Introduction to Finite Volume projection methods. On Interfaces with non-zero mass flux

Transcription

1 Introduction to Finite Volume projection methods On Interfaces with non-zero mass flux Rupert Klein Mathematik & Informatik, Freie Universität Berlin Summerschool SPP 1506 Darmstadt, July 09, 2010

2 Introduction to Finite Volume projection methods The finite-volume principle Upwinding and dissipation for convection-dominated flows Divergence constraints from the zero-mach number limit Projection and auxiliary variable schemes

3 Finite Volume Principle Mass Conservation for arbitrary Ω Ω Ω Ω σ n t v n v t (v n) Change of a volume s mass with time Figure 1.1: Change of a volume s mass with time M(t; Ω) = If, in addition, ϱv satisfies the conditions of Gauß integral theorem, (see appendix A), then Ω (ϱv) n dσ = Ω (ϱv) dv,and Ω ( ) ϱ t + (ϱv) dv =0 for arbitrary t2 Gauß domains Ω. (1.4) Ω M(t 2 ; Ω) M(t 1 ; Ω) = ρ(t, x) dv (ρv) n dσ dt This equation can hold, for continuously differentiablet 1 fields Ω ϱ, v and for arbitrary control volumes Ω only, if pointwise the following partial differential equation is satisfied:

4 Finite Volume Principle General Conservation Law for arbitrary Ω Ω Ω Ω σ n t v n v t (v n) Change of a volume s mass with time Figure 1.1: Change of a volume s mass with time Φ(t; Ω) = If, in addition, ϱv satisfies the conditions of Gauß integral theorem, (see appendix A), then Ω (ϱv) n dσ = Ω (ϱv) dv,and Ω ( ) ϱ t + (ϱv) dv =0 for arbitrary t2 Gauß domains Ω. (1.4) Ω Φ(t 2 ; Ω) Φ(t 1 ; Ω) = ρφ(t, x) dv F φ n dσ dt This equation can hold, for continuously differentiablet 1 fields Ωϱ, v and for arbitrary control volumes Ω only, if pointwise the following partial differential equation is satisfied:

5 Finite Volume Principle General Conservation Law for cells of a grid Change of a volume s mass with time Φ(t; Ω) = Ω Φ(t 2 ; Ω) Φ(t 1 ; Ω) = ρφ(t, x) dv t2 t 1 Ω F φ n dσ dt

6 Finite Volume Principle Tasks: Approximation of fluxes F φ Recovery of detailed φ-distributions from cell averages Robust & accurate time integration Key advantage: Based on fundamental conservation principle valid, in principle, even for under-resolved solutions example: shock capturing schemes for hyperbolic systems Disadvantages: Recovery difficult on general unstructured meshes (but: see, e.g., recent work by M. Dumbser et al.) Rigorous theory less advanced than that of FEM methods (but: see recent developments of Discountinuous Galerkin methods)

7 Introduction to Finite Volume projection methods The finite-volume principle Upwinding and dissipation for convection-dominated flows Divergence constraints from the zero-mach number limit Projection and auxiliary variable schemes

8 Upwinding and Dissipation Burgers Equation or u t + ( ) u 2 /2 = 0 x ( t + u x ) u = 0 smooth solutions: u constant along characteristics dx ch /dt = u. characteristic-based solution u(t, x) = u(0, x t u(t, x)) There cannot be any over/undershoots beyond the initial data!!

9 Upwinding and Dissipation Upwind scheme: F u = ( u 2 /2 ) upwind Solution at time t = 0.1 Solution at time t = 0.2 Solution at time t = q 0 q 0 q x Central scheme: F u = 1 2 Solution at time t = 0.1 [( u 2 /2 ) L + ( u 2 /2 ) R] x Solution at time t = x q q 0 your guess what happens next x x

10 Upwinding and Dissipation Conclusion: Upwinding introduces physically correct dissipation

11 Introduction to Finite Volume projection methods The finite-volume principle Upwinding and dissipation for convection-dominated flows Divergence constraints from the zero-mach number limit Projection and auxiliary variable schemes

12 Divergence Constraints Incompressible flow with variable density Leading order result: p = P + M 2 p Consequences: ρe = p γ 1 + M2ρv2 2 P γ 1 (M 0) (ρe) t + ([ρe + p]v) = 0 v (0) = 0 ρ t + (ρv) = 0 ρ (0) t + v (0) ρ (0) = 0 (ρv) t + (ρv v) + 1 M2 p = 0 (ρv)(0) t + (ρv v) (0) + p = 0 P and p must be maintained as primary variables in a low-mach numerical implementation

