Semi-implicit semi-Lagrangian Scheme Research Articles

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

FVM 1.0: a nonhydrostatic finite-volume dynamical core for the IFS

Abstract. We present a nonhydrostatic finite-volume global atmospheric model formulation for numerical weather prediction with the Integrated Forecasting System (IFS) at ECMWF and compare it to the established operational spectral-transform formulation. The novel Finite-Volume Module of the IFS (henceforth IFS-FVM) integrates the fully compressible equations using semi-implicit time stepping and non-oscillatory forward-in-time (NFT) Eulerian advection, whereas the spectral-transform IFS solves the hydrostatic primitive equations (optionally the fully compressible equations) using a semi-implicit semi-Lagrangian scheme. The IFS-FVM complements the spectral-transform counterpart by means of the finite-volume discretization with a local low-volume communication footprint, fully conservative and monotone advective transport, all-scale deep-atmosphere fully compressible equations in a generalized height-based vertical coordinate, and flexible horizontal meshes. Nevertheless, both the finite-volume and spectral-transform formulations can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude–latitude coordinates, and physics parameterizations, thereby facilitating their comparison, coexistence, and combination in the IFS. We highlight the advanced semi-implicit NFT finite-volume integration of the fully compressible equations of IFS-FVM considering comprehensive moist-precipitating dynamics with coupling to the IFS cloud parameterization by means of a generic interface. These developments – including a new horizontal–vertical split NFT MPDATA advective transport scheme, variable time stepping, effective preconditioning of the elliptic Helmholtz solver in the semi-implicit scheme, and a computationally efficient implementation of the median-dual finite-volume approach – provide a basis for the efficacy of IFS-FVM and its application in global numerical weather prediction. Here, numerical experiments focus on relevant dry and moist-precipitating baroclinic instability at various resolutions. We show that the presented semi-implicit NFT finite-volume integration scheme on co-located meshes of IFS-FVM can provide highly competitive solution quality and computational performance to the proven semi-implicit semi-Lagrangian integration scheme of the spectral-transform IFS.

Real-Time Nonlinear Model Predictive Control of a Transport–Reaction System

Two real-time nonlinear model predictive control (NMPC) algorithms for a transport–reaction system are designed. The system is modeled by a hyperbolic partial differential equation and discretized by means of a two-time-level semi-implicit semi-Lagrangian scheme. For the resulting lumped-parameter system, a constrained optimal control problem is formulated and state constraints are implemented in the form of barrier functions. The NMPC algorithms perform a single step or several steps of an iterative solution routine of the optimal control problem at every sampling point. With this suboptimal solution strategy, a fixed maximum evaluation time and execution in real time are guaranteed. An analysis of the nominal stability is provided for one NMPC scheme. The robustness of the controllers is evaluated for an example problem, where a nonisothermal plug-flow reactor with irreversible exothermic reactions is considered. The control objectives are to limit the maximum reactor temperature (avoid hot spots) and to maximize the process output.

An improved dynamic core for a non-hydrostatic model system on the Yin-Yang grid

A 3D dynamic core of the non-hydrostatic model GRAPES (Global/Regional Assimilation and Prediction System) is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. Three-dimensional Coriolis forcing is introduced to the new core, and full representation of the Coriolis forcing makes it straightforward to share code between the Yin and Yang subdomains. Similar to that in the original GRAPES model, a semi-implicit semi-Lagrangian scheme is adopted for temporal integration and advection with additional arrangement for cross-boundary transport. Under a non-centered second-order temporal and spatial discretization, the dry nonhydrostatic frame is summarized as the solution of an elliptical problem. The resulting Helmholtz equation is solved with the Generalized Conjugate Residual solver in cooperation with the classic Schwarz method. Even though the coefficients of the equation are quite different from those in the original model, the computational procedure of the new core is just the same. The bi-cubic Lagrangian interpolation serves to provide Dirichlet-type boundary conditions with data transfer between the subdomains. The dry core is evaluated with several benchmark test cases, and all the tests display reasonable numerical stability and computing performance. Persistency of the balanced flow and development of both the mountain-induced Rossby wave and Rossby-Haurwitz wave confirms the appropriate installation of the 3D Coriolis terms in the semi-implicit semi-Lagrangian dynamic core on the Yin-Yang grid.

