Projection Methods for Rotating Flow Daniel Arndt Gert Lube Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics IACM - ECCOMAS 2014 Computational Modeling of Turbulent and Complex Flows with Applications I July 24th 2014
Overview 1 Introduction 2 Time discretization 3 Stabilized Spatial Discretization 4 Numerical Results Daniel Arndt Projection Methods for Rotating Flow 2
Overview 1 Introduction 2 Time discretization 3 Stabilized Spatial Discretization 4 Numerical Results Daniel Arndt Projection Methods for Rotating Flow 3
Setting Navier Stokes Equations in an Inertial Frame of Reference u t + (u )u ν u + p = f in Ω (0, T ] Ω R d bounded polyhedral domain u = 0 in Ω (0, T ] Navier Stokes Equations in a Rotating Frame of Reference v t + (v )v ν v + 2ω v + p = f in Ω (0, T ] v = 0 in Ω (0, T ] ω (ω r) = 1 2 (ω r)2 p = p 1 (ω r)2 2 Daniel Arndt Projection Methods for Rotating Flow 4
Setting Navier Stokes Equations in an Inertial Frame of Reference u t + (u )u ν u + p = f in Ω (0, T ] Ω R d bounded polyhedral domain u = 0 in Ω (0, T ] Navier Stokes Equations in a Rotating Frame of Reference v t + (v )v ν v + 2ω v + p = f in Ω (0, T ] v = 0 in Ω (0, T ] ω (ω r) = 1 2 (ω r)2 p = p 1 (ω r)2 2 Daniel Arndt Projection Methods for Rotating Flow 4
Overview 1 Introduction 2 Time discretization 3 Stabilized Spatial Discretization 4 Numerical Results Daniel Arndt Projection Methods for Rotating Flow 5
Pressure-Correction Scheme, Diffusion step Find ũ k+1 such that and satisfying 1 2 t (3ũk+1 4u k + u k 1 ) + (u ũ k+1 + 1 2 ( u )ũ k+1 ) ν 2 ũ k+1 + 2ω ũ k+1 + p k = f(t k+1 ) in Ω ũ k+1 = 0 on Ω. Daniel Arndt Projection Methods for Rotating Flow 6
Pressure-Correction Scheme, Projection step Find (u k+1, p k+1 ) such that and satisfying 1 2 t (3uk+1 3ũ k+1 ) + φ k+1 = 0 in Ω u k+1 = 0, u k+1 n Ω = 0 in Ω u k+1 n = 0 on Ω. The updated pressure is then given by where α {0, 1}. p k+1 = φ k+1 + p k αν ũ k+1 Daniel Arndt Projection Methods for Rotating Flow 7
Standard Pressure-Correction Scheme, Error Estimates Theorem (Rates of convergence for α = 0) Provided (u, p) is sufficiently smooth, (u t, ũ t, p t ) satisfies u u t l 2 ([L 2 (Ω)] d ) + u ũ t l 2 ([L 2 (Ω)] d ) C t 2 p p t l (L 2 (Ω)) + u ũ t l ([H 1 (Ω)] d ) C t Proof. Similar to the proof by Guermond and Shen in [GS04] Remark p k+1 n Ω =... = p 0 n Ω non-physical Neumann boundary condition leads to numerical boundary layer, reducing the order of convergence the splitting-error is of size O( t 2 ) Daniel Arndt Projection Methods for Rotating Flow 8
Rotational Pressure-Correction Scheme, Error Estimates Theorem (Rates of convergence for α = 1) Provided (u, p) is sufficiently smooth, (u t, ũ t, p t ) satisfies Proof. u u t l 2 ([L 2 (Ω)] d ) + u ũ t l 2 ([L 2 (Ω)] d ) C t 2 u u t l 2 ([H 1 (Ω)] d ) + u ũ t l 2 ([H 1 (Ω)] d ) C t 3 2 p p t l 2 (L 2 (Ω)) C t 3 2 Similar to the proof by Guermond and Shen in [GS04] Remark p k+1 n Ω = (f(t k+1 ) ν u k+1 ) n Ω is a consistent pressure boundary condition the splitting-error is of size O( t 2 ) Daniel Arndt Projection Methods for Rotating Flow 9
Overview 1 Introduction 2 Time discretization 3 Stabilized Spatial Discretization 4 Numerical Results Daniel Arndt Projection Methods for Rotating Flow 10
