A Review of Solution Techniques for Unsteady Incompressible Flow

Zeist 2009 p. 1/57 A Review of Solution Techniques for Unsteady Incompressible Flow David Silvester School of Mathematics University of Manchester

Zeist 2009 p. 2/57 PDEs Review : 1966 1999 Update : 2000 2009 Outline

Zeist 2009 p. 3/57 PDEs Outline u t + u u ν 2 u + p = 0 u = 0 Navier Stokes

Zeist 2009 p. 4/57 PDEs Outline u t + u u ν 2 u + p = 0 u = 0 Navier Stokes u t + u u ν 2 u + p = jt u = 0 T t + u T ν 2 T = 0 Boussinesq

Zeist 2009 p. 5/57 Navier-Stokes Equations u t + u u ν 2 u + p = 0 u = 0 in W Ω (0,T) in W Boundary and Initial conditions u = g on Γ D [0,T]; ν u n p n = 0 u( x, 0) = u 0 ( x) in Ω. on Γ N [0,T];

Zeist 2009 p. 6/57 Finite element matrix formulation Introducing the basis sets X h = span{ φ i } n u i=1, Velocity basis functions; M h = span{ψ j } n p j=1, Pressure basis functions. gives the method-of-lines discretized system: ( ) ( ) ( ) ( ) M 0 u N( u) + νa B T u + 0 0 B 0 p t p t with associated matrices N ij = ( u φ i, φ j ), A ij = ( φ i, φ j ), convection diffusion B ij = ( φ j,ψ i ), divergence. = ( ) f 0

Example: Flow over a Step Zeist 2009 p. 7/57

Zeist 2009 p. 8/57 PDEs Review : 1966 1999 Update : 2000 2009 Outline

Zeist 2009 p. 9/57 PDEs Outline Review : 1966 1999 1967 1982 1991 1999

Zeist 2009 p. 10/57 Spatial Discretization I Suppose that Ω R 2. Introducing u = (u x,u y ) gives u x t + u y t + ( u x x + u y ( u x x + u y With a discrete analogue.. ) u x ν 2 u x + p y ) y x = 0 u y ν 2 u y + p y = 0 ( ux x + u ) y = 0 y

Zeist 2009 p. 10/57 Spatial Discretization I Suppose that Ω R 2. Introducing u = (u x,u y ) gives u x t + u y t + ( u x x + u y ( u x x + u y ) u x ν 2 u x + p y ) y x = 0 u y ν 2 u y + p y = 0 ( ux x + u ) y = 0 y With a discrete analogue.. M 0 0 0 M 0 + 0 0 0 α x t α y t α p t F 0 Bx T 0 F By T B x B y 0 α x α y α p = 0 0 0

Zeist 2009 p. 11/57 Spatial Discretization II The method-of-lines discretized system is a semi-explicit system of DAEs: M 0 0 F 0 Bx T 0 M 0 + 0 F By T = 0 0 0 B x B y 0 α x t α y t α p t α x α y α p 0 0 0 The DAEs have index equal to two The discrete problem is nonlinear F := νa + N(α)

Zeist 2009 p. 12/57 Spatial Discretization III... to give an index one DAE system: α x t M 1 F 0 M 1 Bx T α y t + 0 M 1 F M 1 By T 0 B x M 1 F B y M 1 F A p α x α y α p = 0 0 0 The matrix A p := B x M 1 B T x + B y M 1 B T y is the (consistent) Pressure Poisson matrix. Explicit approximation in time gives a decoupled formulation. Diagonally implicit approximation in time gives a segregated (SIMPLE-like) formulation. Implicit approximation in time does not look attractive!

Zeist 2009 p. 13/57 ( ) ( ) M 0 u 0 0 t p t + ( N( u) + νa B T B 0 ) ( ) u p = ( ) f 0 1967

Zeist 2009 p. 14/57 A simple splitting/projection approach I ( ) ( ) M 0 u + 0 0 t p t ( F( u) B T B 0 ) ( ) u p Given a time step dt and a parameter 0 < γ 2 = ( ) f Algorithm: Chorin, 1967; Temam, 1969 for k = 0, 1,... solve M u t + F( u k ) u = f B T p k solve BM 1 B T φ = B u 0 compute compute end u k+1 = u M 1 B T φ p k+1 = p k + (γ/dt)φ

Zeist 2009 p. 15/57 A simple splitting/projection approach II Key Question: Is the projection/splitting consistent?

