The vertical stratification is modeled by 4 layers in which we work with averaged concentrations in vertical direction:

Transcription

1 Numerical methods in smog prediction M. van Loon Center for Mathematics and Computer Science, dep. AW, P.O. Boz ^072 J(%%? GB Ams<erJam, The Netherlands ABSTRACT A smog prediction model has been developed at the Dutch National Institute of Public Health and Environmental protection (RIVM). To improve its performance, at CWI numerical techniques and algorithms are developed for the several sub-processes of the model, in particular for advection and chemistry. In order to obtain more accuracy in areas of interest a grid refinement technique is used. The aim of this paper is to give a short introduction to the model, and then to focus on numerical methods for advection and chemistry. Finally, attention is paid to grid refinement and its implementation in the model. INTRODUCTION TO THE MODEL The purpose of this section is to give an overview of the model. A full model description can be found in [4]. The geographical domain is an area of 28.6 x in shifted pole coordinates representing, roughly speaking, Europe. The coordinate transformation consists of a shift of the equator to 60 Northern latitude, resulting in a physically almost uniform mesh if the computational grid is uniform. In general, this will lead to smaller Courant numbers and a less severe restriction on the time step. The vertical stratification is modeled by 4 layers in which we work with averaged concentrations in vertical direction: surface layer: the lowest 50m of the atmosphere in which emissions by traffic and space heating take place, but also removal of pollutants by dry deposition. mixing layer: the layer between the top of the surface layer and the varying mixing height. Vertical diffusion is modeled as an exchange process between this layer and the surface layer. Exchange with the layer(s) above takes only place by changes in the mixing height. This

2 564 Computer Simulation process is called fumigation. Emissions take partly place in this layer, depending on the actual depth. reservoir layer: The top of this layer is determined by the maximum effective source height. If the mixing height exceeds the top of the reservoir layer, this layer vanishes. * upper layer: The top of this layer is equal to the maximum possible mixing height, taken equal to 3000m. The model includes: advection, horizontal and vertical diffusion, wet and dry deposition, fumigation and chemical reactions. For details, see [4]. To solve the corresponding differential equation, the method of fractional steps is applied. This means that each process is integrated separately. To obtain second order in time, this isfirstdone in the order listed above and then in reversed order. The model takes fixed time steps of 30 minutes and updates the meteorological parameters after each time step. As wind fields the 6- or 12-hourly 1000 mbar and 850 mbar wind fields from the European Centre for Medium range Weather Forecasting (ECWMF) are available. The actual wind fields used in the model are obtained by interpolating the ECWMF fields in time and space and making the resulting wind fields divergence free. The model updates its wind fields every hour. As a result of applying fractional steps, each process is treated separately which allows us to consider each process on its own and to select the best possible candidate. NUMERICAL ADVECTION Aspects In [10] some desirable properties for advection schemes are listed. Apart from accuracy we consider most important from this list for application in atmospheric models Positivity: negative solution values may lead to instabilities when dealing with chemical equations. Therefore we require the scheme to be positive. It will turn out that the positivity property of the selected scheme is closely related to the monotonicity property. Conservativity: especially for long range transport it is important that the scheme does not loose or gain mass. Therefore we will focus on these requirements when selecting a numerical advection scheme. An additional requirement is that the numerical scheme can easily be applied in the context of grid refinement (see below). As this will cause no problems, no further attention is paid to this aspect. The advection equation The advection equation on the sphere in conservation form for concentrations c is given by (see e.g. [10]) dc 1 \d(uc] d(vccos9) dt rcoso I d<j> do (1)

