedynamics and other physical processes, computed priorto the boundary layer. Discretizing Equation (46) using(45) yields:

X∗ − Xt

�t= I1

∂∂z

(K(Xt )∂X∗

∂z

)−E1

∂∂z

(K(Xt )∂Xt

∂z

)+(I1 − E1)S

Xt+�t − X∗�t

= I2∂∂z

(K(Xt )∂Xt+�t

∂z

)−E2

∂∂z

(K(Xt )∂X∗

∂z

)+(I2 − E2)S

, (47)

where I1, I2, E1 and E2 are specified by Equations(38)–(40). Note that the apparent difference in sign of thediffusion terms between the schemes (45) and (47) is dueto the response function (−k2) of the second derivativehaving been absorbed into the K of (45).

In this section we compare the new scheme with thestandard partially-implicit scheme that is used opera-tionally at the Met Office. This latter scheme is definedby:

Xt+�t − Xt

�t= ∂

∂z

(1 + ε

2K(Xt )

∂Xt+�t

∂z

)+

+ ∂

∂z

(1 − ε

2K(Xt )

∂Xt

∂z

)+ S. (48)

Once K(Xt ) has been evaluated, the new scheme doublesthe cost of solving (48), as it requires the solution of twotridiagonal systems. However, in practice the extra costis very small in relation to the total model cost. In global-model forecast tests with the UM on an NEC SX8 vectorsupercomputer, it amounts to less than 1% of the totalCPU time.

4.1. Single-column model experiments

A single-column version of the Met Office’s UM is usedfirst in an assessment of the new scheme. In this version,only physical processes (boundary-layer turbulence, sur-face exchange, radiative heating and cooling, convectionand cloud microphysics) are represented, and dynamicalprocesses are absent. The experiment simulates the diur-nal cycle on 23 September 2003 at a grid point locatedat Cardington, England (52.1 °N, 0.4 °W). That day wasone of clear skies with anticyclonic conditions and amoderate geostrophic wind of order 7 ms−1 (for moredetails see Edwards et al. (2006)). A stretched quadraticgrid (Davies et al., 2005) is used with Charney–Phillipsvertical staggering and 21 vertical levels in the bound-ary layer. The resolution at low levels is close to 20 m,decreasing gradually to 250 m at upper levels.

In Figures 4 and 5, time series of the results using thenew and standard schemes are compared to one anotherand to a control solution. The plotted variable is the u

(zonal) wind component at the lowest model level. Thecontrol solution is obtained using the standard scheme(48) with ε = 1 and a very short time step �t = 9 s.Experiments with this case show that the solution changesvery little when time steps shorter than 60 s are used. Thetime step �t = 9 s for the control solution is chosen to be100 times shorter than the time step �t = 900 s used torun the two schemes, to make the time-truncation errorsnegligible. Note that when the two schemes are run witha short time step of �t = 9 s, they indeed produce almostidentical solutions, so for practical purposes convergenceto a unique solution is achieved.

In Figure 4(a), the result from the standard schemewith ε = 2 is compared to that of the new scheme withP = 3/2. The standard scheme gives a very noisy solu-tion. The noise is essentially a two-time-step oscillation,and typically occurs in regions where the boundary layeris stable, so that the exchange coefficients are stronglydependent on the diffused variable (u here). The observednoise is completely eliminated when the new scheme is


standard (std) scheme vs new scheme

0 5 10 15 20 25 30 35

Local time (hours)

0 5 10 15 20 25 30 35

Local time (hours)

1.0

1.5

2.0

(a) (b)

2.5 z

onal

win

d

1.0

1.5

2.0

2.5



zon

al w

ind

ControlStd scheme: epsilon=2New scheme: P=3/2

new scheme

ControlP=1P=3/2P=2

Figure 4. Time series for the lowest-level wind component u, using: (a) standard scheme with ε = 2 (thin continuous line), and new scheme withP = 3/2 (dashed line); (b) new scheme with P = 1 (thin continuous line), P = 3/2 (dashed line), and P = 2 (dash-dotted line). The control

solutions are plotted with thick continuous lines.

standard (std) vs new scheme

0 5 10 15 20 25 30 35

Local time (hours)

0 5 10 15 20 25 30 35

Local time (hours)