13 Introduction to Finite Volume projection methods The finite-volume principle Upwinding and dissipation for convection-dominated flows Divergence constraints from the zero-mach number limit Projection and auxiliary variable schemes

14 Projection and Aux-Variable schemes Compressible flow equations: ρ t + (ρv) = 0 (ρv) t + (ρv v) + ΓP π = ρgk P t + (P v) = 0 P = p 1 γ = ρθ, π = p/p, Γ = c p /R

15 Projection and Aux-Variable schemes Pseudo-incompressible model: (Durran (1988)) ρ t + (ρv) = 0 (ρv) t + (ρv v) + ΓP π = ρgk + (P v) = 0 P P (z), ρθ = P (z), θ = θ(z) + θ

16 Projection and Aux-Variable schemes Auxiliary variable projection scheme predictor step Solve over t by second-order (upwind FV) scheme: ρ t + (ρv) = 0 (ρv) t + (ρv v) + ΓP π n = ρgk Provides: P t + (P v) = 0 second-order advection of {θ, v} no divergence control imposed! second-order gravity term first-order pressure gradient effect P P n = O(( t) 2 ) incompr. case, 2nd-ord.: Schneider et al., JCP 99; 4th-ord.: Kadioglu, Minion, Klein, JCP 08

17 Projection and Aux-Variable schemes Auxiliary variable projection scheme predictor step Solve over t by second-order (upwind FV) scheme: ρ t + (ρv) = 0 (ρv) t + (ρv v) + ΓP π n = ρgk Provides: P t + (P v) = 0 second-order advection of {θ, v} no divergence control imposed! second-order gravity term first-order pressure gradient effect P P n = O(( t) 2 )

18 Projection and Aux-Variable schemes Auxiliary variable projection scheme predictor step Solve over t by second-order (upwind FV) scheme: ρ t + (ρv) = 0 (ρv) t + (ρv v) + ΓP π n = ρgk Provides: P t + (P v) = 0 second-order advection of {θ, v} no divergence control imposed! second-order gravity term first-order pressure gradient effect P P n = O(( t) 2 )

19 Projection and Aux-Variable schemes Auxiliary variable projection scheme MAC-projection Constraint (P v) n+1 2 = 0 δπ = π n+1 2 π n Advection velocity correction (at cell interfaces) Divergence control (P v) n+1 2 = (P v) t 2 P θ δπ 1 P n (P θ δπ) = 2 1 ( t) 2 P (P P n ) n Post-correction of all advective flux contributions

20 Projection and Aux-Variable schemes Auxiliary variable projection scheme MAC-projection MAC-projection controls advection fluxes across grid cell interfaces Advection velocities result from some interpolation from the cell-centers Danger: pressure-velocity decoupling Previous approaches: Rhie-Chow stabilization Exact projection Approximate projection compromises mass continuity non-compact stencils inexact divergence control

21 Projection and Aux-Variable schemes Alternative route : cell-centered exact projection Exact control of divergence on dual cells Piecewise linear, discontinuous ansatz functions for momentum on primary cells Bi-/trilinear pressure ansatz functions S. Vater, R.K., Num. Math., 113, (2009)

22 Projection and Aux-Variable schemes Stabilizing Second projection : inf-sup-stable Petrov-Galerkin scheme for generalized saddle-point problem Find (u, p) (X 2 M 1 ), such that a(u, v) + b 1 (v, p) = f, v v X 1 b 2 (u, q) = g, q q M 2 where a(u, v) = (u, v) b 1 (v, p) = (v, p) b 2 (u, q) = ( u, q) (result currently available for 2D shallow water / homentropic gasdynamics) S. Vater, R.K., Num. Math., 113, (2009); using Nicolaïdes (1982), Bernardi et al. (1988)

23 Projection and Aux-Variable schemes Summary of auxiliary-variable projection method Predictor auxiliary unconstrained system second-order upwind finite volume scheme well-balanced discretization of gravity term MAC-projection controls advective flux divergence brings P back to P (z) cell-centered projection stabilizes checkerboard p v modes generates second-order accuracy w.r.t. pressure Background nowhere invoked explicitly!!

24 Projection and Aux-Variable schemes Last remark and warning: Projection does not preserve the no-slip wall boundary condition! Therefore, the quality of maintaining this b.c. depends on the accuracy of the predictor step. See BCM (2001) for a detailed discussion. Brown, Cortez & Minion, JCP, 168, (2001)