Oswald Knoth

An Intercomparison of T-REX Mountain Wave Simulations and Implications for Mesoscale Predictability

AbstractNumerical simulations of flow over steep terrain using 11 different nonhydrostatic numerical models are compared and analyzed. A basic benchmark and five other test cases are simulated in a two-dimensional framework using the same initial state, which is based on conditions during Intensive Observation Period (IOP) 6 of the Terrain-Induced Rotor Experiment (T-REX), in which intense mountain-wave activity was observed. All of the models use an identical horizontal resolution of 1 km and the same vertical resolution. The six simulated test cases use various terrain heights: a 100-m bell-shaped hill, a 1000-m idealized ridge that is steeper on the lee slope, a 2500-m ridge with the same terrain shape, and a cross-Sierra terrain profile. The models are tested with both free-slip and no-slip lower boundary conditions.The results indicate a surprisingly diverse spectrum of simulated mountain-wave characteristics including lee waves, hydraulic-like jump features, and gravity wave breaking. The vertical v...

Implicit–explicit Runge–Kutta methods applied to atmospheric chemistry-transport modelling

Abstract Chemistry-transport calculations are highly stiff in terms of time-stepping. Because explicit ODE solvers require numerous short time steps in order to maintain stability, it seems that especially sparse implicit–explicit solvers are suited to improve the numerical efficiency for atmospheric chemistry applications. In the new version of our mesoscale chemistry-transport model MUSCAT [Knoth, O., Wolke, R., 1998a. An explicit–implicit numerical approach for atmospheric chemistry–transport modelling. Atmospheric Environment 32, 1785–1797.], implicit–explicit (IMEX) time integration schemes are implemented. Explicit second order Runge–Kutta methods for the integration of the horizontal advection are used. The stiff chemistry and all vertical transport processes (turbulent diffusion, advection, deposition) are integrated in an implicit and coupled manner utilizing the second order BDF method. The horizontal fluxes are treated as ‘artificial’ sources within the implicit integration. A change of the solution values as in conventional operator splitting is thus avoided. The aim of this paper is to investigate the interaction between the explicit Runge–Kutta scheme and the implicit integrator. The numerical behavior is discussed for a 1D test problem and 3D chemistry-transport simulations. The efficiency and accuracy of the algorithm are compared to results obtained using the Strang splitting approach. The numerical experiments indicate that our second order implicit–explicit Runge–Kutta methods are a valuable alternative to the conventional operator splitting approach for integrating atmospheric chemistry-transport-models. In mesoscale applications and in cases with stronger accuracy requirements the ‘source splitting’ approach shows a better performance than Strang splitting.

Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows

Air quality modeling is numerically extremely expensive and, therefore, it requires fast algorithms and sophisticated numerical software. This is due to the fact that reacting flow calculations are highly stiff for time-stepping. In this paper implicit-explicit time integration schemes are derived which use explicit higher order Runge-Kutta schemes for the integration of the horizontal advection. The stiff chemistry and all vertical transport processes (turbulent diffusion, advection, deposition) are integrated in an implicit and coupled manner by a higher order BDF method. High order accuracy and stability conditions are investigated for this class of implicit-explicit schemes. The numerical behavior of the new integration schemes is discussed for a 1D and a simple 3D chemistry-transport problem.

The parallel model system LM-MUSCAT for chemistry-transport simulations: Coupling scheme, parallelization and applications

Publisher Summary This chapter discusses the parallel model system LM-MultiScale chemistry aerosol transport (MUSCAT) for chemistry-transport simulations: coupling scheme, parallelization, and applications. The physical and chemical processes in the atmosphere are very complex. They occur simultaneously, coupled and in a wide range of scales. These facts have to be taken into account in the numerical methods for the solution of the model equations. The numerical techniques allow the use of different resolutions in space and also in time. Air quality models base on mass balances described by systems of time-dependent, three-dimensional advection-diffusion reaction equations. A parallel version of the multiscale chemistry-transport code MUSCAT is presented, which is based on multiblock grid techniques and implicit-explicit (IMEX) time integration schemes. The meteorological fields are generated simultaneously by the non-hydrostatic meteorological model LM. Both codes run in parallel mode on a predefined number of processors and exchange information by an implemented coupler interface. The ability and performance of the model system are discussed for a “Berlioz” ozone episode.

An explicit–implicit numerical approach for atmospheric chemistry–transport modeling