1.0

1.5

2.0

2.5

(a) (b)



zon

al w

ind

1.0

1.5

2.0

2.5



zon

al w

ind

Controlstd scheme: epsilon=3new scheme: P=3/2new scheme: P(unstab)=1/2, P(stab)=3/2

Controlnew scheme: P(unstab)=1/2, P(stab)=2new scheme: P(unstab)=1/2, P(stab)=3/2new scheme: P(unstab)=0, P(stab)=3/2

new scheme comparisons

Figure 5. As Figure 4, but for: (a) standard scheme with ε = 3 (thin continuous line), and new scheme with Punstable = Pstable = 3/2 (dashedline), and with Punstable = 1/2 and Pstable = 3/2 (dash-dotted line); (b) new scheme with Punstable = 1/2 and Pstable = 2 (thin continuous line),

Punstable = 1/2 and Pstable = 3/2 (dashed line), and Punstable = 0 and Pstable = 3/2 (dash-dotted line).

run with P = 3/2. However, between 10 h and 13 h,when the boundary layer has just switched from beingstable to unstable, there is a noticeable difference in accu-racy, the standard scheme being closer to the control.The same is also true between 34 h and 36 h. Further-more, during this period there seems to be a relationshipbetween the value of P and the accuracy of the solu-tion. This is clear in Figure 4(b), where the new schemeis compared to the control for three different values ofP : P = 1, P = 3/2, and P = 2. As P becomes smaller,the accuracy of the solution improves. Unfortunately, forP = 1, which leads to the most accurate solution, thenoise is not completely eliminated. This behaviour is con-sistent with the analysis presented in the last two sectionsabove. For large values of P the new scheme over-damps.

This loss of accuracy, seen in Figure 4, could lead toloss of forecast skill in an operational context. However,

it is possible to overcome this by making the value ofP a function of the boundary-layer type. The followingmodification is therefore proposed:

P ={

Punstable for an unstable boundary layerPstable for a stable boundary layer

,

where Pstable > Punstable ≥ 0. Suitable values for Pstable

and Punstable can be determined by numerical experimen-tation. A sufficiently large value for Pstable ensures that asmoothly-evolving solution is always obtained in stable-boundary-layer regions, while a small value for Punstable

ensures an accurate solution in unstable-boundary-layerregions. However, care must be taken with just how smallPunstable is set, since a very small value may not guaranteeunconditional stability. To investigate this, in Figure 5(a)the standard scheme with ε = 3 is compared to the newscheme with two settings: Pstable = Punstable = 3/2, and

1568 N. WOOD ET AL.

Punstable = 1/2 and Pstable = 3/2. The standard schemewith ε = 2 is not included here, since its noisy responsewould obscure the details of the other schemes. It is clearthat:

• the standard scheme, even with ε = 3, is still noisy;• the new scheme with Pstable = Punstable = 3/2 addresses

the noise problem of the standard scheme; and• the new scheme with Punstable = 1/2 and Pstable = 3/2

also does so, but is more accurate.

Figure 5(b) shows that reducing the value of Punstable from1/2 to 0, while holding Pstable fixed at 3/2, results ina further improvement in accuracy. It also shows that

changing the value of Pstable from 3/2 to 2 while holdingPunstable = 1/2 has little impact on solution accuracy.This suggests that it may be better to err on the sideof choosing a larger value of Pstable than risk problemswith a smaller value.

4.2. Numerical weather prediction model experiments

We now give an example of the performance of the newscheme using the Met Office’s UM, configured to have:

• approximately 40 km horizontal resolution at mid-latitudes;

0

30S

(a)

(b)

60S

90S

0

30S

60S

90S

45W90W 0

45W90W 0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Atmos u wind incr: bdy layer at 130.0 metresat 0000 20/06/03 from 1200 19/06/03


Figure 6. Increment (ms−1) of the meridional wind component at the 4th-lowest vertical level for the standard scheme with ε = 2: (a) at0000 UTC on 20 June 2003; and (b) at the following time step.


• 70 vertical levels, with the bottom 21 of these beingidentical to those used in the single-column modelexperiments presented above; and

• a time step �t = 15 min.

