an economical second-order advection scheme for numerical weather prediction

Q. J. R. Meteorol. SOC. (1999), 125, pp. 2291-2303

An economical second-order advection scheme for numerical weather prediction

By P. MALGUZZI’* and N. TARTAGLIONE* ’ CNR-FISBAT Italy

2 ~ ~ ~ , Italy

(Received 1 April 1997; revised 15 December 1998)

SUMMARY A very simple second-order Eulerian scheme for the advection equation, based on a forward (backward)

time integration on even (odd) grid points, is studied. The proposed scheme is similar, but not equivalent, to the so-called ‘hopscotch method’, developed in the 1960s and early 1970s for the advection4iffision equation, and is stable up to Courant number 2.0. It is shown that, in the case of the advection equation, the proposed scheme has the same advantages yielded by the forward-backward scheme in the case of the shallow-water equations; in particular, it is equivalent to the application of centred time and space differencing on the Eliassen grid. The new scheme, unlike the classical leapfrog scheme, can be coupled to the forward4ackward integration of the gravity-wave problem in primitive-equation models.

With the aid of the proposed scheme, an explicit version of the atmospheric, Bologna Limited-Area Model (developed in recent years at the FISBAT Institute of the National Council of Research of Italy) is devised, and a comparison with the semi-implicit version of the same model is performed. The explicit version runs with a double time step, achieving the same accuracy with significantly less computer time and storage. It is suggested that the new time scheme is particularly suitable for numerical weather prediction on massively parallel computing machines based on SIMD (Single Instruction Multiple Data) and distributed memory architecture, which strongly penalize non-local algorithms.

KEYWORDS: Numerical methods Stability

1. INTRODUCTION

Over recent years, limited-area numerical weather prediction models (LAMs) have reached the limit of the hydrostatic assumption thanks to the availability of more powerful computing machines at lower costs. In particular, massively parallel computers have recently become available leading to widespread efforts to implement state of the art LAMs on such machines (see, for instance, Engelen and Wolters 1995; Marrocu et al. 1998).

This work originates from the need to develop a LAM suitable to be efficiently parallelized on a QUADRICS massively parallel supercomputer. QUADRICS is a Single Instruction Multiple Data (SIMD) massively parallel supercomputer, that is, it runs a unique flow of instructions executed simultaneously by all the different processors (as many as thousands in some cases). QUADRICS has a three-dimensional (3-D) architecture based on a cubic lattice of nodes, where each node is connected to its six nearest neighbours. In order to obtain the best performance in distributed memory SIMD machines like QUADRICS, the numerical algorithm used to integrate the primitive equations in time must be ‘local’, i.e. the computations performed at a particular physical point must involve variables at the nearest neighbours of that point. Given the above characteristics, LAMs based on semi-implicit and semi-Lagrangian schemes, although performing very efficiently on parallel computers with different architectures, may not end up with the best performance.

Explicit grid-point models can be parallelized easily and efficiently. A competitive alternative to semi-implicit schemes is obtained by splitting the time integration of the gravity-wave terms (the so-called ‘adjustment step’) from the (slower) advective, diffu- sive, and physical contributions in the primitive equations (Gadd 1978). While extra computations are required to solve the Helmholtz problem in semi-implicit models, * Corresponding author: CNR-HSBAT, via Gobetti 101,40129 Bologna, Italy.

229 1

2292 P. MALGUZZI and N . TARTAGLIONE

several adjustment steps for each advection step have to be performed in split models. To minimize the number of adjustment steps, the propagation of gravity waves can be integrated with the forward-backward scheme (Mesinger and Arakawa 1976, hereinafter referred to as MA; Mesinger 1977), which allows us to double the time step with respect to, for instance, the leapfrog scheme (an alternative method for integrating shallow-water equations with twice the leapfrog time step is presented in Brown and Campana (1978)).

