1. Introduction
Dynamical cores of modern numerical weather codes require the numerical solution of the compressible Euler equations. A variety of spatial discretization techniques is available, among them combinations of finite differences and finite volumes. The typical spatial and temporal discretization parameters suggest the application of explicit time integration methods for the advective terms.
Common integration methods in time are split-explicit methods based on different explicit integration methods for the slow advective modes and the fast sound modes. A popular split-explicit method uses a coupling of a three-stage low-storage Runge–Kutta method (RK3) for the slow modes and the forward backward method for the fast modes (Wicker and Skamarock 2002). Versions of that integration scheme are used in modern weather forecast systems like Weather Research and Forecasting Model (WRF), Nonhydrostatic Icosahedral Atmospheric Model (NICAM), or the Consortium for Small Scale Modeling (COSMO). It is applied in there with additional filtering by divergence damping.
We develop here methods with enlarged stability regions compared to established methods in the case when no divergence damping is applied. These methods are designed with respect to the mathematical concept of temporal convergence order that cannot be applied in the presence of divergence damping because the filter parameter depends on the time step size. Thus, we describe a methodology, how to develop methods that are optimal (or optimized) with respect to
an order concept,
a stability concept.
up to 12 free parameters;
nine order conditions, eight of them being nonlinear;
a stability function that is computed as the maximum of the modulus of eigenvalues of matrices that depend on the varying wavenumber;
a stability region we want to maximize, but in which sense?
The outline of this paper is as follows: in section 2 we review multirate infinitesimal step methods (which constitute an extension of split-explicit methods) as well as the order conditions and the stability concept for partitioned equations, we extend our methods and the order concept to nonautonomous equations, and we finally discuss which criteria should be met by an optimized method. In section 3 we construct optimized methods by two different approaches. Section 4 follows with numerical tests and a discussion and outlook in section 5 conclude the paper.
2. Review of split-explicit methods
a. Multirate infinitesimal step methods
The length of the integration interval in each stage is given by di. In an implementation the exact integration procedure for z′ = fc + g(z) is substituted by a integration procedure of sufficiently high order and sufficiently good stability properties using a smaller step size. The applicable small step size is dictated by the stability properties of the method, thus the number of small steps and therefore the computational cost is determined by the total lengths 
b. Order conditions
When g = 0 then the MIS method is equivalent to a classic Runge–Kutta method with coefficient matrix 
The construction of methods that satisfy the full set of nine third-order conditions is undertaken in (Wensch et al. 2009). These methods improve the stability regions of RK3 by a factor of roughly 2. We further show, that for α = γ = 0 there is no additional order condition for order two. Therefore any classic Runge–Kutta method of order two or higher gives an split-explicit method of at least order two provided the fast term integration procedure is of order two.
c. Nonautonomous systems
We know how to apply our methods to autonomous systems, where the right-hand side does not depend explicitly on time. Our methods are designed to work for these systems, the order theory has been developed for these systems. When time-dependent forcing terms enter the system, the right-hand side will depend explicitly on time [cf. y′ = f(y) with y′ = f(t, y)]. For our methods, it is not even clear, at which intermediate time points the forcing terms have to be evaluated. In section 4 below such an application is used as a test case.
d. Split-explicit methods
The splitting methods of Wicker and Skamarock belong to the class above where we have to set α = γ = 0. The classic RK3 method uses β21 = ⅓, β32 = ½, β43 = 1, and vanishing βij otherwise. The total integration length is 
The methods of Knoth and Wolke (Knoth and Wolke 1998) use nonvanishing αij where αi+1,i = 1, but again the γ terms are not present. They analyze the order of the method under the assumption that the exact integration procedure is used. The lowest-order additional order condition they obtain is of order three.
Thus, our methods generalize these methods by allowing for arbitrary α values and additional γ values.
