Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Split-explicit time integration methods in numerical weather prediction Oswald Knoth, J¨ org Wensch HYP 2012 Padova
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Introduction 1 Motivation Dry Euler equations Linearized equations Splitting methods 2 Splitting Split explicit methods 3 Methods Generalized Runge-Kutta 4 Approach Order conditions Numerische Tests 5 Test case of Klein Nonhydrostatic Case of Blossey/Durran Gravity waves with WRF Conclusion 6
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Introduction 1 Motivation Dry Euler equations Linearized equations Splitting methods 2 Splitting Split explicit methods 3 Methods Generalized Runge-Kutta 4 Approach Order conditions Numerische Tests 5 Test case of Klein Nonhydrostatic Case of Blossey/Durran Gravity waves with WRF Conclusion 6
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Motivation Motivation: Atmospheric models contain slow (advection) and fast (gravity and sound wave) modes. Meteorologically important: Medium and low frequencies CFL-number of fast waves restricts time step Pure advection allows larger step sizes CFL ADVECTION / CFL SOUND ≤ 1 / 10 Apply multirate strategy slow processes are integrated by large time steps fast processes are integrated by small time steps where the slow (advective) tendencies are fixed The linearized, discretized, one-dimensional compressible Euler equations serve as the model equation set for examining the stability of the integration schemes
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Dry Euler equations Dry 2D Euler equations in conservative form: ∂ρ ∂ t = − ∂ρ u ∂ x − ∂ρ w ∂ z ∂ρ u = − ∂ρ uu − ∂ρ wu − ∂ p ∂ t ∂ x ∂ z ∂ x ∂ρ w = − ∂ρ uw − ∂ρ ww − ∂ p ∂ z − ρ g ∂ t ∂ x ∂ z ∂ρθ ∂ t = − ∂ρ u θ − ∂ρ w θ ∂ x ∂ z Prognostic variables are density ρ and the products of density with winds u , w and potential temperature θ . Pressure p is a diagnostic variable from the equation of state 1 � R ρθ 1 − κ , � p = p κ 0 with κ = R c p , R gas constant for dry air, c p the heat capacity of dry air at constant pressure and p 0 the pressure at ground, g is the acceleration of gravity.
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Linearized equations Test equations for linear stability analysis The approximate, quasi-Boussinesq linearized equations u t = − c s p x − Uu x w t = − c s p z − Uw x − N θ θ t = − Nw − Uu x p t = − c s ( u x + w z ) − Up x where c s >> U . One dimensional acoustic advection system u t = − c s p x − Uu x p t = − c s u x − Up x
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Introduction 1 Motivation Dry Euler equations Linearized equations Splitting methods 2 Splitting Split explicit methods 3 Methods Generalized Runge-Kutta 4 Approach Order conditions Numerische Tests 5 Test case of Klein Nonhydrostatic Case of Blossey/Durran Gravity waves with WRF Conclusion 6
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Splitting Time integration methods for y = f ( y ) + g ( y ) ˙ with y (0) = y 0 where f represents the energetically relevant slow mode (advection, Rossby waves) and g the fast mode (sound waves, gravity waves). To integrate the fast system, the forward–backward or Stoermer-Verlet method is used. For a symplectic structure u = g u ( p ) ˙ p = g p ( u ) ˙ the FB scheme reads u n +1 = u n + ∆ τ g u ( p n ) p n +1 = v n + ∆ τ g p ( u n +1 ) FB is of second order and in connection with staggered central differences is stable for a CFL–condition c s ∆ τ ∆ x ≤ 1 .