The disadvantage of the forward-backward scheme is that it turns out to be unstable when coupled with the leapfrog integration of the advection terms (see MA and Haltiner and Williams 1980). For this reason, split explicit models often use the first-order Matsuno scheme (like the University of BelgradeNational Meteorological Centre model (Lazic and Telenta 1990)) for advection which, being an iterative procedure, requires the computation of advection twice. The two-step, second-order Lax-Wendroff scheme (Lax and Wendroff 1960) has been used in split models (Gadd 1978). However, the implementation of the Lax-Wendroff scheme implies even higher computational costs.

The present paper proposes a very simple second-order advection scheme which does not suffer from the above hindrances. Section 2 introduces the new scheme and addresses the problem of numerical stability and accuracy. In section 3 the scheme is compared with the classic leapfrog scheme, showing that the same precision is achieved by halving the number of operations. In section 4, a numerical example, consisting of the advection of a passive tracer by a two-dimensional deformation wind, is presented. Section 5 is devoted to a more meteorological application, performed on the semi- implicit Bologna Limited-Area Model (BOLAM) (see, for instance, Buzzi et al. 1994), developed at the FISBAT Institute of the National Council of Research of Italy. It is shown that the fully explicit, time-split version of BOLAM based on the new advection scheme runs with a significantly larger time step. Finally, in section 6, the conclusions are drawn.

2. THE NEW ADVECTION SCHEME

Let us consider the one-dimensional advection problem

a,u + Ca,u = o (1)

where c is a constant phase speed. The independent variables x and t are discretized on a regular grid X i = i Ax, tn = n At , and the dependent variable u is defined on the discrete points xi, i = 1, 2, . . . , N according to:

~1 = u ( x i , t n ) . (2)

Let us then consider the following numerical scheme for Eq. (l), forward in time and centred in space:

LL cAt p, = - Ax u;+1 = u; - 2(u;+l - u;-l), (3)

where p is the Courant number. We now consider the following modification of Eq. (3): the right-hand side (r.h.s.) of Eq. (3) is computed first on the spatial grid points defined by odd i indices, and the updated values of u are subsequently used to compute the new values on even i indices. This particular procedure obviously implies a forward (backward) time stepping for the odd (even) grid points. In the present work the scheme will be referred to as FBAS (Forward-Backward Advection Scheme). The scheme

AN ECONOMICAL SECOND-ORDER ADVECTION SCHEME 2293

is similar to the so called 'hopscotch method', introduced in the late 1960s for the advection-diffusion equation (Gourlay 1970; see also Roache 1976), the difference being that in the hopscotch method the solution is evaluated first at i + n odd (i.e. the role played by odd and even grid points is exchanged at consecutive time steps).

The stability analysis of the FBAS can be derived by the standard application of the Von Neumann method. A possible way to tackle the problem is to name the solution differently over odd and even grid points as follows:

N 2 - Vj = ~ 2 j - 1 , j = 1, 2, . . . ,

(4) Tj=u2j , j = l , 2 , . . . -.

Thus, the scheme defined above is equivalent to the forward-backward scheme applied to the shallow-water equation:

N 2

By applying the Von Neumann method, the numerical solution of Eq. ( 5 ) can be written as:

v? = vo eik(2j-l)An

T,? = hnTo e J

ik2 j Ax

where V,, To, and h are complex constants. Substituting Eq. (6) into Eq. (5 ) gives

(A - l)Vo + ipTo sin kAx = 0 (A - l)To + iLLhVo sin kAx = 0. (7)

Solving for h yields a second-order algebraic equation, whose solutions can be written as:

*

(8) h = X k i d m , X=1- - s in LLL 2 kAx. 2

It can be seen that Ihl = 1 whenever 1x1 < 1, so that the scheme is conditionally neutral. In turn, 1x1 < 1 implies p < 2, as can be readily demonstrated from Eq. (8) for kAx = n/2 (wavelength equal to twice the shortest resolvable wavelength). Thus, the present scheme is stable as long as the Courant number is smaller than 2, allowing a double time step with respect to other common schemes* used for Eq. (1). Another advantage compared with more classic schemes is that only one time level of the solution needs to be stored.