Further, methods based on multistep–multistage methods as slow process integration procedure have been constructed in Jebens et al. (2009). These methods have a much larger stability region than the RK-based methods.
e. Stability
The spatial discretization is done on a staggered C grid with grid width Δx by central differencing of the fast acoustic terms and third-order upwind differencing of the advective term (see Skamarock and Klemp 1992; Wensch et al. 2009).
f. Design criteria
The properties of a method are determined by
Computational cost: Number of stages s, total length of fast process integration intervals
Accuracy: Approximation order of the underlying Runge–Kutta method and the splitting scheme.
Stability: For fixed μ the size CS,max of the stability triangle and the maximum tracer CFL number CA,max. Further, stability has to be checked with respect to both the exact integration procedure and the small step integration procedure.
The full set of nine order conditions is rather restrictive. Further, the fast process integration procedure has to be very simple for reasons of computational efficiency. Thus, it will be of order two at most. A very promising candidate is the Störmer–Verlet method.
We therefore impose the following properties:
The method has s = 3 stages.
The full set of four third-order conditions for the underlying Runge–Kutta method is satisfied because this method is also used for tracer advection.
The additional order condition for order two for partitioned systems is satisfied.
We set the ratio CA/CS = μ = ⅙.
The fast integration procedure is Störmer–Verlet.
3. Construction of methods with improved stability region
We have followed two different approaches to construct three-stage MIS methods of at least full order two with improved stability properties. In particular, we construct one three-stage method (MIS2) with full order two and several three-stage methods with classic order three (MIS3C, TVDMIS3A, and TVDMIS3B). In addition, we construct a four-stage method with full order three (MIS4).
a. Genetic optimization
We use a genetic algorithm to optimize the objective function which combines good stability properties with small integration interval D. The initial population consists of N = 1000 randomly generated second-order three-stage methods. Second-order accuracy is established by a parameterization of second-order methods. In each optimization step a new population is generated as follows:
The method with the best value of the objective function is chosen as parent of the next generation.
The coefficients of the parent are randomly perturbed within a perturbation radius R to generate N − 1 children.
The next generation is formed from the parent and its children.
The perturbation radius R is decreased.
The algorithm is repeated until there is no improvement in 10 successive steps.
The method we have obtained by the genetic optimization process has an extended stability area, but seems to be quite sensitive to the selected number of micro steps.
Therefore, we use this method as a parent method for further line-search-type optimization procedures. In a first step we add constraints guaranteeing a stability triangle of at least CS,max ≥ 12 and minimize the total length D of the fast process integration intervals. The coefficients of the resulting method MIS2 are given in Table 1. The stability region is displayed in Fig. 1. The target region is covered almost completely. The size is approximately CS,max ≈ 17. The number of small steps is taken to be 18. Now, we add the classic third-order accuracy conditions as additional constraints and continue the line-search optimization procedure. The resulting method MIS3 is given in Table 2. It has classic order three but only full order two. The corresponding stability region is depicted in Fig. 2.
Coefficients of the method MIS2 of full order two determined by line-search optimization with the parent method as the starting point.
Stability region of the three-stage method of classic and total second-order MIS2.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Coefficients of the method MIS3C of classic order three determined by line-search optimization with the parent method as the starting point.
Stability region of the three-stage method of classic third-order MIS3C.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Finally, we tried to satisfy all nine order conditions for full order three for partitioned systems. We did not succeed to find a method with three stages with appropriate stability properties. By allowing for a fourth stage we have constructed such a method (MIS4) satisfying all nine order conditions for order three displayed in Table 3. Its stability region is displayed in Fig. 3. Again, roughly the complete target area is covered. The size is CS,max ≈ 17. In addition, we have tried for an optimized method for the case CA/CS = ¼. We call this method MIS4a. Its coefficients are given in Table 4, its stability region is displayed in Fig. 4.
Coefficients of the method MIS4 with full order three and four stages.
Stability region of the four-stage method of full third-order MIS4.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Coefficients of the method MIS4a with full order three and four stages.
Stability region of the four-stage method of full third-order MIS4a.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
b. Methods based on TVD-RK
The underlying RK method is traditionally used in the advection scheme for the transport equations for substances like water, vapor, etc. It is, therefore, natural to consider TVD-RK methods as underlying RK methods. We aim at methods where the overall length 
We have found out that the method in Table 5 is a promising candidate with d2 = c2 = ⅔.