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Splitting, the linearized equation The approximate, quasi-Boussinesq linearized equations u t = − c s p x − Uu x w t = − c s p z − Uw x − N θ θ t = − Nw − Uu x p t = − c s ( u x + w z ) − Up x One dimensional acoustic advection system u t = − c s p x − Uu x p t = − c s u x − Up x
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Splitting, the nonlinear equation Splitting in the dry 2D Euler equation: ∂ρ ∂ t = − ∂ρ u ∂ x − ∂ρ w ∂ z ∂ρ u = − ∂ρ uu − ∂ρ wu − ∂ p ∂ t ∂ x ∂ z ∂ x ∂ρ w = − ∂ρ uw − ∂ρ ww − ∂ p ∂ z − ρ g ∂ t ∂ x ∂ z ∂ρθ ∂ t = − ∂ρ u θ − ∂ρ w θ ∂ x ∂ z y = F ( y , y ) ˙
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Splitting, the nonlinear ”linearized¨ equation Splitting in the dry ”pressure linearized”2D Euler equation: ∂ t = − ∂ρ u ∂ρ ∂ x − ∂ρ w ∂ z ∂ρ u = − ∂ρ uu − ∂ρ wu − ∂ p ∂ρθ ∂ t ∂ x ∂ z ∂ρθ ∂ x ∂ρ w = − ∂ρ uw − ∂ρ ww − ∂ p ∂ρθ ∂ z − ρ g ∂ t ∂ x ∂ z ∂ρθ ∂ρθ ∂ t = − ∂ρ u θ − ∂ρ w θ ∂ x ∂ z y = F ( y ) + A ( y ) y ˙
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Linearized test equation for stability considerations Discretize linear one-dimensional acoustic equation in space Advection: Third order upwinding, Acoustic: Central differences Apply Fourier decomposition We obtain a 2 by 2 linear ODE for each Fourier component ˙ y = Ly + Ny
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Split explicit methods, Wicker/Skamarock Wicker and Skamarock (MWR 2002) used a three-stage Runge-Kutta method as slow integrator: u n +1 / 3 = u n + ∆ t 3 f u ( u n ) p n +1 / 3 = p n + ∆ t 3 f p ( p n ) u n +1 / 2 = u n + ∆ t 2 f u ( u n +1 / 3 ) p n +1 / 2 = p n + ∆ t 2 f p ( p n +1 / 3 ) u n +1 = u n + ∆ tf u ( u n +1 / 2 ) p n +1 = p n + ∆ tf p ( p n +1 / 2 )
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion Split explicit methods, Wicker/Skamarock Resulting splitting scheme: u = u n , p = p n for k = 1 : n s / 3 u = u + ∆ τ g u ( p ) + ∆ τ f u ( u n ) p = p + ∆ τ g p ( u ) + ∆ τ f p ( p n ) end u n +1 / 3 = u , p n +1 / 3 = p , u = u n , p = p n for k = 1 : n s / 2 u = u + ∆ τ g u ( p ) + ∆ τ f u ( u n +1 / 3 ) p = p + ∆ τ g p ( u ) + ∆ τ f p ( p n +1 / 3 ) end u n +1 / 2 = u , p n +1 / 2 = p , u = u n , p = p n for k = 1 : n s u = u + ∆ τ g u ( p ) + ∆ τ f u ( u n +1 / 2 ) p = p + ∆ τ g p ( u ) + ∆ τ f p ( p n +1 / 2 ) end u n +1 = u , p n +1 = p
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion General Runge-Kutta Methods for i = 1 : s + 1 F := � a ij ∆ τ := ∆ t y := y 0 , c i f ( y j ) , n s for k = 1 : c i n s y := y + ∆ τ g ( y ) + ∆ τ F end y i := y end Underlying Runge–Kutta method: 0 c 2 a 21 c i a i 1 . . . a ii − 1 c s a s 1 a ss − 1 1 . . . a s +11 a s +1 s RK3 after L.J. Wicker and W.C. Skamarock: Time-Splitting Methods for Elastic Models Using Forward Time Schemes, MWR, 2002. 0 1/3 1/3 1/2 0 1/2 0 0 1
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion General fast integrator Assume that we can solve the fast part of y = f ( y ) + g ( y ) ˙ analytically Then a split Runge–Kutta method reads: Z ni (0) = y n i − 1 ∂τ Z ni ( τ ) = 1 ∂ � a ij f ( Y nj ) + g ( Z ni ( τ )) c i j =1 Y ni = Z ni ( c i h ) , y n +1 = Y n , s +1 For the nonlinear case Z ni (0) = y n i − 1 ∂τ Z ni ( τ ) = 1 ∂ � a ij F ( Y nj , Z ni ( τ )) c i j =1 Y ni = Z ni ( c i h ) , y n +1 = Y n , s +1
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion General fast integrator Application of a split-explicit Runge-Kutta method to the linear test y = Ly + Ny yields stability matrix M ∈ C 2 × 2 : equation ˙ y n +1 = My n M depends on: wave number k Number of small time steps n s CFL number for advection U ∆ t ∆ x CFL number for sound c s ∆ t ∆ x Spectral radius of M as a function of the two CFL numbers by n s = 10 or n s = inf. Line has slope 1/4, below the line U < c s 4 ≈ 85m/s ≈ 340m/s.
Introduction Splitting methods Split explicit methods Generalized Runge-Kutta Numerische Tests Conclusion General fast integrator Stability plot for RK3, exact fast integration: Resulting CFL restrictions: U ∆ t ∆ x ≤ 1 . 7 → ∆ t ≤ 6 . 8 s c s n s ∆ τ ≤ 3 . 1 → ∆ t ≤ 1 . 8 s ∆ x
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