MOX EXPONENTIAL INTEGRATORS FOR MULTIPLE TIME SCALE PROBLEMS OF ENVIRONMENTAL FLUID DYNAMICS. Innsbruck Workshop October

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1 Innsbruck Workshop October EXPONENTIAL INTEGRATORS FOR MULTIPLE TIME SCALE PROBLEMS OF ENVIRONMENTAL FLUID DYNAMICS Luca Bonaventura - Modellistica e Calcolo Scientifico Dipartimento di Matematica F. Brioschi Politecnico di Milano p. 1/2

2 Acknowledgments Work stimulated by the participation to the ICTP School on Integrable System, Trieste, Italy (June 29) Mostly carried out during IPAM-UCLA programme on Model and data hierarchies for simulating and understanding climate, Los Angeles, USA (March-April 21) Thanks for invitations and financial support to: Filippo Giorgi, Adrian Tompkins (ICTP) Rupert Klein (FUB), IPAM-UCLA Italian Ministry of Research PRIN 28 p. 2/2

3 Outline of the presentation overview of typical multiple time scale problems in environmental fluid dynamics (especially NWP) short review of some basic time discretization methods often employed in this field comparison of these methods to exponential integration techniques: assessment of their potential in applications to geophysical fluid dynamics can they compete with semi-implicit and semi-lagrangian methods? tentative answers, open issues and perspectives for future developments p. 3/2

4 Typical features of atmospheric flows model equations equivalent to Euler equations + turbulence models + source terms (gravity, rotation, radiation) + clouds and water vapour great difference in vertical and horizontal spatial scales time scale of rotational effects, important for planetary scale motions: from hours to days advection dominated problem sound speed 34 m/s, stratospheric jet stream 8 m/s, speed of typical tropospheric winds and internal gravity waves 1 m/s some similar features in mathematical modelling of oceans, coastal basins, rivers p. 4/2

5 Approach to time discretization instrinsically stiff problem early approach (Charney and von Neumann 195): use filtered equations which only represent slower phenomena modern approach: robust time integration techniques semi-implicit methods to avoid time step restrictions due to sound waves semi-lagrangian methods to increase accuracy and efficiency in advection dominated regimes target (UK Met Office): 45 minutes of supercomputer CPU for a 1 day global forecast with O(1 8 ) degrees of freedom massive use of operator splitting p. 5/2

6 A model problem u t η t + (u )u + fk u + g η = + u η + η u = 2D homogeneous fluid layer over flat bottom subject to gravity and Coriolis force u velocity, η depth of fluid layer k normal to the fluid plane, f Coriolis coefficient, g gravity acceleration solutions contain fast gravity waves and slow solutions close to geostrophic equilibrium fk u g η p. 6/2

7 Semi-implicit, semi-lagrangian method u n+1 u n t η n+1 η n t + f 2 k (un+1 + u n ) + g ] [ η n+1 + ( η n ) 2 ] + [ ηn u n+1 + ( u n ) = 2 = semi-implicit: Crank-Nicolson discretization of terms creating gravity waves semi-lagrangian: generalized method of characteristics for advection terms η, u values interpolated along streamlines linearly unconditionally stable, only requires solution of well conditioned linear system for η n+1 at each time step p. 7/2

8 Exponential Euler Rosenbrock method Linearize around initial datum at each timestep du dt = f(u) = f(un ) + A(u u n ) + R(u) u() = u n A = f u (un ) Approximate computation of nonlinear terms yields numerical method u n+1 = u n + tφ(a t)f(u n ) exact for linear, constant coefficient problems, second order for generic nonlinear, linearly unconditionally stable finite difference approximation of Jacobian can be used without harming stability or accuracy p. 8/2

9 Results of some numerical experiments dynamical systems one dimensional advection diffusion equations one dimensional inertial-gravity wave propagation shallow water equations on the sphere results obtained with second order Exponential Rosenbrock integrator exponential matrix computed by Arnoldi decomposition, + Padé approximation of Hessenberg matrix exponential implementation derived from EXPOKIT package (Sidje 1998) p. 9/2

10 The Lorenz system comparison of second order Exponential Rosenbrock method and second order Runge Kutta to reference MATLAB solver solution with small absolute tolerance similar behaviour for short timesteps, superior accuracy of the exponential method for (2 times) longer timesteps 4 3 Ref RK2 ExpR2 4 3 Ref RK2 ExpR p. 1/2