The underlying TVD-RK method.
In the appendix we show that it is optimal in a certain sense, although the optimal method (Gottlieb and Shu 1998) has a larger total variation diminishing (TVD) radius.
Our settings for the optimization procedure fmincon of Matlab are as follows:
For prescribed aij, our optimization variables are αij and γij—a total of six free parameters. The classic order conditions for order three are always satisfied. The additional order condition for order two is added to our optimization problem as a constraint.
An assumed size CS,max of the stability area is always as additional parameter present, either as a fixed parameter or an additional optimization variable. For stability we have to assure that the stability function has modulus not greater than one in the triangular stability region defined by Eq. (11). Because of the maximum principle for holomorphic functions it suffices to assure this on the boundary of the triangular region. The modulus of the two eigenvalues of the amplification matrix is evaluated on a set of representative wavenumbers and on a grid on the boundary of the triangular region. These inequalities are always added as constraints to our optimization problem.
The simultaneous minimization of 
CS,max is a free parameter. The objective function is ϕ = −CS,max. The D is not taken into account.
CS,max is a fixed parameter. The objective function is ϕ = −D.
CS,max is a free parameter. The objective function is ϕ = −CS,max/D.
The stability function is evaluated for a finite number of small steps with Störmer–Verlet. The number is determined such that the small step size is within the stability region of the Störmer–Verlet method, whereas an additional safety factor of 0.9 is applied.
The coefficients of two methods, named TVDMISA and TVDMISB are given in Tables 6 and 7.
Coefficients of method TVDMISA.
Coefficients of method TVDMISB.
Both methods have a stability area size of CS,max ≈ 12. The total small step integration length is D ≈ 1.25. If a macro step with CS = 12 is applied, the suggested small step numbers are nss = 8, 2, 5. In our experiments we found out that the method works nicely if we round this up to small step numbers nss = 9, 3, 6. By taking 18 small steps for CS = 12 this corresponds in fact to D = 1.5.
Below we show the stability regions for the methods MIS2, MIS3C, MIS4, MIS4a, TVDMISA, and TVDMISB in Figs. 1 to 6.
Stability region of the TVD-based method TVDMISA.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Stability region of the TVD-based method TVDMISB.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
All six methods have an exceptionally extended stability region. In fact, we could use CFL numbers almost up to 18 with respect to the speed of sound (although this results in an increased number of small steps in the fast term integration procedure, these steps are cheap compared with the advection procedure). An overview of the methods constructed here is given in Table 8 where RK3 is added as a reference.
Properties of various MIS methods (boldface indicates where MIS4 is stable for CS,max = 16 only for Mach numbers slightly <
4. Numerical tests
Our methods have been designed with respect to a stability model based on the linear acoustics equation. At this point, we test our methods on a selection of examples from the literature that serve to evaluate further qualitative and quantitative properties. In particular, we evaluate scale-selctive damping properties (Vater et al. 2011), order of convergence for nonlinear problems (Durran and Blossey 2012), and the performance on the rising bubble benchmark problem (Wicker and Skamarock 1998) as well as on the cold bubble downburst problem (Straka et al. 1993).
a. Scale-selective damping properties of the MIS method
The damping properties are important with respect to the time integration methods for compressible atmospheric flow with respect to highly oscillatory spatial modes. These modes may be excited by local small-scale heat release (see Vater et al. 2011). They can be removed by implicit time integration methods but then the smooth modes will be damped too. In Vater et al. (2011) blended methods are constructed that damp the short wave modes only but minimize the amplitude and dispersion error for long wave modes. In this section we apply the MIS method to their test problem and show the scale-selective damping properties of the MIS methods.
The MIS method reduces to a classic Runge–Kutta method if the fast part g is absent. Here, we consider the opposite case where f is absent. If the exact solution operator is applied to the differential equation for Zni at each stage, we obtain the exact solution of the differential equation y′ = g(y) in the case of αij and γij = 0. This is especially the case for the RK3 method; thus, scale-selective damping properties cannot be obtained.