3 Computer Simulation 565 where r denotes the radius of the earth in meters, </> and 0 the longitude and latitude coordinates in radians, and u and v the wind velocities in 0 and 0 direction (in m/s). Throughout the paper, it is assumed that the (0,0) domain fi is numerically represented by a uniform N x M grid with mesh widths A</> and A0. The numerical solution in the grid points (or grid cells) is denoted by c^. The method-of-lines approach In [1] we investigated numerical advection schemes based on the /(-schemes by Van Leer [3]. Following the method-of-lines we first approximate the space derivatives of equation (1) while remaining continuous in time. This results into dt r cos 0j A0 (2) where F-^i^ denotes an approximation for the flux uc between cell i,j and 2+lj. Similarly F-^^i denotes an approximation for the flux re cos 0 between cells i,j and z, j-f 1. In spherical coordinates conservation of mass implies c(0,0) cos0dcf)d0 = constant. (3) n If we define the numerical equivalent of expression (3) as N M $^ 5^ Cjj cos 0j = constant, (4) then it can easily be seen that semi-discretization (2) guarantees conservation of mass in the sense of (4). In order to obtain a positive semidiscretization, the fluxes in (2) should be defined such that d ^>0ifc^0. (5) To do this, the computed fluxes are limited such that relation (5) holds. In fact, the limiter can be seen as a nonlinear 'switch' between thefirstorder upwind scheme - which is positive - and a higher order scheme. We will not discuss this further, see [1] for details. The last step in the method-of-lines approach is to numerically integrate (2), thus obtaining a fully discrete solution. For some Runge-Kutta methods, it can be shown (see also [1]) that for reasonable Courant numbers positivity is maintained after time integration. In [1] an example is given showing that this method can be applied successfully when modeling transport over the poles. We therefore consider this method also a suitable method for application in global atmospheric models.

4 566 Computer Simulation A direct approach Recently, a new method closely related to the method described above, has been developed [2]. The reason to call this approach direct is due to the fact that time and space variables are discretized simultaneously, like for example in the Lax-Wendroff method. The method itself uses directional splitting, i.e. in the first step advection is calculated in 0-diiection only, and in the second step advection is performed in the ^-direction only, using the result of the first step. Therefore the method is essentially a ID method. The advection equation in ^-direction only is given by which is integrated by % r cos 0 c"+i = c" -f - '^ ''^ rcos0ja0 The fluxes f^i ^ are made dependent on the Courant number at 0= (?+ )A<p in such a way that the order of consistency is three in case of a constant wind field. In a similar way advection in ^-direction is treated. Flux limiting is done in the same way as in the method-of-lines approach Due to the directional splitting a temporal error of O(r) is introduced. This error can be made O(T^) if the order in which the spatial directions are treated is reversed each time step. Another way to obtain second order in time is to modify the wind fields in such a way that Hist wider error (ernib cancel, see [2j. In case the wind velocities are constant, no split error is made. For a slowly varying wind field the splitting errors are expected to be small. Numerical experiments in [2] show the better accuracy of the direct method over the method-of-lines approach in Cartesian coordinates. Moreover, the direct scheme is (much) cheaper with regard to computational cost. The direct method can easily be adapted such that it is unconditionally stable, i.e. for Courant numbers >1. When dealing with advection on the sphere, this is a large advantage in the neighborhood of the poles. A disadvantage is that in polar regions the errors caused by time splitting can become quite large. However, in our model we do not have poles within the model domain and the direct method is applied without any problem CHEMICAL SOLVER Introduction In atmospheric models, chemical equations often have to be solved with an accuracy level of, say, at most 1%. For solving these equations the so-called or iseilclo uteddy utcite Approximat'ioTi scnemes ^L2o o A /1 ooaj are

5 Computer Simulation 567 quite popular. However, a comparison between these kind of methods and some state-of-the-art implicit solvers from the stiff ODE field showed some severe disadvantages of the QSSA approach, see 8i As Verwer & Simpson [9j point out, these disadvantages are often not seen due to lumping of reactions and tuning of the numerical method for the specific problem. We will not discuss this further. Instead we will give an outline of an alternative approach, called TWOSTEP and developed by Verwer [7. Outline of TWOSTEP This approach exploits the special form of the differential equation descnbing chemical reactions between n species. This form is given by where P is an n- vector specifying the production terms and L an n x n diagonal matrix defining the loss rates. The numerical integration of system (8) is performed by the second order variable step Boc&word Dzj^erenZWrnrz FormWo (BDF) method ^i ^ yn _^ Tr/(L4-l,Z/^\), T - L^-^, (9) with -y = (c-j-l)/(c4-2), c = (^-fn_i)/(4i-^-w arid +l)y-2/"-')/(c' + 2c). (10) Using the special form (8), the scheme (9) can be rewritten as Gauss-Seidel iteration is now applied to the nonlinear system of equations (11). Note that the diagonal form of L makes this process essentially an explicit one. In the model we perform only one Gauss-Seidel iteration per time step. Although L and P are positive for positive concentrations by definition, positivity of the scheme (11) is not guaranteed because Y" may become negative. So far, this has never been a problem. Moreover, the time step strategy can easily be adapted such that it rejects a time step if a negative solution component is created. Another way to get rid of negative values, is to set them to zero. The latter is done presently. Note that for Y" y (the Backward Euler method) positivity is guaranteed. To get the method started, a Backward Euler step is carried out, i.e. y = i/o and 7 =!. For a description of the time step strategy, we refer to [7], where a full description is given. Numerical experiments show that this approach is favorable to the QSSA methods, see [7, 9j.