An alternative cell‐averaged departure point reconstruction for pointwise semi‐Lagrangian transport schemes

Convection‐permitting limited‐area models based on the same spectral semi‐implicit semi‐Lagrangian (SL) techniques which are used in the ECMWF global model, are run operationally in several countries of the ALADIN/HIRLAM consortium. Forecasters have reported a general tendency for these models to produce overestimated precipitation and unrealistic divergent winds at the edges of the cold outflows generated by the precipitation evaporation in the vicinity of convective clouds. These grid‐point storms have been associated with a spurious behaviour of the pointwise interpolation used in the SL scheme, where grid‐scale buoyant updraughts create strong small‐scale convergence near the surface. A modification of the interpolation weights in the SL transport scheme introduces the concept of cell‐averaging into the traditional pointwise SL scheme which improves the conservation property of the scheme and eliminates the spurious mode. The COMAD (COntinuous Mapping about Departure points) correction applied to the standard interpolation weights takes into account the deformation of the air parcels along each direction of interpolation in order to improve the continuity and the conservative property of the re‐mapping between the model grid points and the origin points of the backward trajectories. The method has been validated with the small planet configuration of the Integrated Forecast System at ECMWF and with the limited‐area version of the same dynamics used for the AROME (Météo‐France) and HARMONIE (HIRLAM) models. The pathological behaviour of grid‐scale buoyant flows permitted by these dynamical cores is corrected by the COMAD interpolations. The precipitation forecasts in the convection‐permitting models AROME/HARMONIE which show an overestimation of intense convective precipitation are systematically improved by the new weights.

Theoretical and Numerical Analysis of a Class of Semi-Implicit Semi-Lagrangian Schemes Potentially Applicable to Atmospheric Models

Abstract In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangian methods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear term in the second step makes the proposed class of schemes competitive in terms of efficiency compared to some well-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi-Lagrangian semi-implicit schemes and semi-implicit predictor–corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies.

A Compressible Nonhydrostatic Cell-Integrated Semi-Lagrangian Semi-Implicit Solver (CSLAM-NH) with Consistent and Conservative Transport

Abstract A cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic solver for the fully compressible moist Euler equations in two-dimensional Cartesian (x–z) geometry is presented. The semi-implicit CISL solver uses the inherently conservative semi-Lagrangian multitracer transport scheme (CSLAM) and a new flux-form semi-implicit formulation of the continuity equation that ensures numerically consistent transport. The flux-form semi-implicit formulation is based on a recent successful approach in a shallow-water equations (SWE) solver (CSLAM-SW). With the new approach, the CISL semi-implicit nonhydrostatic solver (CSLAM-NH) is able to ensure conservative and consistent transport by avoiding the need for a time-independent mean reference state. Like its SWE counterpart, the nonhydrostatic solver presented here is designed to be similar to typical semi-Lagrangian semi-implicit schemes, such that only a single linear Helmholtz equation solution and a single call to CSLAM are required per time step. To demonstrate its stability and accuracy, the solver is applied to a set of three idealized test cases: a density current (dry), a gravity wave (dry), and a squall line (moist). A fourth test case shows that shape preservation of passive tracers is ensured by coupling the semi-implicit CISL formulation with existing shape-preserving filters. Results show that CSLAM-NH solutions compare well with other existing solvers for the three test cases, and that it is shape preserving.

A mass-conserving and multi-tracer efficient transport scheme in the online integrated Enviro-HIRLAM model

Abstract. In this paper a new advection scheme for the online coupled chemical–weather prediction model Enviro-HIRLAM is presented. The new scheme is based on the locally mass-conserving semi-Lagrangian method (LMCSL), where the original two-dimensional scheme has been extended to a fully three-dimensional version. This means that the three-dimensional semi-implicit semi-Lagrangian scheme which is currently used in Enviro-HIRLAM is largely unchanged. The HIRLAM model is a computationally efficient hydrostatic operational short-term numerical weather prediction model, which is used as the base for the online integrated Enviro-HIRLAM. The new scheme is shown to be efficient, mass conserving, and shape preserving, while only requiring minor alterations to the original code. It still retains the stability at long time steps, which the semi-Lagrangian schemes are known for, while handling the emissions of chemical species accurately. Several mass-conserving filters have been tested to assess the optimal balance of accuracy vs. efficiency.