A numerical consequence of high vertical resolutionin the boundary layer is that the vertical diffusionproblem becomes more acute, and a numerical schemethat is conditionally stable may then produce very noisysolutions. An example of this can be seen in Figure 6,where increments, i.e. time tendencies multiplied by thetime step, from the model’s boundary-layer scheme aredisplayed at two consecutive time steps for the meridionalwind component u at the 4th-lowest vertical level. Theseplots are obtained from an integration of the UM usingthe standard scheme (48) with ε = 2. For large areas overthe ocean (off both coasts of southern South America

and to the south of Africa), the sign of the incrementreverses (warm colours change to cold ones and viceversa), indicating a time-decoupled behaviour. Such anabrupt change of sign over large areas at two consecutivetime steps is physically unrealistic. The boundary layeris stable in the area where this spurious numericalphenomenon occurs. We now discuss further this time-decoupled behaviour at specific grid points.

In Figures 7 and 8, time series over 24 h of incre-ments from the model’s boundary-layer scheme for themeridional wind component u and the temperature T areplotted for a point at (49.9 °S, 14.1 °E), which is in theSouth Atlantic, and the boundary layer remains of stabletype throughout the forecast period. There are two lineson each plot: the time series at the lowest level are plot-ted with a continuous line, and those at the adjacent levelwith a dashed line. Figure 7(a) displays the time series

12 18 0 6 12Universal time (hours)




−2

1

0

1

2

3(a)

(b)



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−2

1

0

1

2

3



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−0.50

−0.25

0.00

0.25

0.50<

T>

tem

pera

ture

BL

tend

ency

*dt (

K)

−0.50

−0.25

0.00

0.25

0.50

<T

> te

mpe

ratu

re B

L te

nden

cy*d

t (K

)

Standard scheme: physics timestep=15 min, epsilon=2


Figure 7. Time series for the increments of wind component u (left panels) and temperature T (right panels), at the lowest model level (solidline) and the next-lowest model level (dashed line), for the point at (49.9 °S, 14.1 °E), using the standard scheme with ε = 2 and �t = 15 min:

(a) without substepping, and (b) with three substeps of the fast physics.

1570 N. WOOD ET AL.





−2

−1

0

1

2

3(a)

(b)



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−2

−1

0

1

2

3



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−0.50

−0.25

0.00

0.25

0.50

<T

> te

mpe

ratu

re B

L te

nden

cy*d

t (K

)

−0.50

−0.25

0.00

0.25

0.50<

T>

tem

pera

ture

BL

tend

ency

*dt (

K)

New scheme: physics timestep=15 min, P=3/2

New scheme: physics timestep=15 min, P=2

Figure 8. As Figure 7, but for the new scheme, with: (a) P = 3/2, and (b) P = 2.

obtained when the standard scheme is used with ε = 2.The solution is very noisy, and very large oscillations areobserved, both in the meridional wind and in the temper-ature. The oscillations persist even when the fast physicsprocesses are fully substepped three times – i.e. the dif-fusion coefficients, the boundary-layer tendencies and theconvection-scheme tendencies are successively computedthree times with a time step �t = 5 min, which is one-third of the model time step �t = 15 min. This behaviouris seen in Figure 7(b): although there is a reduction in theamplitude, the problem is still very noticeable. The reduc-tion is much larger when the new scheme is used withP = 3/2. For this value, however, the time series are stillsomewhat noisy, as shown in Figure 8(a). This temporaldecoupling is virtually eliminated when P is increased to2 (Figure 8(b)).

The above example demonstrates the effectivenessof the new scheme in eliminating oscillations in an

operational forecast model. In this example the newscheme has been used with P = Pstable = Punstable. Usinginstead Pstable = 2 and Punstable = 1/4 has the same effect(not shown) for the plotted grid point. This is to beexpected, given that the value of P at this grid pointremains constant and equal to 2 during the integration,since the boundary layer remains of stable type.

