COSMO Priority Project ‚Runge-Kutta‘ Working group 2: Dynamics and Numerics report ‘Oct – Sept. 2007’ COSMO General Meeting, Athen Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

COSMO Priority Project ‚Runge-Kutta‘ Alternative discretizations (due to alternative grids) [RENUMBERED] Remove grid redundancy by Serendipity Grids DWD: Steppeler 09/05 Report The serendipity grids should be investigated, which reduce the redundancy of the interpolation procedures. In this way they achieve more accuracy and more efficiency [NEW] Higher order discretization on unstructured grids using Discontinuous Galerkin methods DWD: Baldauf, Univ. Freiburg: Kroener, Dedner, NN start, 2011 report In the DFG priority program 'METSTROEM' a new dynamical core for the COSMO-model will be developed. It will use Discontinuous Galerkin methods to achieve higher order, conservative discretizations. Currently the building of an adequate library is under development. The work with the COSMO-model will start probably at the end of This is therefore base research especially to clarify, if these methods can lead to efficient solvers for NWP.

COSMO Priority Project ‚Runge-Kutta‘ Radiative upper boundary condition DWD: Herzog 09/05 Report The Klemp Durran boundary is further developed [NEW] Radiative upper boundary condition; non-local in time NN report in 06/2009 At the University Freiburg a Radiative upper boundary condition was developed. It is non-local in time, but nevertheless can be implemented efficiently into non- hydrostatic models. This radiation condition will be further developed during the DFG priority program METSTROEM.

COSMO Priority Project ‚Runge-Kutta‘ Implementation of neglected diabatic terms in p'-equation MPI-H: Petrik, DWD: Herzog 2.10 Diagnostic tools [NEW] Application of the integration tool to energy, mass balance DWD: Baldauf, MPI-H: Petrik The integration tool to calculate balance equations by volume integrations of densities and surface integrations of fluxes developed in the Priority project 'Runge-Kutta', Task 3 will be applied to questions of energy and mass budgets.

COSMO Priority Project ‚Runge-Kutta‘ COSMO Priority Project: Further developments of the Runge-Kutta Time Integration Scheme report ‘Oct – Sept. 2007’ COSMO General Meeting, Athen Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

COSMO Priority Project ‚Runge-Kutta‘ Tasks of the Priority Project ‚Runge-Kutta‘: 1.Looking at pressure bias 2.Continue RK case studies 3.Conservation 4.Advection of moisture quantities in conservation form 5.Investigation of convergence 6.Deep valleys 7.(Different filter options for orography) (finished) 8.Higher order discretization in the vertical for Runge Kutta scheme 9.Physics coupling scheme 10.Testing of alternative fast wave scheme 11.Development of a more conservative dynamics (planned) 12.Development of an efficient semi-implicit solver in combination with RK time integration scheme (planned) 13.NEW: Divergence damping in a truely 3D-version 14.NEW: DFI for RK

COSMO Priority Project ‚Runge-Kutta‘ List of people contributing to the project (Sept Aug. 2007): (alphabetical order) Michael Baldauf (DWD, D) Gabriella Ceci (CIRA, I) Guy deMorsier (MeteoCH, CH) Jochen Förstner (DWD, D)finishes work in COSMO Almut Gassmann (Univ. Bonn, D)finishes work in COSMO Lucio Torrisi (CNMCA, I) Pier Luigi Vitagliano (CIRA, I) new: Gdaly Rivin (Roshydromet, RU) Additional meeting of PP-RK-Group during the LM-User-Workshop, Langen, 07./

COSMO Priority Project ‚Runge-Kutta‘ Task 1: Looking at pressure bias (Torrisi) Goals: verifications of LM 7 km runs showed a higher positive pressure bias for the RK core than for the Leapfrog core, whereas other variables show comparable behaviour. Reasons and solutions? Leapfrog RK starting point of the task:

COSMO Priority Project ‚Runge-Kutta‘ Status the measurements p'T'-dynamics dynamical bottom boundary condition (DBBC) brought a certain improvement Intermediate question: Why is RK sensitive to DBBC whereas Leapfrog is not? LME parallel run at DWD shows even reduced pressure bias compared to Leapfrog in period ' ' (but is this a somewhat special period because there was mostly a high pressure situation in Europe ?) actual status of the problem is not clear no further work done Task 1: Plans: Test idealised mountain flow with RK/Leapfrog --> information, if a pressure bias is caused by dynamics or by parameterizations (T. Davies proposal) (G. Ceci) diabatic terms in pressure equation (R. Petrik (MPI-Hamburg), H. Herzog (retired) ) see WG2, Task determine mass conservation with conservation inspection tool from task 3 (Petrik, Baldauf) see WG2, Task