Weak Formulation Find U = (u, p) : (0, T ) V Q = [H0 1(Ω)]d L 2 0 (Ω), such that where ( t u, v) + A G (u, U, V) = (f, v) V = (v, q) V Q A G (w; U, V) := a G (U, V) + c(w; u, v) a G (U, V) := ν( u, v) + (2ω u, v) (p, v) + (q, u) c(w, u, v) := ((w )u, v) ((w )v, u) 2 Daniel Arndt Projection Methods for Rotating Flow 11
Local Projection Stabilization Idea Separate discrete function spaces into small and large scales Add stabilization terms only on small scales. Notations and prerequisites Family of shape-regular macro decompositions {M h } Let D M [L (M)] d denote a FE space on M M h for u h. For each M M h, let π M : [L 2 (M)] d D M be the orthogonal L 2 -projection. κ M = Id πh u fluctuation operator Averaged streamline direction u M R d : u M C u L (M), u u M L (M) Ch M u W 1, (M) Daniel Arndt Projection Methods for Rotating Flow 12
Local Projection Stabilization Idea Separate discrete function spaces into small and large scales Add stabilization terms only on small scales. Notations and prerequisites Family of shape-regular macro decompositions {M h } Let D M [L (M)] d denote a FE space on M M h for u h. For each M M h, let π M : [L 2 (M)] d D M be the orthogonal L 2 -projection. κ M = Id πh u fluctuation operator Averaged streamline direction u M R d : u M C u L (M), u u M L (M) Ch M u W 1, (M) Daniel Arndt Projection Methods for Rotating Flow 12
Assumptions Assumption (A.1) Consider FE spaces (V h, Q h ) satisfying a discrete inf-sup-condition: inf sup q Q h \{0} v V h \{0} ( v, q) v L 2 (Ω) q L 2 (Ω) β > 0 V h div := {v h V h ( v h, q h ) = 0 q h Q h } {0} Assumption (A.2) The fluctuation operator κ M = id π M provides the approximation property (depending on D M and s {0,, k}): κ M w L 2 (M) Ch l M w W l,2 (M), w W l,2 (M), M M h, l s. A sufficient condition for (A.2) is P s 1 D M. Daniel Arndt Projection Methods for Rotating Flow 13
Assumptions Assumption (A.3) Let the FE space V h satisfy the local inverse inequality v h L2 (M) Ch 1 M v h L2 (M) v h V h, M M h. Assumption (A.4) There are (quasi-)interpolation operators j u : V V h and j p : Q Q h such that for all M M h, for all w V [W l,2 (Ω)] d with 2 l k + 1: w j u w L 2 (M) + h M (w j u w) L 2 (M) Ch l M w W l,2 (ω M ) and for all q Q H l (M) with 2 l k on a suitable patch ω M M: q j p q L2 (M) + h M (q j p q) L 2 (M) Ch l M q W l,2 (ω M ) v j u v L (M) Ch M v W 1, (M) v [W 1, (M)] d. Daniel Arndt Projection Methods for Rotating Flow 14
Stabilization Terms LPS Streamline upwind Petrov-Galerkin (SUPG) s h (w h ; u, v) := τ M (w M )(κ M ((w M )u), κ M ((w M )v)) M M M h div-div t h (w h ; u, v) := M M h γ M (w M )( u, v) M LPS Coriolis stabilization a h (w h ; u h, v h ) := α M (w M )(κ(ω M u h ), κ(ω M v h )) M M M h Daniel Arndt Projection Methods for Rotating Flow 15
Stability and Existence Stabilized Problem Find U h = (u h, p h ) : (0, T ) V h div Q h, s.t. V h = (v h, q h ) V h div Q h ( tu h, v h ) + A G (u h ; U h, V h ) + (s h + t h + a h )(u h ; u h, v h ) = (f, v h ) Stability Define for V V Q the norm V 2 LPS := ν v 2 + (s h + t h + a h )(V, V). Then the following stability result holds: u h (t) 2 L 2 (Ω) + t 0 U h (s) 2 LPS ds u h (0) 2 L 2 (Ω) + 3 f 2 L 2 (0,T ;L 2 (Ω)) Corollary (using the generalized Peano theorem) discrete solution u h : [0, T ] V h div for the LPS model. Daniel Arndt Projection Methods for Rotating Flow 16