The existence of two roots in the stability analysis indicates that a computational solution is present. The computational solution given by the plus sign in Eq. (8) corresponds to an upstream propagation with phase speed -c in the limit of At going to zero. This stems from the formal equivalence between Eq. (3) and the shallow-water problem Eq. (5 ) , which is known to correspond (see MA) to two advection equations with phase speeds of fc. * The hopscotch scheme is also unstable when the Courant number is greater than 1.0, while it is unconditionally stable for the diffusion equation.

2294 P. MALGUZZI and N. TARTAGLIONE

The presence of a computational mode in the FBAS deserves further examination. Let us consider, after Eq. (7), the ratio of the amplitudes of even to odd points:

To i(h - 1) V, p sin kAx' -- -

For the physical mode:

(9)

. p sin kAx 2 (zr = i ~ - l - i d - = J p sin kAx

1 - - p2 4 sin2 kAx - 1 = e-ie (10)

where

p sin kAx e = arcsin ( ) . Similarly, for the computational mode:

In the limit of small time steps and/or long wavelengths the above expression tends to - 1, indicating that the computational mode changes sign going from odd to even points. The removal of this mode can easily be accomplished by any kind of scale-selective dissipation.

The existence of coherent solutions at odd and even grid points is linked to the proximity to 1 of the ratio To/ V,. Equation (10) indicates that the physical mode presents a shift between the solutions at odd and even points. This is particularly evident in the limiting case of p = 2 where, for any wavelength, the ratio To/Vo tends to e(-ikAx); by substituting this ratio in Eq. (6) it turns out that VY = TY, implying a 'two by two staircase' shape of the solution. This should not be interpreted as loss of effective spatial resolution; rather, as will be shown in the following section, it is exactly the shift given by Eq. (1 1) that makes the accuracy of the FBAS of second order in time.

3. THE FBAS AND THE LEAPFROG SCHEME

We will show that one FBAS step is equivalent to two steps of the leapfrog scheme

(13)

where p~ is the Courant number based on the leapfrog time step A k . By applying the Von Neumann method to Eq. (13), the well known result follows (see MA):

with the Courant number halved. The leapfrog scheme applied to Eq. (1) reads:

m + l - m-1 ui - ui - PL(UY+n+l - Uy-1)

(14)

which implies Ih~1 = 1 whenever p~ < 1. Let us consider the stable case, where the physical modes of FBAS and leapfrog can be written, respectively, as follows:

(15) - - e-iO hL = e-i@L


Figure 1. (a) Space-time grid used by the leapfrog scheme (see text). (b) Time-staggered grid for the leapfrog scheme equivalent to the scheme in Eq. (5). (See text.)

where, after Eqs. (8) and (14),

p2 sin2 kAx O(p) = arccos

O L ( ~ L ) = arcsin(pL sin kAx). (16)

2@L(P/2) = @(PI. (17)

(18) which is clearly satisfied by Eq. (16). Thus, applying the leapfrog scheme twice with half the time step leads to the same dispersion relation as applying the FBAS once. Hence, although each of the two parts of Eq. ( 5 ) are only first order in time, FJ3AS is of second-order accuracy.

The above result should not come as a surprise. It is well known that the economy achieved by the forward-backward integration of the shallow-water equations is equivalent to that obtained with centred time differencing by the half density grid known

It is now easy to show that h ~ ( p / 2 ) ~ = h ( p ) or, equivalently,

In fact, by taking the cosine of both sides of Eq. (17) it follows that:

1 - 2 sin2 0L(p/2) = cos ~ ( p )


as the Eliassen grid (see MA). The same remarks apply to the advection equation. In fact, if we consider the space-time grid sketched in Fig. l(a), the solution given by the leapfrog scheme in Eq. (13) on points marked by crosses is uncoupled from the solution computed on points marked by dots. Therefore, the original space-time grid can be considered as the superposition of two more elementary subgrids staggered in time, and the scheme in Eq. (13) simultaneously computes the solution on both grids. Let us now take the leapfrog scheme applied to only one subgrid, where the solution and the space and time indices are renamed as in Fig. l(b); it is clear that two steps on such a grid are equivalent to one step of the FBAS in Eq. (5) with p = 2cAqJAx = 2 p ~ .