COSMO Priority Project ‚Runge-Kutta‘ Task 1: RMSE

COSMO Priority Project ‚Runge-Kutta‘ Task 1: BIAS

COSMO Priority Project ‚Runge-Kutta‘ Task 1: RMSE

COSMO Priority Project ‚Runge-Kutta‘ Task 1: BIAS

COSMO Priority Project ‚Runge-Kutta‘  k+1 -  k-1 z k+1 - z k-1  k+1 -  k-1 2( z k+1/2 - z k-1/2 ) Intermediate question: Why is RK sensitive to DBBC whereas Leapfrog is not? Possible answer: Discretization of vertical gradients in p‘-equation RK: Leapfrog: Both versions are consistent and even of the same order of accuracy. But the first version seems to better coincide with DBBC. With the LF approach a slight improvement in MSLP bias is found

COSMO Priority Project ‚Runge-Kutta‘ Task 2: Continue RK case studies (Torrisi, deMorsier) extensive verification of the other tasks Work to do: verifications should be continued Status: CNMCA: COSMO-ITA tests with LF / RK / RK+Semi-Lagr. (results see task 4) COSMO-ITA tests (RK+SL) with nudging and 3D-VAR interpolated initial state

COSMO Priority Project ‚Runge-Kutta‘ Status: MeteoSwiss: since ~ runs a 2.2 km version with assimilation cycle. Several unstable cases found in previous winter period (e.g. ‚13. Jan. 2004‘) most of them could be simulated with Semi-Lagrange Adv. for moisture variables Winter storms Kyrill (' ') and Lothar (' ') simulated with MeteoCH new pre-operational model chain (2.2 km and 6.6 km):  new configuration: 1.) WRF-like RK3 used (instead of TVD-RK3) (as found at DWD for the Kyrill case) 2.) Semi-Lagrange-Adv. for moisture (instead of Bott-scheme) 3.) new level distribution especially in boundary layer (cures problems with TKE scheme)

COSMO Priority Project ‚Runge-Kutta‘ New level distribution compared to DWD-models New level distribution L60.2 in pre-operational COSMO-S2 now similar to COSMO-DE / IFS in the boundary layer. This prevents instabilities (2*dz structures for TKE)

COSMO Priority Project ‚Runge-Kutta‘ New level distribution compared to DWD-models

COSMO Priority Project ‚Runge-Kutta‘ km TVD-RK3 2.2km 'WRF'-RK3 nrdtau=5,  t=10 sec. nrdtau=5,  t=20sec not possible with 20 sec. Storm Lothar, forecast, 24h wind gusts

COSMO Priority Project ‚Runge-Kutta‘ balance equation for scalar  : Task 3: Conservation (Baldauf) Tool for inspection of conservation properties will be developed. temporal change flux divergence sources / sinks integration area = arbitrarily chosen cuboid (in the transformed grid, i.e. terrain-following) Status: available in LM 3.23: Subr. init_integral_3D: define cuboid (in the transformed grid!), prepare domain decomp. Function integral_3D_total: calc. volume integral  V  ijk  V ijk Subr. surface_integral_total: calc. surface integrals   V j ijk *  A ijk prelimineary idealised tests were carried out report finished; will be published in the next COSMO-Newsletter Nr. 7 (2007) Task is finished (Study of conservation properties will be continued in collaboration with MPI-Hamburg, see WG2 Task )

COSMO Priority Project ‚Runge-Kutta‘ (M n -M n-1 ) /  t total surface flux total moisture mass M =   x dV Weisman-Klemp (1982)-test case without physical parameterisation (only advection & condensation/evaporation) Semi-Lagrange-Adv. for q x with multiplicative filling  x :=  (q v + q c ) Res. timestep violation in moisture conservation (?) Task 3:

COSMO Priority Project ‚Runge-Kutta‘ total moisture mass M =   x dV (M n -M n-1 ) /  t total surface flux Res. Weisman-Klemp (1982)-test case with warmer bubble (10 K) without physical parameterisation, without Condensation/Evap. Semi-Lagrange-Adv. for q x with multiplicative filling  x :=  (q v + q c ) Residuum  0  advection seems to be ‚conservative enough‘ possible reasons for conservation violation: saturation adjustment conserves specific mass (and specific energy) but not mass (and energy) itself ! timestep Task 3:

COSMO Priority Project ‚Runge-Kutta‘ Task 4: Advection of moisture quantities in conservation form (Förstner, Baldauf) implementation of the Bott (1989)-scheme into the Courant-number independent advection algorithm for the moisture densities (Easter, 1993, Skamarock, 2004, 2006) Status: Task was finished in Sept because implemented schemes (Bott-2, Bott-4) behaved well But in the meanwhile: stability problems occured in some cases  revival of the task necessary!

COSMO Priority Project ‚Runge-Kutta‘ Problems found with Bott (1989)-scheme in the meanwhile: 2.) Strang-splitting ( 'x-y-z' and 'z-y-x' in 2 time steps) produces 2*dt oscillations Solution: proper Strang-Splitting ('x-y-2z-y-x') in every time step solves the problem, but nearly doubles the computation time 1.) Directional splitting of the scheme: Parallel Marchuk-splitting of conservation equation for density can lead to a complete evacuation of cells Solution: Easter (1993), Skamarock (2004, 2006), mass-consistent splitting 3.) metric terms prevent the scheme to be properly mass conserving Schär–test case of an unconfined jet and ‚tracer=1‘ initialisation (remark: exact mass conservation is already violated by the 'flux-shifting' to make the Bott-scheme Courant-number independent)

COSMO Priority Project ‚Runge-Kutta‘ COSMO-ITA 2.8 km: comparison RK+Bott / RK+Semi-Lagrange RK+SL for light precipitation: TS is larger, whereas FBI is smaller than that for RK+Bott. Moreover, RK+SL has slightly less domain-averaged precipitation and larger maximum prec. values than RK. L. Torrisi

COSMO Priority Project ‚Runge-Kutta‘ Marchuk - splitting formulation of the mass fluxes with: –weighted contravariant velocities –moist air density clipping of density near bottom boundary with a value of kg/m 3 Problems arise if mass is completely evacuated from a grid cell in a single 1D substep, e.g. Cr z = 1 at first layer above ground. mass consistent splitting (Easter 1993, Skamarock 2006) formulation of the mass fluxes with: –contravariant velocities –transformation-adjusted dry density: clipping of density near bottom boundary with a value of 0.5 kg/m 3 Advection of Moisture Quantities in Bott-Scheme Old VersionNew Version Task 4:

COSMO Priority Project ‚Runge-Kutta‘ COSMO-ITA: RK+SL / RK+new Bott RK+new Bott has a larger FBI for all precipitation thresholds than RK+SL (= COSMO-ITA operational run). Moreover, RK+new Bott has a deterioration in MSLP bias and RMSE after T+12h. SL Bott

COSMO Priority Project ‚Runge-Kutta‘ Moisture transport DWD:COSMO-DE: Bott-scheme used COSMO-EU: SL scheme planned operationally MeteoCH: COSMO-S2 and COSMO-S7: SL scheme used pre-operationally CNMCA: COSMO-ITA 2.8: SL-scheme used pre-operationally SL is not positive definite  clipping necessary  'multiplicative filling' combines clipping with global conservation Work done: 'multiplicative filling uses global summation (summation of REAL not associative and therefore not reproducible) New subroutine sum_DDI( f(:,:) ) (with accuracy estimation) reproducibility for 'multiplicative filling' is now reached Semi-Lagrangian advection in COSMO-model

COSMO Priority Project ‚Runge-Kutta‘ Task 5: investigation of convergence (Ceci, Vitagliano) Goals: determination of the spatial and temporal order of convergence of the RK- scheme in combination with advection schemes of higher order. Planned test cases: linear, 2D, hydrostatic mountain flow (h=10 m, a=10 km) linear, 2D, non-hydrostatic mountain flow (h=10 m, a=500 m) nonlinear, 2D mountain flows (dry case) (h=500 m, a=10 km) linear, 3D mountain flow nonlinear mountain flows with precipitation