A Cell-Integrated Semi-Lagrangian Semi-Implicit Shallow-Water Model (CSLAM-SW) with Conservative and Consistent Transport

Abstract A Cartesian semi-implicit solver using the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme is constructed and tested for shallow-water (SW) flows. The SW equations solver (CSLAM-SW) uses a discrete semi-implicit continuity equation specifically designed to ensure a conservative and consistent transport of constituents by avoiding the use of a constant mean reference state. The algorithm is constructed to be similar to typical conservative semi-Lagrangian semi-implicit schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The accuracy and stability of the solver is tested using four test cases for a radially propagating gravity wave and two barotropically unstable jets. In a consistency test using the new solver, the specific concentration constancy is preserved up to machine roundoff, whereas a typical formulation can have errors many orders of magnitude larger. In addition to mass conservation and consistency, CSLAM-SW also ensures shape preservation by combining the new scheme with existing shape-preserving filters. With promising SW test results, CSLAM-SW shows potential for extension to a nonhydrostatic, fully compressible system solver for numerical weather prediction and climate models.

Coupling a mass‐conserving semi‐Lagrangian scheme (SLICE) to a semi‐implicit discretization of the shallow‐water equations: Minimizing the dependence on a reference atmosphere

AbstractIn a recent paper, a conservative semi‐Lagrangian mass transport scheme SLICE has been coupled to a semi‐implicit semi‐Lagrangian scheme for the shallow‐water equations. The algorithm involves the solution at each timestep of a nonlinear Helmholtz problem, which is achieved by iterative solution of a linear ‘inner’ Helmholtz problem; this framework, as well as the linear Helmholtz operator itself, are the same as would be used with a non‐conservative interpolating semi‐Lagrangian scheme for mass transport. However, in order to do this, a reference value of geopotential was introduced into the discretization. It is shown here that this results in a weak dependence of the results on that reference value. An alternative coupling is therefore proposed that preserves the same solution framework and linear Helmholtz operator but, at convergence of the nonlinear solver, has no dependence on the reference value. However, in order to maintain accuracy at large timesteps, this approach requires a modification to how SLICE performs its remapping. An advantage of removing the dependence on the reference value is that the scheme then gives consistent tracer transport. Copyright © 2010 Royal Meteorological Society and Crown Copyright. Published by John Wiley & Sons, Ltd.

Well-Posed Lateral Boundary Conditions for Spectral Semi-Implicit Semi-Lagrangian Schemes: Tests in a One-Dimensional Model

Abstract The aim of this paper is to investigate the feasibility of well-posed lateral boundary conditions in a Fourier spectral semi-implicit semi-Lagrangian one-dimensional model. Two aspects are analyzed: (i) the complication of designing well-posed boundary conditions for a spectral semi-implicit scheme and (ii) the implications of such a lateral boundary treatment for the semi-Lagrangian trajectory computations at the lateral boundaries. Straightforwardly imposing boundary conditions in the gridpoint-explicit part of the semi-implicit time-marching scheme leads to numerical instabilities for time steps that are relevant in today’s numerical weather prediction applications. It is shown that an iterative scheme is capable of curing these instabilities. This new iterative boundary treatment has been tested in the framework of the one-dimensional shallow-water equations leading to a significant improvement in terms of stability. As far as the semi-Lagrangian part of the time scheme is concerned, the use of a trajectory truncation scheme has been found to be stable in experimental tests, even for large values of the advective Courant number. It is also demonstrated that a well-posed buffer zone can be successfully applied in this spectral context. A promising (but not easily implemented) alternative to these three above-referenced schemes has been tested and is also presented here.

