C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 1 An implicit solver based on dual time stepping and finite volumes for meteorological applications Pier Luigi Vitagliano CIRA COSMO WG2 Conservative Dynamic Core

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 2 OUTLINE Motivation Mathematical model Numerical schemes Test case Future work

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 3 MOTIVATIONS AND GOALS Improve numerical efficiency Improve conservation properties Improve capability to deal with steeper orography Test a time integration scheme for meteorological applications Test spatial schemes based on finite volumes Issue recommendations on future implementation in COSMO

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 4 MATHEMATICAL FORMULATION W =W = F y =F x =F z =B=B= EULER EQUATIONS IN CONSERVATIVE VARIABLES

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 5 SPATIAL DISCRETISATION Finite Volumes approach Integral form allows discontinuities in the flow field Conservation laws applied to each sub-domain (cell) Variables stored at cell centers Fluxes approximated at cell face centers  (W)/  t + R(W) = 0 R(W) = Q – B – D Q = fluxes D = k∆ 4 W artificial dissipation B = source terms

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 6 SPATIAL DISCRETISATION Example of flux evaluation Q m = F mk = ½ (F m + F k ) Conservation laws applied to each sub-domain (cell) Variables stored at cell centers Fluxes approximated at cell face centers km

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 7 DUAL TIME STEPPING  W n+1 /  + ½(3W n+1 - 4W n + W n-1 )/  t + R(W n+1 ) = 0 add a pseudo-time  derivative to the unsteady equation advance the solution in  until the residual of the unsteady equation is negligible formulation is A-stable and damps the highest frequency very large physical time step  t can be used iterations in  are performed by explicit Runge-Kutte scheme convergence acceleration techniques can be adopted without loss of time accuracy: residual averaging, local time stepping, multigrid Jameson, A., 1991: Time Dependent Calculations Using Multigrid,with Applications to Unsteady Flows Past Airfoils and Wings. AIAA Paper 91–1596  (W)/  t + R(W) = 0

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 8 DUAL TIME STEPPING Example of time integration with DTS: a norm of the residuals of mass transport equations is monitored

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 9 PRECONDITIONING Improve convergency in dual time for low Mach number flows Correct ill-behaved artificial viscosity fluxes at low Mach Difficulties rise from large ratio between acoustic wave speed and fluid speed Premultiplying the time derivative changes the eigenvalues of the system and accelerates the convergence to steady state. P·  W/  + R(W) = 0 Turkel, E., 1999: Preconditioning techniques in computational fluid dynamics. Annu.Rev.Fluid Mech. 1999,31: Venkateswaran, S., P. E. O. Buelow, C. L. Merkle, 1997: Development of linearized preconditioning methods for enhancing robustness and efficiency of Euler and Navier-Stokes Computations, AIAA Paper

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 10 PRECONDITIONING Example of convergence to steady solution with and without Preconditioning

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 11 Discretisation of the gravity force term Field initialisation Effect of mesh skewness Flux – force unbalance

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 12 Flow over a gaussian mountain simulated with a test code based on finite volumes conservative schemes. Vertical velocity component. The dashed line shows the lower boundary of the Rayleigh damping layer, which prevents the wave reflection. TEST CASE MOUNTAIN FLOW

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 13 Mesh for test on complex orography with cold bubble

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 14 CONCLUSIONS COMPUTER CODE FOR TEST RUN READY INITIAL TESTS ON STEADY MOUNTAIN FLOW STUDY ON COMPLEX OROGRAPHY STARTED

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 15 FUTURE WORK TEST CASES: 1)Atmosphere at rest with deformed mesh (Zaengl (2004)) 2)Cold bubble (Straka et al. (1993)) 3)Mountain test cases: (Schaer et al (2002) sect. 5b) (Bonaventura(2000)) (Klemp,Wilhelmson) 4)Linear gravity waves (Skamarock-Klemp (1994),Giraldo(2008))

C M C C Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO General Meeting - September 8th, 2009 COSMO WG 2 - CDC 16 TEST CASE COLD BUBBLE Initial Field Density contour. Step Δρ/ρ SL = Initial Field U-Velocity contour