For methods with αij and γij ≠ 0 there is not necessarily an exact integration of y′ = g(y) performed, even if the exact solution operator is applied in each stage. Thus, these methods may show scale-selective damping.
The linearized one dimensional Euler equation in Eq. (10) serves as a test equation with CFL number 10 and background velocity U = 0. In the test case from Vater et al. (2011) a high- and a low-frequency signal are overlaid by amplitude modulation. The exact solution is a signal traveling with constant velocity. A numerical discretization should in the ideal case
resolve long waves accurately,
filter inaccurately represented wavelength fast.
In Fig. 7 we display the initial solution and the numerical results given by the MIS method when Störmer–Verlet is applied as fast term integrator. The number of small steps is chosen such that the stability requirements for this method are met. In Fig. 8 the results from RK3 are given: the top image shows the results with Störmer–Verlet as fast term integrator. The bottom image shows the results when in addition divergence damping is applied.
(top) Exact solution and (bottom) exact solution computed by TVDMISA with Störmer–Verlet.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Classic RK3 (top) with Störmer-Verlet and (bottom) with additional divergence damping.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
For RK3 the damping properties of the fast term integration procedure determine the damping properties of the method. There is no damping of the short-wave modes. This can be achieved by additional divergence damping, which gives satisfactory results too. The MIS methods exhibit scale-selective damping properties without divergence damping.
b. Numerical order tests
Besides the theoretical prediction of the order of our methods in case of exact or finite small step integration a numerical test of the achieved orders of temporal convergence seems appropriate now.
In this section we evaluate the performance of split-explicit methods in a nonlinear test case. We give a shortened description here taken literally from the original contribution Durran and Blossey (2012).
To estimate the order of the temporal discretization we follow the methodology applied in Durran and Blossey (2012). A high accuracy solution has been computed with the classic Runge–Kutta method of fourth order with a time step size of 10−4 in order to estimate the discretization error. The same spatial discretization is used in all our tests. The time step size (in seconds) varies over the set {1, 2.5, 5, 10, 15}. The results are displayed in Fig. 9. The ideal slope for a method of order one, two, three is indicated by dotted, slash–dotted, and slashed lines. The method MIS4 with for stages and predicted order three clearly shows third-order error behavior. The methods MIS3C and TVDMISB as well as RK3 show second order of convergence. Finally, we display order results for the method RK3TVD from Baldauf (2010) that has been derived from a third-order TVD method, too. This method also fits in the class of our methods. It displays clearly the first order of convergence. This again reinforces the fact that without a careful order analysis it is difficult to derive higher-order methods of splitting type.
Error vs step size for a variety of methods.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
c. The rising bubble benchmark example
We apply our methods to the test case from Wicker and Skamarock (1998) where a warm bubble rises driven by buoyancy in an isentropic atmosphere.
The test conditions are exactly the same as in Wensch et al. (2009). We display the results from there (methods RK3 and MIS3A) as well as results on the methods constructed in this paper (MIS3C, MIS4, and TVDMISA) in Table 9. We have solved the problem with the methods above with several different time step sizes. Divergence damping is not applied. The maximum step size for which a stable integration was possible is displayed in Table 9. Whereas MIS3A improves the maximum macro step size by a factor of almost 2, the newly constructed methods allow even 4 to 5 times larger macro step sizes.
Maximum time step size for the rising bubble test case. All methods are applied without divergence damping.
d. The cold bubble downburst benchmark example
In contrast to the example above we consider now a cold bubble falling down. We use the test example from Straka et al. (1993) with the modifications that have been made in Wicker and Skamarock (2002). On the ground the bubble generates winds of magnitude 30 m s−1, several Kelvin–Helmholtz waves are generated. Snapshots of the solution at t = 0, 450, and 900 s are displayed in Fig. 10. A grid width of 100 m is used in both directions. With a background wind of 20 m s−1 present the maximum Mach number is slightly above ⅙. The MIS integrations schemes have been implemented in the All Scale Atmospheric Model (ASAM). The compressible Euler equations are discretized on a staggered grid, where the advection operator is approximated by third-order upwinding; all other operators are approximated by second-order central differencing. We have solved the problem with a macro step size of Δt = 1, 2, and 3 s. The corresponding maximum micro step size is ⅙ s. Note, that RK3 failed on these step sizes without divergence damping. Therefore, RK3 is applied in combination with divergence damping [see Wicker and Skamarock (1998), with α = 0.025].