Stability and Existence Stabilized Problem Find U h = (u h, p h ) : (0, T ) V h div Q h, s.t. V h = (v h, q h ) V h div Q h ( tu h, v h ) + A G (u h ; U h, V h ) + (s h + t h + a h )(u h ; u h, v h ) = (f, v h ) Stability Define for V V Q the norm V 2 LPS := ν v 2 + (s h + t h + a h )(V, V). Then the following stability result holds: u h (t) 2 L 2 (Ω) + t 0 U h (s) 2 LPS ds u h (0) 2 L 2 (Ω) + 3 f 2 L 2 (0,T ;L 2 (Ω)) Corollary (using the generalized Peano theorem) discrete solution u h : [0, T ] V h div for the LPS model. Daniel Arndt Projection Methods for Rotating Flow 16
Stability and Existence Stabilized Problem Find U h = (u h, p h ) : (0, T ) V h div Q h, s.t. V h = (v h, q h ) V h div Q h ( tu h, v h ) + A G (u h ; U h, V h ) + (s h + t h + a h )(u h ; u h, v h ) = (f, v h ) Stability Define for V V Q the norm V 2 LPS := ν v 2 + (s h + t h + a h )(V, V). Then the following stability result holds: u h (t) 2 L 2 (Ω) + t 0 U h (s) 2 LPS ds u h (0) 2 L 2 (Ω) + 3 f 2 L 2 (0,T ;L 2 (Ω)) Corollary (using the generalized Peano theorem) discrete solution u h : [0, T ] V h div for the LPS model. Daniel Arndt Projection Methods for Rotating Flow 16
Theorem (A., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Let be u h (0) = j uu 0. Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νrem 2 + τ M u M 2 + dγ M + α M ω 2 L (M)hM 2 ( ) + τ M u M 2 + α M hm ω 2 2 L (M) h 2(s k) M u(τ) 2 W s+1,2 (ω M ) ] + tu(τ) 2 W k,2 (ω M ) dτ ) u(τ) 2 W k+1,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) Daniel Arndt Projection Methods for Rotating Flow 17
Theorem (A., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Let be u h (0) = j uu 0. Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νrem 2 + τ M u M 2 + dγ M + α M ω 2 L (M)hM 2 ( ) + τ M u M 2 + α M hm ω 2 2 L (M) h 2(s k) M u(τ) 2 W s+1,2 (ω M ) ] + tu(τ) 2 W k,2 (ω M ) dτ ) u(τ) 2 W k+1,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) Daniel Arndt Projection Methods for Rotating Flow 17
Theorem (A., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Let be u h (0) = j uu 0. Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νrem 2 + τ M u M 2 + dγ M + α M ω 2 L (M)hM 2 ( ) + τ M u M 2 + α M hm ω 2 2 L (M) h 2(s k) M u(τ) 2 W s+1,2 (ω M ) ] + tu(τ) 2 W k,2 (ω M ) dτ ) u(τ) 2 W k+1,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) Daniel Arndt Projection Methods for Rotating Flow 17
Choice of Parameters and Projection Spaces We achieve a method of order k provided τ M u M 2 h 2(s k) M νre 2 M C h M C ν u L (M) h 2(k s) M C τ M τ 0 u M 2 max{ 1 γ M, γ M } C γ M = γ 0 α M ω 2 L (M)h 2(1+s k) M h 2(k s 1) M C α M α 0 ω 2 L (M) Examples for suitable projection spaces One-Level: Q k /Q k 1 /Q t, P k /P k 1 /P t, Q k /P (k 1) /P t t k 1 Two-Level: P k /P k 1 /P t, Q k /Q k 1 /Q t, Q k /P (k 1) /P t t k 1 Daniel Arndt Projection Methods for Rotating Flow 18
Overview 1 Introduction 2 Time discretization 3 Stabilized Spatial Discretization 4 Numerical Results Daniel Arndt Projection Methods for Rotating Flow 19
Couzy Testcase 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Choose f such that the following pair is a solution in Ω = [0, 1] 2 : ( ( ) 1 u(x) = sin (πt) cos 2 πx ( ) 1 p(x) = π sin 2 πx sin ( 1 2 πy ( 1 sin 2 πy ) sin (πt). ) ( ) ( )) T 1 1, sin 2 πx cos 2 πy Daniel Arndt Projection Methods for Rotating Flow 20
