Construction of very high order Residual Distribution Schemes for steady problems

Transcription

1 Construction of very high order Residual Distribution Schemes for steady problems Rémi Abgrall, Mario Ricchiuto, Cédric Tavé, Nadège Villedieu and Herman Deconinck Mathématiques Appliquées de Bordeaux, Projet Scalapplix, INRIA FutURs, Talence, France von Karman Institute for Fluid Dynamics, Rhodes St Genese, Belgium Construction of very high order Residual Distribution Schemesfor steady problems p.1/44

2 Overview 1. RDS schemes 2. Construction on scalar advection 3. Numerical examples Scalar Systems Construction of very high order Residual Distribution Schemesfor steady problems p.2/44

3 Forewords This lecture foccusses on steady problems. Several JCP papers on second order unsteady. Existing work on coupling hyperbolic/convective terms to viscous one (Villedieu Ricchiuto) Discussion on scalar problems. Construction of very high order Residual Distribution Schemesfor steady problems p.3/44

4 λ u = 0 u = g x Ω x Γ unstructured triangular meshes, triangles T, u σ u(m σ ), σ d.o.f scheme u n+1 σ = u n σ ω i T,σ T steady solution i.e. n +. Φ T σ. Construction of very high order Residual Distribution Schemesfor steady problems p.4/44

5 Second order σ : vertices of the triangles u h piecewise linear interpolant of {u σ }, Conservation relation Φ σ = λ u h dx := Φ T σ T T = kσ T u σ k σ = σ T T λ Nσ Construction of very high order Residual Distribution Schemesfor steady problems p.5/44

6 Example : piecewise interpolation Upwind finite volume scheme, N scheme Local Rusanov, Φ σ = k σ + ( uσ ũ ) ( ) 1 ( ũ = Φ σ = 1 3 σ k σ ( Φ T + α σ T σ σ,σ T k σ u σ ) ) (u σ u σ ) Construction of very high order Residual Distribution Schemesfor steady problems p.6/44

7 Example : piecewise interpolation The upwind FV scheme, N scheme and Rusanov scheme are monotone schemes Φ σ = σ σ c σσ (u σ u σ ), c σσ 0. Provides a local maximum principle on the solution Construction of very high order Residual Distribution Schemesfor steady problems p.7/44

8 Summary Conservation property Φ σ = σ T T λ u h dx := Φ T Second order approximation of the flux (here because u h = u + O(h 2 ). LP condition (Φ T σ (u) = O(h 3 ) for the exact solution) Monotonicity preserving scheme (Φ σ = c σσ (u σ u σ )) σ technique for going from first order to second order. scheme : Φ T σ = 0 T,σ T Construction of very high order Residual Distribution Schemesfor steady problems p.8/44

9 How to get really high order, compact, monotonicity preserving schemes on general meshes? Use of P k interpolation, Improved conservation relation Probably more general than this (Hermite, spectral, etc) Construction of very high order Residual Distribution Schemesfor steady problems p.9/44

10 Other contributions Abgrall-Roe, J. Scientific Computing, 2001 Hubbard (Computer and Fluids, 2005) Ricchiuto et al (VKI LS, 2005) Andrianov et al. (IJNM, 2005) De Palma et al. (JCP in press) Construction of very high order Residual Distribution Schemesfor steady problems p.10/44

11 Notations degrees of freedom mesh τ, triangles T, verticesm j, we seek a solution that is piecewise polynomial of degree k in each triangle, need to provide (k + 1)(k + 2)/2 degrees of freedom : Scheme σ T Φ T σ = 0 Construction of very high order Residual Distribution Schemesfor steady problems p.11/44

12 Example, k = 2 Degree of freedom P 2 interpolation. Construction of very high order Residual Distribution Schemesfor steady problems p.12/44

13 Structural condition : Φ T := T Lax Wendroff like result+ high order div F h (u h )dx = + standard assumption Lax Wendroff solution of T λ u h dx = σ T Φ T σ (u h ), div F(u) = 0 + boundary conditions High order if Φ T σ (u h ) = O(h 2+k ) for smooth solutions+ regular meshes Construction of very high order Residual Distribution Schemesfor steady problems p.13/44

14 Monotonicity condition Ensures stability in L with Φ σ = c σσ (u σ u σ ) σ σ,σ T c σσ 0 dependant on the solution. Construction of very high order Residual Distribution Schemesfor steady problems p.14/44

15 Previous try ( Abgrall-Roe, JSC, 2001) define sub-triangulation use a reference monotone first order scheme Φ L,T σ sub triangulation define sub residuals in each sub-triangle accordingly u n+1 σ = u n σ ω i T,σ T Ψ T σ, Ψ T σ = T T T,σ T T in each ( ) Φ T T σ Construction of very high order Residual Distribution Schemesfor steady problems p.15/44

16 First try We can construct the N scheme on the sub triangles of T : Φ T ξ σ for ξ = I, II, III, IV Consider clear that linear Φ ξ = T ξ λ u h. Φ ξ Φ T ξ σ σ T ξ quadratic but we still can use the N scheme for a comparison purpose and want. Ψ T ξ σ = βσφ ξ ξ Construction of very high order Residual Distribution Schemesfor steady problems p.16/44

17 First try Main problem : Φ L,T σ σ T T λ u h = σ T ( ) Φ T T σ Problem is not conservation but algebraic Construction of β T σ needs this condition. Construction of very high order Residual Distribution Schemesfor steady problems p.17/44