Abstract We present the main features of a new atmospheric chemistry–transport code. The employed concepts satisfy essential requirements of third generation atmospheric–transport models. It is shown that our approach can reduce the computational work load of current chemistry transport models by as much as 70–80%. In this paper, we focus on a new temporal integration scheme which is applied to the spatially discretized transport equations. The spatial discretization is performed on a terrain–following grid which allows on-line coupling to existing mesoscale models. Our time integration scheme is of explicit–implicit type. The horizontal advection is integrated explicitly with a large time step and acts as an artificial source in the coupled implicit integration of all vertical transport processes as well as the chemistry. The second-order BDF method is applied for the implicit integration. The linear algebra core in the LSODE code is replaced by a block Gauss–Seidel iteration. This method exploits the sparsity structure of the chemistry–transport problem. We find that two or three iterations suffice, and therefore make the code even faster than the traditional QSSA method. In contrast to the traditional operator splitting, the new approach is free of a transient phase during each implicit integration step. Together with our non–standard starting procedure, large step sizes are maintained throughout integration, with the exception of sunrise and sunset. Large parts of the code are vectorized by loops about the grid cell dimension. Due to memory limitations, a decomposition of the horizontal grid into rectangular subdomains is implemented. The implicit integration is performed with its own time-step selection for each subdomain. The computational efficiency of our approach is investigated with a realistic scenario in Saxony and is compared to the efficiency obtained by an operator splitting approach in combination with a QSSA solver for the chemistry. The sensitivity of our model to three different mechanisms is discussed briefly. Lastly, a technique is introduced with which chemical reaction mechanisms can be easily incorporated into chemistry–transport models.

The Chemistry-Transport Modeling System lm-Muscat : Description and citydelta Applications

Air quality models base on mass balances described by systems of time-dependent, three-dimensional advection-diffusion-reaction equations. The solution of such systems is numerically expensive in terms of computing time. This requires the use of fast parallel computers. Multiblock grid techniques and implicit-explicit (IMEX) time integration schemes are suited to take benefit from the parallel architecture. A parallel version of the multiscale chemistry-transport code MUSCAT (MUiltiScale Chemistry Aerosol Transport) is presented which is based on these techniques (Wolke and Knoth, 2000).

Partially implicit peer methods for the compressible Euler equations

When cut cells are used for the representation of orography in numerical weather prediction models this leads to very small cells. On one hand this results in very harsh time step restrictions for explicit methods due to the CFL criterion. On the other hand cut cells only appear in a small region of the domain. Therefore we consider a partially implicit method: In cut cells the Jacobian incorporates advection, diffusion and acoustics while in the full cells of the free atmosphere only the acoustic part is used, i.e. the method is linearly implicit in the cut cell regions and semi-explicit in the free regions and computes with time step sizes restricted only by the CFL condition in the free atmosphere. Furthermore we use a simplified Jacobian in the cut cell regions in order to save storage and gain computational efficiency. While the method retains the order independently of the Jacobian we present a linear stability theory which takes the effects of the simplifications of the Jacobian on stability into account. The presented method is as stable and accurate as the underlying split-explicit method but furthermore it can compute with cut cells with nearly no additional effort.

Explicit Two-Step Peer Methods for the Compressible Euler Equations

A new time-splitting method for the integration of the compressible Euler equations is presented. It is based on a two-step peer method which is a general linear method with second-order accuracy in every stage. The scheme uses a computationally very efficient forward-backward scheme for the integration of the high-frequency acoustic modes. With this splitting approach it is possible to integrate stably the compressible Euler equations without any artificial damping. The peer method is tested with the dry Euler equations and a comparison with the common split-explicit second-order three-stage Runge-Kutta method by Wicker and Skamarock shows the potential of the class of peer methods with respect to computational efficiency, stability and accuracy.

Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations

AbstractThe compressible Euler equations exhibit wave phenomena on different scales. A suitable spatial discretization results in partitioned ordinary differential equations where fast and slow modes are present. Generalized split-explicit methods for the time integration of these problems are presented. The methods combine explicit Runge–Kutta methods for the slow modes and with a free choice integrator for the fast modes. Order conditions for these methods are discussed.Construction principles to develop methods with enlarged stability area are presented. Among the generalized class several new methods are developed and compared to the well-established three-stage low-storage Runge–Kutta method (RK3). The new methods allow a 4 times larger macro step size. They require a smaller integration interval for the fast modes. Further, these methods satisfy the order conditions for order three even for nonlinear equations. Numerical tests on more complex problems than the model equation confirm the enhanced sta...

Numerical methods for the solution of large kinetic systems

Abstract The photochemical reaction mechanisms used in air pollution models usually consider 40 to 100 pollutant species and more than 150 reactions. The equations resulting from these chemical mechanisms are nonlinear, highly coupled and extremely stiff depending on the time of the day. Therefore, the simulation time of the models is determined to a large degree by the computational burden associated with the solution of the chemistry equations. In recent years, the Quasi Steady State Approximation (QSSA) method is favored for solving the chemistry equations. In the QSSA the solution of large linear systems is not necessary. In this paper we investigate a new approach based on the BDF which solves the sparse linear equations during the Newton iteration by linear Gauss-Seidel iterations. The Jacobian is computed explicitly and not by finite differences. The effect of different numbers of Gauss-Seidel sweeps is investigated. In addition, sparsing techniques proposed by Nowak (1992) and Zlatev (1991) are tested with respect to their efficiency in our algorithms. Our method is compared with respect to its accuracy as well as computational speed with a method of Verwer (1993) which is also based on the two-step BDF combined with nonlinear Gauss-Seidel iterations to approximately determine the implicitly defined solution. The results show that our BDF code reaches the nonlinear Gauss-Seidel approach of Verwer (1993) with respect to the computational speed.