Zeist 2009 p. 15/57 A simple splitting/projection approach II Key Question: Is the projection/splitting consistent? Practictioners say yes... Chorin (1967), Temam(1969), Kim & Moin (1985), Van Kan (1986), Bell, Colella & Glaz (1989), Gresho & Chan (1984, 1990), Perot (1993), Turek (1997). Mathematicians say sometimes... Guermond (1984), Rannacher (1992), E & Liu (1995, 1996), Shen (1992, 1996), Prohl (1997, 2007 ).

Zeist 2009 p. 16/57 ( ) ( ) M 0 u 0 0 t p t + ( N( u) + νa B T B 0 ) ( ) u p = ( ) f 0 1967 1982

Zeist 2009 p. 17/57 A Stokes splitting/projection approach I ( M 0 0 0 ) ( ) u + t p t ( N( u) + νa B T B 0 ) ( ) u p Given a time step dt, discretise the total derivative: Algorithm: Pironneau, 1982; Hansbo, 1982 for k = 0, 1,... compute X m ( x) the solution at τ k = kdt of = ( ) 0 0 dx dτ = u k (X,τ), X(τ k+1 ) = x; end interpolate u k = u k (X m ( x)) ( ) ( ) 1 solve dt M + νa B T u k+1 B 0 p k+1 = ( ) 1 dt M u k 0

Zeist 2009 p. 18/57 A simple splitting/projection approach II Key 1 Question: Is the total derivative approximation stable numerically?

Zeist 2009 p. 18/57 A simple splitting/projection approach II Key 1 Question: Is the total derivative approximation stable numerically? Mathematicians say yes, but care is needed... Hansbo (1982), Pironneau (1982), Morton, Priestley & Suli (1988,1989, 1994), Bermejo & Staniforth (1991, 1992).

Zeist 2009 p. 19/57 ( ) ( ) M 0 u 0 0 t p t + ( N( u) + νa B T B 0 ) ( ) u p = ( ) f 0 1967 1982 1991

A clever splitting/projection approach I ( M 0 0 0 ) ( ) u + t p t ( N( u) + νa B T B 0 ) ( ) u p Given dt and parameters θ = 1 1/ 2, α = 1 2θ 1 θ Algorithm: Glowinski, 1986, 1991 for k = 0, ( 1,... ) ( ) 1 solve θdt M + ανa BT u k+θ = B 0 p k+θ ( ) 1 solve solve end (1 2θ)dT M + N( u k+1 θ) + βνa ) ( ) M + ανa BT u k+1 = B 0 p k+1 ( 1 θdt ( f k 0 ) = ( ) f 0,β = 1 α. u k+1 θ = f k+θ ( f k+1 θ 0 ) Zeist 2009 p. 20/57

Zeist 2009 p. 21/57 A clever splitting/projection approach II Key 1 Question: Is the algorithm over-complicated?

Zeist 2009 p. 21/57 A clever splitting/projection approach II Key 1 Question: Is the algorithm over-complicated? Mathematicians say no,... Glowinski & Dean (1984, 1993), Bristeau et al. (1985), Rannacher (1989, 1990), Turek (1996), Smith & Silvester (1997).

Zeist 2009 p. 22/57 Update : 2000 2009 Philip Gresho & David Griffiths & David Silvester Adaptive time-stepping for incompressible flow; part I: scalar advection-diffusion, SIAM J. Scientific Computing, 30: 2018 2054, 2008. David Kay & Philip Gresho & David Griffiths & David Silvester Adaptive time-stepping for incompressible flow; part II: Navier-Stokes equations. MIMS Eprint 2008.61.

Zeist 2009 p. 23/57 Smart Integrator (SI) definition Optimal time-stepping: time-steps automatically chosen to follow the physics. Black-box implementation: few parameters that have to be estimated a priori. Solver efficiency: the linear solver convergence rate is robust with respect to the mesh size h and the Reynolds number 1/ν.