In view of the formal equivalence between the FBAS and the leapfrog scheme on the Eliassen grid, it is also clear that the shift between the FBAS solution on even and odd grid points (as shown at the end of the previous section) can be interpreted as a temporal shift of half a time step. In fact, the ratio between the amplitude of the solution at even and odd points, given by Eqs. (10) and (1 l), is formally identical to the amplification factor of the leapfrog scheme with a halved time step (Eqs. (15) and (16)).

4. A TWO-DIMENSIONAL ADVECTION EXAMPLE

As a simple test of the performance of the FBAS, we consider the two-dimensional example depicted in Fig. 2, where a scalar quantity, say q ( x , y, t), is advected by the deformation wind sketched in Fig. 2(a) and defined by:

u = x(2y - l), v = y(l - y), 0 < x, y < 1. (19)

As initial conditions, we assume that q at time t = 0 is ‘cone shaped’ with a maximum amplitude of 100.0 units and radius 0.1 at the base. Spatial resolution is set to Ax = Ay = 0.01 (units are dimensionless). The analytic solution of the following problem:

atq + a,q + ayq = o (20) can be easily found, and it is depicted for reference in Figs. 2(b), (c) and (d). Figures 3 and 4 show the leapfrog and FBAS approximation of Eq. (20) at the same verification times. The leapfrog scheme uses a time step of 0.01, so that the Courant number attains the maximum value of 1 .O at the upper and lower right comers of the integration domain. The FBAS has been implemented with a double time step, At = 0 . 0 2 ; in both schemes a further increase of 10% in the time step results in numerical instability. A careful inspection of the results indicates that the accuracies of the two schemes are identical, the FBAS requiring half the computer time and the storage of only one time level of the dependent variable.

A weak fourth-order diffusion term has been added to the r.h.s. of Eq. (20), with a coefficient v = 1.OE-8 in order to reduce the occurrence of spurious negative values in both schemes. Its contribution has been computed with forward time stepping, split from the advection computation.

It is interesting to look at the performance of the leapfrog scheme with half spatial resolution. Figure 5 represents such a solution, obtained with At = 0 . 0 2 and the same value of the diffusion coefficient. The accuracy is clearly greatly reduced, and the solution appears much more dispersed with a strong decay of the maximum value. This example definitely shows that the gain in computer time by the FBAS is not obtained at the price of reducing the effective horizontal resolution. In fact, as explained in the previous section, the FBAS computes the solution only on one of the two independent Eliassen grids simultaneously integrated by the classical leapfrog scheme, cutting the unnecessary computation of the physically identical solution described by the other grid.


I . c c c CCCL

Figure 2. (a) Initial conditions and velocity defined by Eq. (19) (see text). Contour interval is 20.0 units. Contour plots of the analytical solutions of Eq. (20) (see text) at t = 1.0, t = 2.0, and t = 3.0 units are shown in (b). (c),

and (d), respectively.

5 . THE FBAS IMPLEMENTED ON THE BOLAM

The BOLAM used in the following simulations, is a semi-implicit, primitive- equation, hydrostatic, three-dimensional, grid-point model in sigma coordinates. It integrates the equations of conservation of momentum, energy and mass, describing the evolution in time of meteorological fields, with variable spacing in the vertical to give higher resolution near the surface. The main prognostic variables are the zonal and meridional wind components, potential temperature, specific humidity and surface pressure. Horizontal discretization is performed on the Arakawa C-grid, with rotated latitude and longitude as independent variables. Vertical discretization is of the Lorenz type; vertical velocity is defined at intermediate levels between prognostic variables.