In Figure 9, time series from another grid point areplotted. This point has coordinates (55.5 °S, 90.6 °E), andis located in the Southern Pacific to the southeast of theSouth American continent. For this point, the boundarylayer switches from stable to unstable during the 24 hforecast period: it is stable during the first 6 h andthen unstable. The standard scheme is very noisy duringthe first 6 h (Figure 9(a)). In contrast, there is virtuallyno noise in the solution computed by the new schemewith Pstable = 2 and Punstable = 1/4 (Figure 9(b)). Beyondthe first 6 h, the two schemes yield similar solutions,

−1.0

−0.5

0.0

0.5

1.0(a)

(b)



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−1.0

−0.5

0.0

0.5

1.0



zon

al w

ind

BL

tend

ency

*dt (

m/s

)

−0.25

0.00

0.25

<T

> te

mpe

ratu

re B

L te

nden

cy*d

t (K

)

−0.25

0.00

0.25<

T>

tem

pera

ture

BL

tend

ency

*dt (

K)


New scheme: physics timestep=15 min, P(unstab)=1/4, P(stab)=2

Figure 9. As Figure 7, but at the point (55.5 °S, 90.6 °E): (a) standard scheme with ε = 2; (b) new scheme with Punstable = 1/4 and Pstable = 2.

although the new scheme seems to give less pronouncedextrema.

Figure 10 displays increments of the meridional windcomponent at the 4th-lowest level obtained from the newscheme with Pstable = 2 and Punstable = 1/4. Comparisonwith the corresponding plots of Figure 6 shows that:

• the change in sign, from one time step to the next,seen in Figure 6 for the standard scheme, is absentfrom Figure 10 for the new scheme (the colours inthe regions off both coasts of southern South America,and to the south of Africa, now remain coherent atconsecutive time steps);

• the increments for the new scheme at the two consec-utive time steps are very similar not only in sign butalso in magnitude, as expected; and

• there is much less horizontal noise seen in the newscheme than in the standard one.

Finally, note that results (not shown) for verificationforecast tests with the UM using this new scheme showa neutral impact on accuracy; so addressing the time-decoupling problem with the new scheme is achievedwithout loss of forecast accuracy.

5. Conclusions

We have developed a new two-time-level discretizationof the diffusion equation that, for constant diffusioncoefficients (the linear case), satisfies the requirementsof:

1. unconditional stability;2. second-order accuracy;3. monotonic damping; and4. maintenance of steady-state solutions.

1572 N. WOOD ET AL.

0

30S

(a)

(b)

60S

90S

0

30S

60S

90S45W90W 0

45W90W 0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2



Figure 10. As Figure 6, but for the new scheme with Punstable = 1/4 and Pstable = 2.

The result is a multi-step implicit–explicit–implicitscheme. It works by dynamically adapting the off-centring according to the value of the diffusion coefficientand the time step. Thus for small coefficients and timesteps it is close to the Crank–Nicolson scheme, but forlarger coefficients and time steps it tends to the fully-off-centred scheme.

We have discussed the extension of the scheme tothe nonlinear case, for which the diffusion coefficientvaries with the field being diffused. The resulting schemeconsists of two semi-implicit steps, with the diffusioncoefficient ‘frozen’ during the time step.

The nonlinear scheme requires an estimate of the non-linearity of the diffusion coefficient, i.e. of the parameterP . If the estimate is too small, the scheme risks being

non-monotonic and exhibiting oscillatory behaviour. Onthe other hand, if the estimate is too large, accuracy willbe lost. For applications in numerical weather prediction,boundary-layer-diffusion values of either P = 3/2 orP = 2 have been found to work well. However, accuracyis improved if P is taken to depend on the stability of theboundary layer. The choice Punstable = 1/2 and Pstable = 2gives good results, as confirmed by both single-columnand global-forecast experiments using the Met Office’sUM. This approach, which requires only the zeroth-orderphysical characterization of whether the boundary layer isstable or unstable, means that the scheme can be applieddirectly with any formulation of the diffusion coefficientK. This is in contrast to the approach of Girard and



Delage (1990), which requires knowledge of the particu-lar functional form of the diffusion coefficient.

An additional advantage of the proposed scheme isthat its response function (which is inversely propor-tional to the damping rate of the scheme) tends to zeroas the dimensionless diffusion coefficient increases, inde-pendently of the assumed value of P . This is in contrast tothe standard off-centred schemes with constant ε, exceptfor the particular choice ε = 1 + 2P .

Acknowledgements

It is a pleasure to acknowledge useful discussions withDr Adrian Lock, and also his assistance in setting up thesingle-column model experiments.

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a monotonically-damping second-order-accurate unconditionally-stable numerical scheme for diffusion

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