Couzy Testcase, unstabilized 10 6 Ro=10 0 10 7 Ek=10 0 Ek=10 2 Ek=10 3 Ek=10 4 h 2 h 3 L 2 (u) error 10 8 10 9 10 10 10 2 10 1 h Daniel Arndt Projection Methods for Rotating Flow 21
Couzy Testcase, stabilized 10 6 Ro=10 0 10 7 Ek=10 0 Ek=10 2 Ek=10 3 Ek=10 4 h 2 h 3 L 2 (u) error 10 8 10 9 10 10 10 2 10 1 h Daniel Arndt Projection Methods for Rotating Flow 22
Couzy Testcase, unstabilized 10 4 Ro=10 0 10 5 Ek=10 0 Ek=10 2 Ek=10 3 Ek=10 4 h 1 h 2 H 1 (u) error 10 6 10 7 10 8 10 2 10 1 h Daniel Arndt Projection Methods for Rotating Flow 23
Couzy Testcase, stabilized 10 4 Ro=10 0 10 5 Ek=10 0 Ek=10 2 Ek=10 3 Ek=10 4 h 1 h 2 H 1 (u) error 10 6 10 7 10 8 10 2 10 1 h Daniel Arndt Projection Methods for Rotating Flow 24
Numerical Results, Rotating Poiseuille Flow Ω = [ 2, 2] [ 1, 1] { (1 y 2, 0) T, x = 2 u(x, y) = (0, 0) T, ( u n)(x = 2, y) = 0, y = 1 u 0 = 0, p 0 = 0, f = 0 ω = (0, 0, 100), ν = 10 3 Flow for the parameters ω = (0, 0, 1), ν = 10 1 Daniel Arndt Projection Methods for Rotating Flow 25
Rotating Poiseuille Flow, div-div Daniel Arndt Projection Methods for Rotating Flow 26
Rotating Poiseuille Flow, div-div Daniel Arndt Projection Methods for Rotating Flow 27
Rotating Poiseuille Flow, div-div adaptive Daniel Arndt Projection Methods for Rotating Flow 28
Rotating Poiseuille Flow, div-div adaptive Daniel Arndt Projection Methods for Rotating Flow 29
Rotating Poiseuille Flow, Coriolis Daniel Arndt Projection Methods for Rotating Flow 30
Rotating Poiseuille Flow, Coriolis Daniel Arndt Projection Methods for Rotating Flow 31
Rotating Poiseuille Flow, SUPG Daniel Arndt Projection Methods for Rotating Flow 32
Rotating Poiseuille Flow, SUPG Daniel Arndt Projection Methods for Rotating Flow 33
Rotating Poiseuille Flow, SUPG Coriolis Daniel Arndt Projection Methods for Rotating Flow 34
Rotating Poiseuille Flow, SUPG Coriolis Daniel Arndt Projection Methods for Rotating Flow 35
Rotating Poiseuille Flow, SUPG Coriolis Adaptive Daniel Arndt Projection Methods for Rotating Flow 36
Rotating Poiseuille Flow, SUPG Coriolis Adaptive Daniel Arndt Projection Methods for Rotating Flow 37
Summary Temporal Discretization Using a BDF2 approach Second order convergence in the velocity w.r.t l 2 ([L 2 (Ω)] d ) Convergence of order 3/2 in the velocity w.r.t l 2 ([H 1 (Ω)] d ) and in the pressure w.r.t l 2 ([L 2 (Ω)] d ) Spatial Discretization The Local Projection Stabilization approach provides Stability and Existence Quasi-optimal error estimates for standard discretizations (e.g. Taylor-Hood) Numerical results confirm analytical estimates Daniel Arndt Projection Methods for Rotating Flow 38
References A. ; Dallmann, Helene ; Lube, Gert: Local Projection FEM Stabilization for the Time-dependent Incompressible Navier-Stokes Problem. In: submitted to Numerical Methods for Partial Differential Equations (2014) Guermond, J ; Shen, Jie: On the error estimates for the rotational pressure-correction projection methods. In: Mathematics of Computation 73 (2004), Nr. 248, S. 1719 1737 Daniel Arndt Projection Methods for Rotating Flow 39
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