Zeist 2009 p. 24/57 Trapezoidal Rule (TR) time discretization We subdivide [0,T] into time levels {t i } N i=1. Given ( un,p n ) at time level t n, k n+1 := t n+1 t n, compute ( u n+1,p n+1 ) via 2 k n+1 u n+1 + w n+1 u n+1 ν 2 u n+1 + p n+1 = f n+1 u n+1 = 0 in Ω u n+1 = g n+1 ν u n+1 n p n+1 n = 0 on Γ D on Γ N with second-order linearization f n+1 = 2 k n+1 u n + ν 2 u n u n u n p n w n+1 = (1 + k n+1 k n ) u n k n+1 k n u n 1

Zeist 2009 p. 25/57 Saddle-point system In R 2 the discretized Oseen system (*) is: F n+1 0 Bx T α x,n+1 0 F n+1 By T α y,n+1 = B x B y 0 α p,n+1 f x,n+1 f y,n+1 f p,n+1 F n+1 := 2 k n+1 M + νa + N( w n+1 h ) The vector f is constructed from the boundary data g n+1, the computed velocity u n h at the previous time level and the acceleration un h t The system can be efficiently solved using appropriately preconditioned GMRES...

Zeist 2009 p. 26/57 Preconditioned system ( F B T B 0 ) P 1 P ( ) α u α p = ( ) f u f p A perfect preconditioner is given by ( ) ( ) F B T F 1 F 1 B T S 1 B 0 } 0 S 1 {{ } P 1 = ( ) I 0 BF 1 I with F = 2 k n+1 M + νa + N and S = BF 1 B T.

Zeist 2009 p. 27/57 For an efficient preconditioner we need to construct a sparse approximation to the exact Schur complement S 1 = (BF 1 B T ) 1 See Chapter 8 of Howard Elman & David Silvester & Andrew Wathen Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics Oxford University Press, 2005. Two possible constructions...

Zeist 2009 p. 28/57 Schur complement approximation I Introducing the diagonal of the velocity mass matrix M M ij = ( φ i, φ j ), gives the least-squares commutator preconditioner : (BF 1 B T ) 1 (BM 1 B T } {{ } amg ) 1 (BM 1 FM 1 B T )(BM 1 B T } {{ } amg ) 1

Zeist 2009 p. 29/57 Schur complement approximation II Introducing associated pressure matrices M p ( ψ i, ψ j ), A p ( ψ i, ψ j ), N p ( w h ψ i,ψ j ), F p = 2 k n+1 M p + νa p + N p, mass diffusion convection convection-diffusion gives the pressure convection-diffusion preconditioner : (BF 1 B T ) 1 Mp 1 F p A 1 p }{{} amg

Zeist 2009 p. 30/57 Adaptive Time Stepping AB2 TR Consider the simple ODE u = f(u) Manipulating the truncation error terms for TR and AB2 gives the estimate T n = u n+1 u n+1 3(1 + k n k n+1 ) Given some user-prescribed error tolerance tol, the new time step is selected to be the biggest possible such that T n+1 tol u max. This criterion leads to k n+2 := k n+1 ( tol umax T n ) 1/3

Zeist 2009 p. 31/57 Stabilized AB2 TR To address the instability issues: We rewrite the AB2 TR algorithm to compute updates v n and w n scaled by the time-step: u n+1 u n = 1 2 k n+1v n ; u n+1 u n = k n+1 w n. We perform time-step averaging every n steps: u n := 1 2 (u n+u n 1 ); u n+1 := u n + 1 4 k n+1v n ; u n+1 := 1 2 v n. Contrast this with the standard acceleration obtained by inverting the TR formula: u n+1 = 2 k n+1 (u n+1 u n ) u n = v n u n

Zeist 2009 p. 32/57 Stabilized AB2 TR 10 2 10 2 10 1 10 1 10 0 10 0 10 1 10 1 t 10 2 t 10 2 10 3 10 3 10 4 10 4 10 5 10 5 10 6 10 6 10 4 10 2 10 0 10 2 t 10 6 10 6 10 4 10 2 10 0 10 2 t Advection-Diffusion of step profile on Shishkin grid. tol = 10 3 tol = 10 4

Zeist 2009 p. 33/57 Stabilized AB2 TR Spin up driven cavity flow with ν = 1/100.

Zeist 2009 p. 34/57 Adaptive Time-Stepping Algorithm I The following parameters must be specified: time accuracy tolerance tol (10 4 ) GMRES tolerance itol (10 6 ) GMRES iteration limit maxit (50)

Zeist 2009 p. 34/57 Adaptive Time-Stepping Algorithm I The following parameters must be specified: time accuracy tolerance tol (10 4 ) GMRES tolerance itol (10 6 ) GMRES iteration limit maxit (50) Starting from rest, u 0 = 0, and given a steady state boundary condition u( x, t) = g, we model the impulse with a time-dependent boundary condition: u( x,t) = g(1 e 5t ) on Γ D [0,T].