Potential temperature deviation of background state at the beginning, middle, and end of the simulation. Contours range from −10 to 5 (°C) in increments of 0.5.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
Contour plots for the result at t = 900 s using a macro step size of Δt = 1 s are given in Fig. 11. Further, the maximum and minimum values for perturbation potential temperature are displayed in the top left of each diagram . There is a good agreement in these values for all five methods. To evaluate the accuracy we display the deviation from symmetry on the upper right in each diagram. There is also a good agreement between all five methods. The five methods for which the results are displayed are (from top to bottom) RK3, MIS4a, TVDMISB, MIS3C, and MIS2. The properties of TVDMISB, MIS3C, and MIS2 can be reviewed in Table 8. We point out, that the four-stage method MIS4 failed on this example. Instead, we display in the second image the results obtained with another method with four stages designed to be stable for Mach numbers up to ¼. The other three MIS methods as well as RK3 (with divergence damping) work stable with this step size. We have increased the step size to Δt = 2 and 3 s. In Fig. 12 we display the results for the latter case. Even with Δt = 3 s all of the methods work stable, where again divergence damping has to be applied for RK3. Deviations from symmetry are larger than in the case Δt = 1 s.
Contour plot of perturbation potential temperature at t = 900 s obtained with step size Δt = 1 s for (top to bottom) RK3 (with divergence damping), MIS4a, TVDMISB, MIS3C, and MIS2. Displayed levels range from −10 to 0.5 (°C) in increments of 0.5. Minimum and maximum values for perturbation potential temperature are displayed in the top left of each panel and the top right shows the deviation from symmetry.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
As in Fig. 11, but for a step size Δt = 3 s.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00068.1
5. Discussion and outlook
We have constructed multirate infinitesimal step splitting methods for wave-type equations typically occurring in numerical weather prediction. By generalizing the split-explicit methods, the newly constructed methods provide a substantially enlarged stability region. They have desirable scale-selective damping properties.
Further, the additional order conditions for splitting methods are satisfied. The complex structure of these order conditions suggests that a resolution of these order conditions by analytic methods is impractical, if not impossible. Thus, numerical optimization procedures have to be applied to derive methods with desirable order and stability properties. Even this is a nontrivial task. We have coupled several strategies based on genetic algorithms, underlying TVD methods, and conventional optimization procedures.
In this paper we have optimized these methods for a setting where CA/CS = ⅙. The procedure proposed here can be applied to other settings, too. An example is given by the method MIS4a, which has been optimized for the case CA/CS = ¼.
The connection between the shape of the stability region and the choice of coefficients is currently not well understood. It is even surprising that an optimization with six free parameters (this is the case for the TVD-based methods) produces methods with a stability region quite close to the prescribed region. Further, it is not clear how much the stability region changes if the fast process integrator is replaced by an implicit method or if the spatial discretization of the fast or the slow waves is changed. The stability properties have to be checked for each choice of discretization techniques separately. A general theory for these questions seems out of reach at the moment.