Externalizing the lateral boundary conditions from the dynamic core in semi-implicit semi-Lagrangian models

Research is still undertaken to develop so-called transparent lateral boundary conditions (LBC) for limited-area numerical weather prediction models. In the widely used semi-implicit semi-Lagrangian schemes, this naturally leads to LBC formulations that are intrinsically intertwined with the numerics of the dynamic core. This may have profound consequences for the implementation and the maintenance of future model codes. For instance, scientific development on the dynamics may be hindered by constraints coming from today’s choices in the LBC formulation and vice versa. Building further on the work of Aidan McDonald, this paper proposes an approach where (1) the LBCs can be imposed by an extrinsic numerical scheme that is fundamentally different from the one used for the dynamic core in the interior domain and (2) substituting one such LBC scheme for another is possible in a manner that leaves the Helmholtz solver unmodified. It is argued that this concept may provide the necessary frame for formulating transparent boundary conditions in spectral limited-area models. Since this idea touches all aspects of the LBC problem, its feasibility can only be established by a rigorous systematic approach. As a first step, this paper provides promising experimental support in a one-dimensional shallow-water model.

Semi-Lagrangian semi-implicit time-splitting scheme for a regional model of the atmosphere

Semi-Lagrangian semi-implicit (SLSI) method is currently one of the most efficient approaches for numerical solution of the atmosphere dynamics equations. In this research we apply splitting techniques in the context of a two-time-level SLSI scheme in order to simplify the treatment of the slow physical modes and optimize the solution of the elliptic equations related to implicit part of the scheme. The performed numerical experiments show the accuracy and computational efficiency of the scheme.

Externalizing the lateral boundary conditions from the dynamic core in semi-implicit semi-Lagrangian models

Research is still undertaken to develop so-called transparent lateral boundary conditions (LBC) for limited-area numerical weather prediction models. In the widely used semi-implicit semi-Lagrangian schemes, this naturally leads to LBC formulations that are intrinsically intertwined with the numerics of the dynamic core. This may have profound consequences for the implementation and the maintenance of future model codes. For instance, scientific development on the dynamics may be hindered by constraints coming from today’s choices in the LBC formulation and vice versa. Building further on the work of Aidan McDonald, this paper proposes an approach where (1) the LBCs can be imposed by an extrinsic numerical scheme that is fundamentally different from the one used for the dynamic core in the interior domain and (2) substituting one such LBC scheme for another is possible in a manner that leaves the Helmholtz solver unmodified. It is argued that this concept may provide the necessary frame for formulating transparent boundary conditions in spectral limited-area models. Since this idea touches all aspects of the LBC problem, its feasibility can only be established by a rigorous systematic approach. As a first step, this paper provides promising experimental support in a one-dimensional shallow-water model.

Conservative semi-implicit semi-Lagrangian scheme for simulation of shallow flows

A fully conservative semi-Lagrangian (SL) scheme is presented to solve for the shallow-water equations. Existing inherently conservative SL schemes only ensure the conservation of mass while momentum is not fully conserved. The gravity terms, which are mainly responsible for the wave structure in dam break flows, are then discretized by using traditional non-conservative Eulerian schemes. In the presence of large variations in water surface (e.g., dam-break type flows), such an approach leads to incorrect shock speed and highly oscillatory results. Indeed, if the conservation of the gravity terms is forced, the use of existing schemes will be restricted to small time steps. In this paper we present a fully conservative scheme which can accurately simulate the shallow flows with a large time step. In our approach, both convective and gravitational terms are treated in a conservative manner, which ensures an accurate shock speed. The fully conservation property improves considerably the performance of common SL schemes for a wide range of practical applications.

Finescale Topography and the MC2 Dynamics Kernel

Abstract The Canadian Mesoscale Compressible Community (MC2) model provided daily forecasts across the Alps at 3-km resolution during the Mesoscale Alpine Programme (MAP) field phase of 1999. Among the results of this endeavor, some have had an immediate impact on MC2 itself as it increasingly became evident that the model was spuriously too sensitive to finescale orographic forcing. The model solves the Euler equations of motion using a semi-implicit semi-Lagrangian scheme in an oblique terrain-following coordinate. To improve model behavior, typical approaches were tried at first. These included a generalization of the coordinate transformation to make the terrain influence decay much more quickly with height as well as the introduction of nonisothermal basic states to diminish the amplitude of numerical truncation errors. The concept of piecewise-constant finite elements was invoked to reduce coding arbitrariness. But it was later pointed out that the problem was very specific and due to a numerical inconsistency. The true height of model grid points is fixed and known in height-based coordinates. Nevertheless, it was discovered that for this semi-Lagrangian scheme to be consistent, the departure height is an unknown that must be obtained in the same manner as the other unknowns.