COSMO Priority Project ‚Runge-Kutta‘ Work done: use of LM 3.21 (  p'T'-dynamics) Problems with unrealistic vertical winds at lateral boundaries solved: Introduction of an appropriate reference state  p', T'=0 at the beginning upper Rayleigh damping layer: idealised test cases need a properly chosen damping constant. The range for it reduces by decreasing the number of levels in damping layer (  compare new WG2 task: 2.3.2, radiative upper boundary condition, non-local in time) Calculation of L 1, L  – error norms for the dry 2D test cases Additionally calculation of mountain drag coefficient, and vertical momentum flux analytical solution for linear mountain flow (Klemp, Lilly, 1978) programmed Task 5: investigation of convergence Plans: determine L 2, L  – errors of KE, w,..., dependent from  x,  t,... for the test cases

COSMO Priority Project ‚Runge-Kutta‘ Spurious vertical velocity at lateral boundaries comparison between the original and modified standard reference atmosphere. Task 5: original standard reference atmosphere and p*, T* modified to get constant Brunt-Väisälä frequency. Problems with unrealistic vertical winds at lateral boundaries solved The effect gets larger as the time step decreases

COSMO Priority Project ‚Runge-Kutta‘ Comparison with analytical solution Analytical solution following Klemp-Lilly (J.Atmos.Sc. 35, , 1978) Task 5: investigation of convergence solution with a damping layer of 85 levels and n R Δt=200.

COSMO Priority Project ‚Runge-Kutta‘ CONVERGENCE OF VERTICAL VELOCITY w L 1 = average of errors L  = maximum error Convergence slightly less than 2. order. (2. order at smaller scales?)

COSMO Priority Project ‚Runge-Kutta‘ NON LINEAR HYDROSTATIC FLOW: STREAMLINES Stable and stationary solution of this non- linear case!

COSMO Priority Project ‚Runge-Kutta‘ CONVERGENCE OF VERTICAL VELOCITY w L 1 = average of errors L  = maximum error

COSMO Priority Project ‚Runge-Kutta‘ Task 6: deep valleys (Förstner, Torrisi, Reinhardt, deMorsier) Goal: detection of the reason for the unrealistic ‚cold pools‘ in Alpine valleys Status (old): The reason for the cold pools was identified: metric terms of the pressure gradient Dynamical Bottom boundary condition (DBBC) (A. Gassmann (2004), COSMO-Newsl.) and a slope-dependent orography-filtering cures the problem to a certain extent. Proposal for future work: inspect the limitations of the terrain following coordinate for steeper slopes, e.g. for future LMK ~1 km horizontal resolution for application of aLMo 2 (MeteoCH) in Alpine region Does the strong conservation form (Gassmann-scheme) delivers advantages for steep terrain?

COSMO Priority Project ‚Runge-Kutta‘ Task 7: Different filter options for orography (Förstner) Status: the orography filtering is now sufficiently weak for DWD-LMK applications (max. slopes 30% allowed) finished

COSMO Priority Project ‚Runge-Kutta‘ Improved vertical advection for the dynamic var. u, v, w, T (or T‘), p‘ motivation: resolved convection vertical advection has increased importance => use scheme of higher order (compare: horizontal adv. from 2. order to 5. order) => bigger w (~20 m/s) => Courant-crit. is violated => implicit scheme or CNI-explicit scheme up to now: implicit (Crank-Nicholson) advection 2. order (centered differences) new: implicit (Crank-N.) advektion 3. order  LES with 5-banddiagonal-matrix but: implicit adv. 3. order in every RK-substep; needs ~ 30% of total computational time!  planned: use outside of RK-scheme (splitting-error?, stability with fast waves?) Task 8: Higher order discretization in the vertical for RK-scheme (Baldauf) Status: no further work done Work to do: best combination with time integration scheme?

COSMO Priority Project ‚Runge-Kutta‘ Task 9: Physics coupling scheme (deMorsier, Förstner, Baldauf, NN) original idea: problems with reduced precipitation could be due to a nonadequate coupling between physics scheme and dynamics Status: no further work done Work to do: what are the reasons for the failure of the WRF-PD-scheme in LM? (turbulence scheme?) Test different sequences of dynamics and physics (especially physics after dynamics)  test tool (Bryan-Fritsch-case) is developed in PP ‚QPF‘, task 4.1 Problems in new physics-dynamics coupling (NPDC): Negative feedback between NPDC and operational moist turbulence parameterization (not present in dry turbulence parameterization) 2-  z - structures in the specific cloud water field (q c ) 2-  z - structures in the TKE field, unrealistic high values, where q c > 0