Future research focuses on methods where the more restrictive ratio CA/CS = ¼ is used or where implicit fast process integrators are applied. Finally, an open problem is the question whether methods with appropriate order and stability properties and 
APPENDIX
Optimality of the Underlying TVD-RK Method
We want the underlying Runge–Kutta method of our splitting scheme to have the TVD property. To speed up the optimization process we want to fix the TVD scheme in advance. Besides good stability properties of the method we want to minimize the total fast process integration interval 
The method with optimal TVD radius is the method of Gottlieb and Shu (see Table A1). Unfortunately, we have d2 = c2 = 1 in this case, thus D = 1 + d3 + d4. In order obtain D values close to one we use a smaller value of c2. The parameter choice c2 = ⅔ seems to be a good compromise. This implies that c3 = ⅔ and b1 = ¼. There is a further advantage of the choice c2 = ⅔: in general the coefficients of a method with three stages are determined when c2, c3 are fixed. In the case c2 = c2 = ⅔ there is a free parameter left, namely, b3.
(left) TVD method of Gottlieb and Shu and (right) the method with c2 = ⅔.
We seek the method with maximal TVD step size among all methods with underlying RK matrix 
The TVD property with TVD radius r implies: if an explicit Euler step is TVD with step size h ≤ H, the TVD method is TVD for h < rH. Our method has TVD radius r = ¾, which is only slightly below the radius r = 1 of the optimal TVD method from Gottlieb and Shu (Gottlieb and Shu 1998).
We observe first that 
To determine optimal coefficients we substitute β = 
From stage i = 2 we obtain r ≤ α21/β21 = 3/2, which poses no restriction on the coefficients because the maximal TVD radius is r = 1.
REFERENCES
Baldauf, M., 2010: Linear stability analysis of Runge–Kutta-based partial time-splitting schemes for the Euler equations. Mon. Wea. Rev., 138, 4475–4496, doi:10.1175/2010MWR3355.1.
Durran, D. R., and P. N. Blossey, 2012: Implicit-explicit multistep methods for fast-wave–slow-wave problems. Mon. Wea. Rev., 140, 1307–1325, doi:10.1175/MWR-D-11-00088.1.
Gottlieb, S., and C.-W. Shu, 1998: Total variation diminishing Runge–Kutta schemes. Math. Comput., 67, 73–85, doi:10.1090/S0025-5718-98-00913-2.
Jebens, S., O. Knoth, and R. Weiner, 2009: Explicit two-step peer methods for the compressible Euler equations. Mon. Wea. Rev., 137, 2380–2392, doi:10.1175/2008MWR2671.1.
Knoth, O., and R. Wolke, 1998: Implicit–explicit Runge–Kutta methods for computing atmospheric reactive flows. Appl. Numer. Math., 28, 327–341, doi:10.1016/S0168-9274(98)00051-8.
Purser, R. J., 2007: Accuracy considerations of time-splitting methods for models using two-time-level schemes. Mon. Wea. Rev., 135, 1158–1164, doi:10.1175/MWR3339.1.
Skamarock, W., and J. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127, doi:10.1175/1520-0493(1992)120<2109:TSOTSN>2.0.CO;2.
Straka, J., R. Wilhelmson, L. Wicker, J. Anderson, and K. Droegemeier, 1993: Numerical solutions of a non-linear density current: A benchmark solution and comparisons. Int. J. Numer. Methods Fluids, 17, 1–22, doi:10.1002/fld.1650170103.
Vater, S., R. Klein, and O. Knio, 2011: A scale-selective multilevel method for long-wave linear acoustics. Acta Geophys., 59, 1076–1108, doi:10.2478/s11600-011-0037-x.
Wensch, J., O. Knoth, and A. Galant, 2009: Multirate infinitesimal step methods for atmospheric flow simulation. BIT Numer. Math.,49, 449–473, doi:10.1007/s10543-009-0222-3.
Wicker, L. J., 2009: A two-step Adams–Bashforth–Moulton split-explicit integrator for compressible atmospheric models. Mon. Wea. Rev., 137, 3588–3595, doi:10.1175/2009MWR2838.1.
Wicker, L. J., and W. Skamarock, 1998: A time-splitting scheme for the elastic equations incorporating second-order Runge–Kutta time differencing. Mon. Wea. Rev., 126, 1992–1999, doi:10.1175/1520-0493(1998)126<1992:ATSSFT>2.0.CO;2.
Wicker, L. J., and W. Skamarock, 2002: Time splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, doi:10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.