6 568 Computer Simulation Conservativity In case the exact solution of system (8) is mass conserving in the sense that one or more relations of the kind constant (12) hold, with w an n-vector with constant weights, we would like the numerical method to satisfy this relation as well. It is well-known that QSSA schemes do not have this property. In [6] a proof is given that the exact solution of (9) or (11) does satisfy this property. However, since Gauss-Seidel iteration is applied to (11), we will not obtain the exact solution arid it can easily be shown that relation (12) does not hold for the numerical solution. Of course, if one iterates (almost) until convergence, no conservation error will be made. Experiments we have done so far show that the TWO STEP scheme conserves mass quite well, even if only one Gauss-Seidel iteration per time step is applied. GRID REFINEMENT In this section, we give an outline of the grid refinement technique used in our model. For details we refer to [5] where a detailed description is given. As computational grid a uniform grid of.55 x.55 (base grid) is used. Such a grid is too coarse to represent local phenomena well enough. A typical ill ii 11 Figure 1: Example of grid refinement with two grid levels example in the context of smog prediction are emissions. To represent such

7 Computer Simulation 569 local phenomena, a much finer grid is necessary. However, a much finer uniform grid would be too expensive in terms of computation time. Yet we need more resolution, at least in certain areas. The solution to this problem is provided by the concept of grid refinement. The basic idea behind this technique is that we only need a finer grid in areas with large solution gradients where the error is expected to be large, and in areas of special interest for the user. To measure this error,, an heuristic space monitor is computed for each grid point. If the value of this monitor (based upon an expression for the curvature of the solution) is considered too large in a certain grid point, the grid is refined around this point. All the refined areas form the new, fine grid and the integration is redone on this grid. This process can be repeated in a recursive way, thus creating a sequence of nested finer and finer grids. An example of possible grid refinement is given in Figure 1. In our model we work with a maximum number of grid levels equal to 4. This gives mesh widths of about 7.5 kilometer. This can still be considered as coarse, but a compromise has to be made between accuracy and computational cost. Acknowledgement The research reported belongs to the project EUSMOG which is carried out in cooperation with the Air Laboratory of the RIVM - The Dutch National Institute of Public Health and Environmental Protection. The RIVM is acknowledged for financial support. REFERENCES 1. W. Hundsdorfer, B. Koren, M. van Loon, and J.G. Verwer. A Positive Finite-Difference Advection Scheme, submitted to J. Comput. Phys. 2. W. Hundsdorfer and R.A. Trompert. Method of lines and direct discretization: a comparison for linear advection. App. Num. Math., 13: , B. van Leer. Upwind-difference methods for aerodynamic problems governed by the Euler equations. In B.E. Engquist, S. Osher, and R.C.J. Somerville, editors, Large-scale computations in fluid mechanics, pages AMS Series, American Mathematical Society, Providence, RI, M. van Loon. Numerical smog prediction I: The physical and chemical model. Report to appear, CWI, Amsterdam, M. van Loon. Numerical smog prediction II: The numerical approach. Report to appear, CWI, Amsterdam, 1994.

8 570 Computer Simulation 6. J.S. Rosenbaum. Conservation Properties of Numerical Integration Methods for Systems of Ordinary Differential Equations. J. Comput. 6., 20: , J,G. Verwer. Gauss-Seidel iteration for stiff ODEs from chemical kinetics. Report NM-R9315, CWI, Amsterdam, J.G. Verwer and M. van Loon. An evaluation of explicit pseudo-steadystate approximation schemes for stiff ODEs from chemical kinetics. Report NM-R9312, CWI, Amsterdam, J.G. Verwer and D. Simpson. Explicit methods for stiff ODEs from atmospheric chemistry. Report NM-R9409, CWI, Amsterdam, D.L. Williamson. Review of numerical approaches for modeling global transport. In H. van Dop and G. Kallos, editors, Air pollution Modeling and its applications IX, pages , New York, Plenum press.