18 An always defined solution Start (for example) from the Rusanov scheme limit residuals update Construction of very high order Residual Distribution Schemesfor steady problems p.18/44

19 Rusanov scheme Φ T σ = 1 6 ( T [ λ n]u h dx + α T σ T ) (u σ u σ ) with monotone scheme. α T max σ T T λ Nσ dx. Construction of very high order Residual Distribution Schemesfor steady problems p.19/44

20 limiting procedure For Rusanov scheme, Φ T σ = σ T define x σ = Φ R σ /Φ T, T [ λ n]u h dx = Φ T. set β T σ = x+ σ σ x + σ and ΦT σ := βt σ ΦT. always defined because σ Φ H,T σ = Φ T implies σ x σ = 1 so that σ x + σ 1. Construction of very high order Residual Distribution Schemesfor steady problems p.20/44

21 Properties If there exists a unique solution to T σ u σ = g Φ H,T σ = 0 σ Ω ( λ u = 0) on Γ then the scheme is third order accurate because λ u h dl = O(h 3+1 ) T Construction of very high order Residual Distribution Schemesfor steady problems p.21/44

22 Numerical experiments convection Construction of very high order Residual Distribution Schemesfor steady problems p.22/44

23 Numerical experiments convection solid rotation Construction of very high order Residual Distribution Schemesfor steady problems p.22/44

24 Same second order convection solid rotation Construction of very high order Residual Distribution Schemesfor steady problems p.23/44

25 Why : existence of mild spurious mode Analysis : see Abgrall, Essentially non oscillatory Residual Distribution schemes for hyperbolic problems, JCP, Interpretation Construction of very high order Residual Distribution Schemesfor steady problems p.24/44

26 Interpretation First order scheme λ Construction of very high order Residual Distribution Schemesfor steady problems p.25/44

27 Interpretation Φ σ = β σ Φ T λ This problem does not exist for genuinely upwind schemes Construction of very high order Residual Distribution Schemesfor steady problems p.26/44

28 Fix Force some upwinding : add Θ(u h )h T )( ) ( λ Nσ λ u h Construction of very high order Residual Distribution Schemesfor steady problems p.27/44

29 Fix Scheme T σ u σ = g ( ) Φ H,T σ = 0 σ Ω ( λ u = 0) on Γ with ( ) ( Φ H,T σ = βσ T T ) λ udx + Θ(u h )h T )( ) ( λ Nσ λ u h Still third order accurate, not any more (formaly) monotonicity preserving but essentially non oscillatory Construction of very high order Residual Distribution Schemesfor steady problems p.28/44

30 Numerical experiments convection solid rotation Construction of very high order Residual Distribution Schemesfor steady problems p.29/44

31 Fourth order Same. Replace quadratic element by cubic elements (10 dof/element) Construction of very high order Residual Distribution Schemesfor steady problems p.30/44

32 Accuracy : convection Grid Convergence LN LFLS_P1 LFLS_P2 LFLS_P Construction of very high order Residual Distribution Schemesfor steady problems p.31/44

33 Accuracy/ # of dof -1 LFLS_P1 LFLS_P2 LFLS_P log( L1-Error ) e+06 log( #DOF ) Construction of very high order Residual Distribution Schemesfor steady problems p.32/44

34 Burgers equation (4th order) u t u 2 x = 0, u(x, y) = 1.5 2x on inflow boundaries bu_methode_lxfp3_maillage_p desc Construction of very high order Residual Distribution Schemesfor steady problems p.33/44

35 Burgers equation (2nd/3rd order) same number of dof. bu_methode_lxfp3_maillage_p desc bu_methode_lxfp1_maillage_p desc Construction of very high order Residual Distribution Schemesfor steady problems p.34/44

36 Comparison, Fourth order Second order Fourth order Second order Construction of very high order Residual Distribution Schemesfor steady problems p.35/44

37 Comparison, 2 2nd order 4th order Construction of very high order Residual Distribution Schemesfor steady problems p.36/44

38 Fluid Mechanics examples Stabilisation procedure : on characteristic waves. Similar as Abgrall-Mezine, Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys., 195(2): , additionnal stabilisation : formal extension to systems Construction of very high order Residual Distribution Schemesfor steady problems p.37/44

39 Jet (3rd order) ρ u v p Construction of very high order Residual Distribution Schemesfor steady problems p.38/44

40 Jet (3rd order) without dissipation test desc Construction of very high order Residual Distribution Schemesfor steady problems p.39/44

41 O1/O2/O3, same dof O3 O2 O1 Construction of very high order Residual Distribution Schemesfor steady problems p.40/44

42 4 state shock tube Construction of very high order Residual Distribution Schemesfor steady problems p.41/44

43 4 state shock tube Construction of very high order Residual Distribution Schemesfor steady problems p.42/44

44 Conclusions, perpectives. Residual distribution of high order (3rd, 4th), Essentially non oscillatory, Preliminary results for fluid mechanics Construction of very high order Residual Distribution Schemesfor steady problems p.43/44

45 To be done Avoid additional stabilisation? (system) Boundary treatement Efficiency Unsteady Construction of very high order Residual Distribution Schemesfor steady problems p.44/44

46 To be done Avoid additional stabilisation? (system) Boundary treatement Efficiency Unsteady ADIGMA! Construction of very high order Residual Distribution Schemesfor steady problems p.44/44