The time integration scheme is leapfrog with a semi-implicit treatment of the terms that describe gravity-wave propagation and an Asselin filter (Asselin 1972) to


Figure 3. Same as Figs. 2(b), (c) and (d), but for the numerical solution obtained with the leapfrog scheme. (See text.)

suppress time splitting. The 3-D Helmholtz problem is solved with a direct solver which uses very efficient fast Fourier transforms in longitude based on the CRAY scientific library routine MCFFT and tridiagonal matrix inversion in the latitudinal direction. A fourth-order horizontal diffusion is applied on sigma surfaces to all prognostic variables except surface pressure. A diffusion operator is also applied to the divergence of horizontal velocity to reduce gravity-wave noise. Both horizontal diffusion and divergence damping are computed with a split Euler step.

Lateral boundary conditions are based on the relaxation scheme of Davies (1976), modified by Lehmann (1993), applied to all prognostic variables. In the experiment shown here, initial (and lateral boundary) 3-D variables are obtained from interpolation of ECMWF (European Centre for Medium-Range Weather Forecasts) analyses, available at 0.75" resolution every 6 hours. Boundary values at all time steps are obtained by linear interpolation in time. The topography and landsea mask are obtained by interpo- lating the US Navy 1 / 6 degree topography on the model grid.

2299 AN ECONOMICAL SECOND-ORDER ADVECTION SCHEME

Figure 4. Same as Fig. 3 , but for the Forward-Backward Advection Scheme (see text) with doubled time step.

Starting from the semi-implicit version, a new version of BOLAM has been con- structed. A forward-backward scheme has been implemented for those terms describing fast-moving gravity modes. This ‘adjustment’ process implies the following steps: (i) the integral of horizontal divergence is computed, and the continuity equation is

integrated forward in time, obtaining the new value of the surface pressure; (ii) a provisional value of virtual temperature (due to the omega-alpha term in the

thermodynamic equation) is computed; (iii) geopotential at the new time level is diagnosed from the virtual temperature and

the hydrostatic relationship; (iv) the pressure-gradient force is computed, and the equations of momentum are

integrated in time. Steps (i) to (iv) are repeated several times for each advection time step.

The leapfrog advection of BOLAM has been replaced by FBAS, both for the horizontal and vertical advection of momentum, virtual temperature, and specific humidity.


E E(b)

1 1 1 1 1 1 1 1 1 1

t A

t

E

E

F I I I I I I I I I I I I

Figure. 5. Same as Fig. 3, but with halved spatial resolution.

At any particular temporal step, only one time level is used to store prognostic variables. The fourth-order horizontal diffusion of BOLAM, as well as the (second-order) divergence damping routines, are unchanged in the new version.

In order to test the new adiabatic formulation, a 24 hour hindcast was performed without calling the physical package but with vertical diffusion included. In the following experiment, the BOLAM was run on a CRAY 90 computer in a configuration with 24 vertical levels and 96 by 96 horizontal grid points, with a resolution of 0.27" x 0.27" corresponding to a 30 km by 30 km grid (in rotated latitude-longitude coordinates). The domain of integration (see Fig. 6) is centred at 7"E, 42'N and contains regions characterized by high orography and complex coastlines. The initial condition is 12 UTC on 4 November 1994. This is a well-documented case (e.g. Buzzi and Tartaglione 1995) where the passage of a slow-moving cold front over the western Mediterranean caused intense precipitation and damaging floods over Piedmont, a region in northern Italy.