11 The coupled Lorenz system slow/fast coupled Lorenz systems: idealized models of atmosphere-ocean interaction, see e.g. M. Pena and E. Kalnay,Nonlin. Proc. Geoph., 11, , 24 exponential Rosenbrock methods and MATLAB stiff solvers compared to reference MATLAB solver robustness of exponential method: results analogous to those of stiff solvers of comparable order 5 6 Stiff ExR3 Ref 45 Stiff ExR3 Ref MO X p. 11/2

12 Advection linear advection equation, but nonlinear flux/slope limiters necessary for high order and monotonicity exponential Rosenbrock methods combined with monotonic 2nd order MUSCL scheme: monotonicity violated at high Courant number at Courant numbers below 1, second order Exponential Rosenbrock method combined with monotonic MUSCL scheme performs better than second order TVD Runge Kutta p. 12/2

13 Coupling to reaction terms system of 2 coupled advection diffusion reaction equations, simple Lotka-Volterra like population dynamics inital state with species at stable equilibrium in each single cell: stable equilibrium to be maintained by advection-diffusion process second order Exponential Rosenbrock method: reaction+advection (left), reaction+advection+diffusion (right), t beyond stability limits for explicit discretizations of both advection and diffusion p. 13/2

14 Inertial gravity waves, 1D linear, 1d shallow water equations with rotation, finite differences on 1D staggered mesh, horizontal scale 1 4 km, unbalanced initial datum with fluid at rest and bump in free surface comparison of exponential method, explicit backward-forward and semi-implicit centered Crank-Nicolson low Courant number solutions (left) vs. high Courant number solutions (right) 5 4 Exp BF SI 5 4 Exp BF SI x x 1 6 p. 14/2

15 Shallow water model on the sphere basis of the new generation MPI-Hamburg climate model triangular mesh derived by refinement of icosahedron mimetic spatial discretization proposed in L.B.,T. Ringler, MWR, 133, , 25 preserving mass and potential enstrophy comparison of second order exponential Rosenbrock method with semi-implicit discretizations error assessment using time dependent analytical solution of M. Läuter, D. Handorf, K. Dethloff, JCP, 21, , 25 2D inertial gravity waves test case 5 of Williamson test suite preliminary evaluation of accuracy and computational cost p. 15/2

16 Accuracy in test with analytical solution analytical solution of M. Läuter et al., 25 simulated up to t = 24 h. simulation on grid levels gl4 ( x 39 km ) to gl6 ( x 8 km), Courant number gh t 3.5 x L 2 errors on η and u : improvements by one order of magnitude in computations with Exponential Rosenbrock method η error u error SI EX SI EX gl4 4.85e-3 1.7e-3.13e- 2.28e-2 gl5 2.48e e e e-3 gl6 1.26e e e e-3 p. 16/2

17 Inertial gravity waves, 2D Courant number approx. 3.5, well resolved Rossby deformation radius, x 9 km initial datum with atmosphere at rest and large bump in free surface at North Pole (nonlinear regime) comparison of exponential method and semi-implicit leapfrog semi-implicit leapfrog solution (right) shows less accurate approximation of gravity wave at time t = 48 h. 3 E 6 E 9 E 12 E 15 E 18 E 15 W 12 W 9 W 6 W 3 W 9 N 3 E 6 E 9 E 12 E 15 E 18 E 15 W 12 W 9 W 6 W 3 W 9 N 75 N 6 N N 6 N N 45 N 3 N 3 N 15 N 15 S 15 N 15 S 3 S 45 S 6 S 75 S S 45 S 6 S 75 S S 9 S p. 17/2

18 Test case 5, I zonal flow over large idealized mountain Courant number approx. 6, well resolved Rossby deformation radius, x 9 km comparison of height fields computed by exponential method and NCAR spectral method at t = 15 days p. 18/2

19 Test case 5, II Courant number approx. 1, well resolved Rossby deformation radius, x 9 km comparison of height fields computed by exponential method and semi-implicit Crank Nicolson method at t = 15 days clear degradation of solution computed by the semi-implicit method p. 19/2

20 Conclusions... exponential integrators: an attractive technique for time discretization of PDEs describing environmental flows stability features similar to standard semi-implicit and semi-lagrangian methods, but clear potential for increased accuracy exactness for linear terms is especially appealing for environmental flows possibility to avoid operator splitting also seems a major advantage naive application of exponential integrators improves accuracy, but not yet efficiency: semi-implicit approach still faster p. 2/2

21 ...and open issues optimal choice of technique for computation of matrix exponentials and of the exponential integrator advantages do not apply immediately to monotonic advection: monotonicity is violated at high Courant numbers higher computational cost due to larger matrix to be handled with respect to semi-implicit methods application to high order spatial discretization methods possible use as predictor combined with more standard time integration techniques p. 21/2