COSMO Priority Project ‚Runge-Kutta‘ Task 10: Testing of alternative fast wave scheme (Gassmann, Torrisi, Förstner) Goals: p‘T‘-RK-scheme ‚shortened-RK2‘-scheme (Gassmann) this allows the use of the ‚radiative upper boundary condition‘ (RUBC) Properties of A. Gassmann dyn. core: Splitting up of vertical advection of p*/T into fast/slow mode equations and consistent boundary conditions Vertical average to half levels: mass weighted mean (in RK simple mean) and base-state consistent formulation of the discrete w-equation Different horizontal pressure gradient discretization Divergence in conservative flux-form Slightly different buoyancy term No divergence damping

COSMO Priority Project ‚Runge-Kutta‘ Status: The fast waves part (Gassmann) is combined with the Leapfrog scheme in LM 3.21 Original Gassmann dynamical core poses stability problems in several cases! Gassmann fast waves part in RK3 worked in only 1 case ‚shortened RK2‘-scheme (Gassmann (2002), Gassmann and Herzog (2007)) is implemented into LM 3.21 using the fast waves solver of RK3 and the RK3 advection/physics subroutines Preliminary investigation of this dynamical core (L. Torrisi) tested in real cases for a five days period: similar results to the RK3 splitting method Separate inspection of divergence in conservation form and vertical staggering Implementation questions pointed out: Splitting of contravariant vertical velocity poses problems in formulation of lower boundary conditions Work to do: Further tests within CLM-community (dx=18 km) (B. Früh, Univ. Karlsruhe)

COSMO Priority Project ‚Runge-Kutta‘ Gassmann Time-Splitting Method

COSMO Priority Project ‚Runge-Kutta‘ Task 13: Divergence damping in a truely 3D-version (NEW) (Baldauf) Description: Cases occured, where the up to now used 'quasi-2D' divergence filtering lead to unstable results. But a complete abandoning of the divergence filtering (as proposed by A. Gassmann for her dynamical core) also leads to several instabilities. This was also shown by stability analyses of the RK-core by M. Baldauf. P. Prohl (DWD) could demonstrate, that the Bryan-Fritsch- test case of a rising warm bubble is unstable with 'quasi- 2D' divergence damping but becomes stable only with a full 3D (=isotropic) version (realised with a preliminary explicit formulation). For operational use an implicit version of 3D divergence damping is necessary. (this task was defined as a subtask in Task 10; but Task 10 concentrates more and more to the A. Gassmann dynamical core)

COSMO Priority Project ‚Runge-Kutta‘ Task 14: DFI for RK NEW (L. Torrisi) Description: Bug of DFI has been repaired in the Leapfrog-version of the COSMO-model. This has still to be done in RK (L. Torrisi already investigated some work)

COSMO Priority Project ‚Runge-Kutta‘ List of people contributing to the WG2 (‚Numerics‘): (alphabetical order) Euripides Avgoustoglou(GR)LM-Z Michael Baldauf (DWD, D) Heinz-Werner Bitzer(MetBW, D)LM-Z, but mostly Meteorol. Analysis Gabriella Ceci (CIRA, I) Guy deMorsier (MeteoCH, CH) Jochen Förstner (DWD, D)(has left LMK-comm.  ICON) Almut Gassmann (MPI, Hamburg)(has left LMK-comm.  ICON) Hans Herzogretired Gdaly Rivin (Roshyd., RU) Uli Schättler(DWD, D) Jürgen Steppelerretired Lucio Torrisi (CNMCA, I) Pier Luigi Vitagliano (CIRA, I)  ‚Numerics‘-group has reduced personal in future!

