Strang Splitting Versus Implicit-Explicit Methods in Solving Chemistry Transport Models. A contribution to subproject GLOREAM. O. Knoth and R.

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1 Strang Splitting Versus Implicit-Explicit Methods in Solving Chemistry Transport Models A contribution to subproject GLOREAM O. Knoth and R. Wolke Institutfur Tropospharenforschung, Permoserstr. 15, Leipzig, Germany Introduction Usually transport-chemistry models are solved by operator-splitting methods where the chemistry and the vertical transport processes are integrated in a coupled way by implicit methods of order two or higher with automatic error control. In mesoscale applications the integration intervals are relatively short and small initial transients occur for the chemistry-diffusion operator which are an artefact of the operator-splitting approach. Integration methods with error control are quite sensitive to this initial transients which results in rather small step sizes to start up the integration. However to take into account effectively changes inside the chemistry-diffusion process (sunset, sunrise, change of the mixing height) an error control is held indispensible. Implicit-explicit methods recently introduced which are of the same complexity as operator-splitting methods do not suffer from this drawback which means that the initial transients are much smaller in comparison to the operatorsplitting method. Furthermore, these new methods allow an effective use of information from previous time levels to speed up the computation on the next time level. A numerical comparison is presented of the Strang operator splitting method to the new implicit-explicit methods in connection with a specially tuned implicit integrator for the coupled solution of the chemistry diffusion operator. Both artificial as real life problems show a smoother behaviour of our new approach with respect to the step sizes and a decreased number of integration steps and function evaluations of the implicit integrator. Editors: P.M. Borrell and P. Borrell 1999: WITPRESS, Southampton

2 Strang Splitting 525 Source splitting with Runge-Kutta methods After the spatial discretisation the transport-chemistry equations can be represented as a huge system of ordinary differential equations by: y = AW + /w (i) where /E (y) represents the horizontal advection and fi(y) includes the vertical transport processes and the chemistry. Starting from the initial value j/o the implicit-explicit integration method (Knoth and Wolke, 1998) is defined for a given step size h by dt where r= The coefficients 5L and c, are free parameters and have to be derived from the order conditions. The stage values 7, are the solution of a differential equation and any suitable solution method can be used there. In the case where fi (y) = 0 the classical Runge-Kutta scheme is obtained. Two classical Runge- Kutta methods which are of special interest in our context are the first order Euler forward method and Heun's second order method (Hairer et #/., 1993). Comparison of source splitting and Strang splitting Symmetric operator splitting which is also called Strang splitting applied to equation (1) requires the solution of three differential equations, two with the explicit operator ^ (y) and in between one with the implicit operator fj (y). When the two equations with the explicit operator are solved for instance by the explicit second order Heun method and only one step is performed one Strang splitting cycle needs the same number of evaluations of the function fa as two integration steps of the implicit-explicit Runge-Kutta method used also in conjunction with the Heun method. Since both methods are of second order the comparison depends on the efficient solution of the equation which involve the implicit operator. The following simple example will illustrate the differences. Let us take^ (y) = A = 0.3 and^ (y) = -y.

3 526 O. Knoth and R. Wolke For this example Strang splitting consists of the following three steps for a given approximation v* at time / Step 2: Step 3: Solve ** ** dt v_, = Our source splitting method is reduced for this example to the solution of the original problem in a sequence of intervals [*, r, + h\. In Fig. 1 we have plotted all output points of the implicit integrator (Step 2 in the Strang splitting approach) including the starting and end point. Note that for our method the starting point for the next interval is the end point from the last integration. In the Strang splitting method these points differ which results in the zig-zag curve Fig. I: Output of the implicit integrator.

4 Strang Splitting 527 This zig-zag curve shows the transient behaviour which has to be resolved by the implicit integration method during every integration interval. Therefore the integrator LSODE (Hindmarsh 1980) which is based on the BDF method requires four times more steps and function evaluations inside the Strang splitting approach than in the source-splitting method. Both methods are available in the transport code MUSCAT (Knoth and Wolke, 1998). The implicit integrator (Knoth and Wolke. 1995) used is an adapted version of the LSODE programme. The order of the BDF is restricted to two. The linear system solver has been replaced by a Gauss-Seidel or block Gauss- Seidel method The Jacobian matrices are stored in a sparse format and their computation time is comparable to that of one function evaluation. The information from stored backward values is exploited in continuing the implicit integration for a new submterval. The spatial discretisation is of second order The necessary meteorological information is taken online from a mesoscale meteorological model which can be driven with its own step size control in time. In Table 1 both methods are compared for a summer smog situation. The model area covers a region of 160 km x 140 km with a horizontal resolution of 4 km and 27 non-equidistant vertical layers. The simulation time is 24 hours. The chemical reaction mechanism involves 71 species and 192 reactions. The tolerance rtol is an input to the implicit integrator and controls the local error. The stepsize h is a multiple of the horizontal Courant number h<$. Both methods show a larger increase in the number of function evaluations than in the number of successful steps for the tighter tolerance. However this increase is much more pronounced for Strang splitting where the automatic error control leads to a decrease in the stepsize at the beginning of every new integration cycle. Table 1 : Comparison for a summer si- nog episc»de (CPU time in hours:minuteis) Source splitting (h = 0.65 w Stratig splitting (h = 1.3 \ w rtol steps functions Jacobians CPU steps functions Jacobian CPU lo-* : : : :12

5 528 O. Knoth and R. Wolke References E. Hairer, S.P. N0rsett, G. Wanner, Solving ordinary differential equations I, Nonstiff problems, Second Edition, Springer Verlag, Heildelberg A.C. Hindmarsh, LSODE and LSODI, two new initial value ordinary differential equation solver, ACM-SIGNUM Newsletter 15 (1980) O. Knoth, R. Wolke, Numerical methods for the solution of large kinetic systems, Appl. Numer.Math 18(1995) O. Knoth, R. Wolke, An explicit-implicit numerical approach for atmospheric chemistry-transport-modelling, Atmos. Environ, 32 (1998) O. Knoth, R. Wolke, Implicit-explicit Runge-Kutta methods for computing atmospheric reactiveflows,appl Numer. Math. 28 (1998)