On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere

We have investigated the accuracy ofthe semi-implicit semi-Lagrangian (SISL) method in simulating internal gravity wave (IGW) motion. We have focused on the relative accuracy of the hydrostatic, and nonhydrostatic IGW solutions. The analysis is based on a linearized model and a Global Circulation Model-Dynamic Core (GCM-DC) with a stretched grid.The nonhydrostatic version of the GCM-DC model produces the familiar IGW train disturbance anchored to an isolated hypothetical mountain. The wave has a distinct tilt away from the vertical direction, which is consistent with classical theory. For the hydrostatic version of the model, the axis of the resulting IGW train rests nearly perpendicular to the mountain top, thus again consistent with classical theory. Increasing the time step from 10 s; Courant number (Cn) = 0.5; to 60 s (Cn = 3.0), results in stable solutions for both the hydrostatic and nonhydrostatic versions of the model. The nonhydrostatic solution is in close agreement with the control run however, the hydrostatic solution exhibits large phase truncation errors.The solutions for the one-dimensional linearized SISL model confirm the GCM-DC results that the nonhydrostatic IGW train is less damped and shifted by the SISL scheme than the corresponding hydrostatic IGW motion. The linear solutions indicate very high accuracy of the physical mode of the solution, but it rapidly deteriorates when Cn exceeds unity. As Δt → 0 the amplitude of the computational mode tends to zero and its frequency to infinity. However, as Δt → ∞, the frequency of the computational SISL mode asymptotically approaches the value of the frequency of the corresponding SISL physical mode. Furthermore, the amplitude of the SISL computational mode is directly proportional to the size of the time step. Therefore, at large time steps, the amplification of the computational mode could offset some of the numerical damping of the physical mode by the SISL scheme.

Impact of semi-Lagrangian trajectories on the discrete normal modes of a non-hydrostatic vertical-column model

The impact of one- and two-term extrapolated-trajectory schemes on the stability properties of centred semi-implicit semi-Lagrangian schemes is analysed by examining the behaviour of the discrete (acoustic) normal modes of a prototypical column model. It is found that, in the absence of any controlling mechanism, both extrapolated-trajectory schemes are unstable. Additionally, they can significantly distort the vertical structure of the modes. Though not studied herein, the analogous distortion of Rossby and gravity waves could be expected to be deleterious to a forecast model. © Crown copyright, 2005. Royal Meteorological Society

Limited-Area Forecasting Model Based on Semi-Lagrangian Semi-Implicit Scheme Leading to Solving the Separable Elliptic Equations

A significant number of atmospheric models use a semi-implicit algorithm in order to carry out the time integration. In the semi-implicit scheme, the size of the time step is no longer restricted by the phase speed of external gravity waves; it is limited mainly by the magnitude of the wind associated with advection. Combining a semi-Lagrangian scheme with the semi-implicit technique introduces the possibility of integrating atmospheric models using relatively large time steps. In this paper, the semi-implicit algorithm in association with the semi-Lagrangian technique is presented. To solve the equations of semi-implicit schema we have used method of dimension reduction which reduces the three-dimensional problem to two-dimensional problems. The original contribution of this way is that semi-implicit approximation leads to the solution of two-dimensional separable partial differential equations. A detailed description is given and the model is tested by using the Lambert projection and the initial data provided by the ARPEGE model. The results are very satisfactory.

Progresses of researches on numerical weather prediction in China: 1999–2002

The recent progresses in the research and development of (NWP) in China are reviewed in this paper. The most impressive achievements are the development of direct assimilation of satellite irradiances with a 3DVAR (three-dimentional variational) data assimilation system and a non-hydrostatic model with a semi-Lagrangian semi-implicit scheme. Progresses have also been made in model physics and model application to precipitation and environmental forecasts. Some scientific issues of great importance for further development are discussed.