COSMO Priority Project ‚Runge-Kutta‘

COSMO Priority Project ‚Runge-Kutta‘

COSMO Priority Project ‚Runge-Kutta‘ Status (until Sept. 2006): 5-day verifications were done for several model configurations only little impact on PMSL by: physics coupling advection of q x (Bott, Semi-Lagrange) most significant impact on PMSL: new dynamical bottom boundary condition (DBBC) (  Task 6) p‘T‘-dynamics in the RK-core both measurements reduce the pressure bias Task 1: verification area (L. Torrisi)

COSMO Priority Project ‚Runge-Kutta‘ positive impact of p‘T‘-dynamics Task 1:

COSMO Priority Project ‚Runge-Kutta‘ positive impact of DBBC Task 1:

COSMO Priority Project ‚Runge-Kutta‘ Status: verifications carried out Work to do: pressure bias improved by another fast waves solver?? ( --> task 10) Task 1:

COSMO Priority Project ‚Runge-Kutta‘ Transport of Tracer in a Real Case Flow Field init Bott (2 nd ) “Flux Form - DIV” + Clipping Bott (2 nd ) “Conserv. Form” semi- Lagrange ( tri-cubic) + Clipping PP RK

COSMO Priority Project ‚Runge-Kutta‘ Advection of Moisture Quantities in Conservation Form Old VersionNew Version Task 4:

COSMO Priority Project ‚Runge-Kutta‘ h Precipitation - date: old version (current settings): moist turbulence ( icldm_turb=2 ) default asymptotic mixing length ( tur_len=500.0 ) old version (test settings): dry turbulence ( icldm_turb=-1 ) smaller asymptotic mixing length ( tur_len=150.0 ) new version (test settings): dry turbulence ( icldm_turb=-1 ) smaller asymptotic mixing length ( tur_len=150.0 ) Task 4:

COSMO Priority Project ‚Runge-Kutta‘ starting point after 1 h modified version: pressure gradient on z-levels, if |metric term| > |terrain follow. term| cold pool – problem in narrow valleys is essentially induced by pressure gradient term T (°C) Task 7:

COSMO Priority Project ‚Runge-Kutta‘ Dynamic Bottom Boundary Condition for metric pressure gradient term in equation for u- and v- component. Gaßmann (COSMO Newsletter No. 4) “(Positive) Pressure Bias Problem” blue:Old Bottom Boundary Cond. red:Dynamic Bottom Boundary Cond. ( Figures by Torrisi, CNMCA Rom) Task 7:

COSMO Priority Project ‚Runge-Kutta‘ Task 7:

COSMO Priority Project ‚Runge-Kutta‘ Task 7:

COSMO Priority Project ‚Runge-Kutta‘ Idealized 1D advection test analytic sol. implicit 2. order implicit 3. order implicit 4. order C= timesteps C= timesteps Task 8: Improved vertical advektion for dynamic var. u, v, w, T, p‘ Task 8:

COSMO Priority Project ‚Runge-Kutta‘ case study ‚ , 00 UTC‘ total precipitation sum after 18 h with vertical advection 2. order difference total precpitation sum after 18 h ‚vertical advection 3. order – 2. order‘ Task 8: Improved vertical advektion for dynamic var. u, v, w, T, p‘ Task 8:

COSMO Priority Project ‚Runge-Kutta‘ contours: vertical velocity isolines: potential temperature Runge-Kutta old p*-T-dynamics Runge-Kutta new p*-T*-dynamics Task 10:

COSMO Priority Project ‚Runge-Kutta‘ Choose CN-parameters for buoyancy in p‘T‘-dynamics from stability analysis  =0.5 (‚pure‘ Crank-Nic.)  =0.6  =0.7  =0.8  =0.9  =1.0 (purely implicit)  choose  =0.7 as the best value Task 10:

COSMO Priority Project ‚Runge-Kutta‘ Physics (I) Radiation Shallow Convection Coriolis force Turbulence Dynamics Runge-Kutta [  (phys) +  (adv)  fast waves ] ‚Physics (I)‘-Tendencies:  n (phys I) Physics (II) Cloud Microphysics Physics-Dynamics-Coupling  n = (u, v, w, pp, T,...) n  n+1 = (u, v, w, pp, T,...) n+1  * = (u, v, w, pp, T,...) * ‚Physics (II)‘-Tendencies:  n (phys II) +  n-1 (phys II) -  n-1 (phys II) Descr. of Advanced Research WRF Ver. 2 (2005) Task 9:

COSMO Priority Project ‚Runge-Kutta‘

COSMO Priority Project ‚Runge-Kutta‘ Task 5: Plans Convergence in dependence of  x and  t independently (reason: internal diffusion of 5. order advection ~C 4 ) (L 2 error norms) Non-hydrostatic linear test case at smaller resolutions Deliver test bed to the COSMO-community Why is the non-linear flow stationary and stable now? p‘T‘-dnamics? Correct inflow boundary conditions/ reference state?