Horizontal diffusion and divergence damping are also included in the test. The coefficients of horizontal diffusion and divergence damping have been fixed at the same


Figure 6. (a) 24-hour forecast of the mean sea-level pressure obtained from the semi-implicit version of the Bologna Limited-Area Model (BOLAM) (see text) valid at 12 UTC on 5 November 1994. Contour interval is

2.0 hPa. (b) Same as (a), but for the Forward-Backward Advection Scheme version of BOLAM.


values, respectively, in the old and new versions. Water-loading effects on the dynamics have been excluded from both versions.

Figure 6(a) shows the mean sea-level pressure obtained from the semi-implicit and leapfrog version with a time step of 120 s, while Fig. 6(b) shows the same field obtained from the FBAS version and a time step of 300 s (adjustment time step of 60 s). It can be readily seen that the two fields are almost identical, the root mean square difference being 0.25 hPa after one day of integration. The same consideration holds for the vertical velocity field (not shown), which is very sensitive to the details of the horizontal wind. The computer time needed by the FBAS version is reduced by a factor of 4, and memory requirements are reduced by 40%. The substantial computer time reduction is only partly accounted for by the time step increase. The absence of the Asselin filter and the reduced number of memory-to-CPU (central processing unit) passages may account for the rest. In view of the completely different nature of the time schemes involved, the above comparison gives a clear indication that numerical errors associated with time truncation are negligible in short-term weather predictions with high-resolution models, at least for time steps of the order of a few minutes.

The above results are better than expected, since the time step was more than doubled. However, the maximum time step for the semi-implicit version is a little smaller than other models running at similar resolution, which are in the range of 2.5 to 3 minutes. The reason is not completely clear, but it has to do with the existence of an unstable, two-step computational mode which requires a very strong Asselin filter (Asselin coefficient equal to 0.35) to be kept under control.

6. CONCLUSIONS

In this work, an explicit, Eulerian advection scheme based on the forward (backward) time integration on even (odd) grid points was studied. The scheme (referred to as FBAS) requires the storage of just one time level of the dependent variable, and is stable up to Courant number 2.0. It was shown that the proposed scheme offers, in the case of the advection equation, the same advantages provided by the forward-backward scheme in the case of the shallow-water equations; in particular, it is equivalent to the application of centred time and space differencing on the Eliassen grid.

The FBAS, unlike the classic leapfrog scheme, can be coupled to the forward- backward integration of the gravity-wave problem in primitive-equation models. With the aid of this new advection scheme, an explicit version of the weather prediction model BOLAM was devised. The new version uses a time step significantly longer than the old, semi-implicit version of the same model, and requires less computer storage.

Although the scheme proposed in this work is second order in time, the fractional step coupling with other schemes (even if second order) leads to a first-order truncation error. This is not a serious hindrance, since the error associated with time discretization turns out to be negligible for time steps of the order of a few minutes.

ACKNOWLEDGEMENTS

Financial support was provided under the Commission of the European Commu- nities ENVIRONMENT project EV5VCT940442 (‘ANOMALIA’) and by the Ente Nazionale per le Nuove Technologie 1’Energia e 1’Ambiente (ENEA). The authors wish to thank Dr Andrew Staniforth and the anonymous reviewer for suggestions that con- tributed to the improvement of the manuscript.


Asselin, R.

Brown Jr., J. A. and

Buzzi, A., Fantini, M., Malguzzi, P.

Buzzi, A. and Tartaglione, N.

Davies, H. C.

Engelen, R. and Wolters, L.

Campana, K. A.

and Nerozzi, F.

Gadd, A. J.

Gourlay, A. R.

Haltiner, G. J. and Williams, R. T.

Lax, P. D. and Wendroff, B.

Lazic, L. and Telenta, B.

Lehmann. R.

Mmocu, M., Scardovelli, R. and

Mesinger, F. and Arakawa, A.

Mesinger, F.

Roache, P. J.

Malguzzi, P.

1972

1978

1994

1995

1976

1995

1978

1970

1980

1960

1990

1993

1998

1976

1977

1976

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an economical second-order advection scheme for numerical weather prediction

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