Zeist 2009 p. 35/57 Adaptive Time-Stepping Algorithm II Setup the Oseen System ( ) and compute [α x,n+1,α y,n+1 ] using GMRES(maxit, itol).

Zeist 2009 p. 35/57 Adaptive Time-Stepping Algorithm II Setup the Oseen System ( ) and compute [α x,n+1,α y,n+1 ] using GMRES(maxit, itol). Compute the LTE estimate e v,n+1

Zeist 2009 p. 35/57 Adaptive Time-Stepping Algorithm II Setup the Oseen System ( ) and compute [α x,n+1,α y,n+1 ] using GMRES(maxit, itol). Compute the LTE estimate e v,n+1 If e v,n+1 > (1/0.7) 3 tol, we reject the current time step, and repeat the old time step with k n+1 = k n+1 ( tol e v,n+1 )1/3.

Zeist 2009 p. 36/57 Example Flow Problem I (ν = 1/1000) -1.2-1.4-1.6 1 0-1 -1 0 1

Time step evolution Zeist 2009 p. 37/57

Linear solver performance Zeist 2009 p. 38/57

Example Flow Problem II (ν = 1/100) Zeist 2009 p. 39/57

Lift Coefficient Zeist 2009 p. 40/57

Time step evolution Zeist 2009 p. 41/57

Zeist 2009 p. 42/57 Bouyancy driven flow u t + u u ν 2 u + p = jt u = 0 T t + u T ν 2 T = 0 in W Ω (0,T) in W in W Boundary and Initial conditions u = 0 on Γ [0,T]; u( x, 0) = 0 in Ω. T = T g on Γ D [0,T]; ν T n = 0 on Γ N [0,T]; T( x, 0) = T 0 ( x) in Ω.

Zeist 2009 p. 43/57 Finite element matrix formulation Introducing the basis sets X h = span{ φ i } n u i=1, Velocity basis functions; M h = span{ψ j } n p j=1, Pressure basis functions. T h = span{φ k } n T k=1, Temperature basis functions; gives the method-of-lines discretized system: M 0 0 F B T M u 0 0 0 + B 0 0 p 0 0 M 0 0 F T u t p t T t = 0 0 g with a (vertical ) mass matrix: ( M ) ij = ([0,φ i ],φ j )

Zeist 2009 p. 44/57 Preconditioning strategy F B T B 0 0 0 0 F M P 1 P α u α p α T = Given S = BF 1 B T, a perfect preconditioner is given by F B T M F 1 F 1 B T S 1 F 1 M 1 B 0 0 0 S 1 0 0 0 F } 0 0 {{ F 1 } P 1 = f u f p f T I 0 0 I BF 1 M F 1 0 0 I BF 1

Example: Natural Convection in 1:16 cavity Zeist 2009 p. 45/57

Zeist 2009 p. 46/57... with efficient linear algebra 35 Average number of Krylov iterations (AMG) 10 2 Evolution of the time step size 30 10 0 25 10 2 N it 20 15 10 t 10 4 10 6 5 Picard Newton 0 0 50 100 150 200 250 300 350 time step 10 8 10 10 0 50 100 150 200 250 300 350 time step

Zeist 2009 p. 47/57 What have we achieved? Black-box implementation: few parameters that have to be estimated a priori. Optimal complexity: essentially O(n) flops per iteration, where n is dimension of the discrete system. Efficient linear algebra: convergence rate is (essentially) independent of h. Given an appropriate time accuracy tolerance convergence is also robust with respect to ν

References Semi-Implicit, Fractional-Step, Pressure Projection I A.J. Chorin A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2:12 26, 1967. J. Kim & P. Moin Application of a fractional step method to incompressible Navier Stokes equations, J. Comput. Phys., 59:308 323, 1985. J. Van Kan A second order accurate pressure correction scheme for viscous incompressible flow, SIAM J. Sci. Stat. Comput., 7(3):870 891, 1986. J.B. Bell, P. Colella & H.M. Glaz A second order projection method for the incompressible Navier Stokes equations, J. Comput. Phys., 85:(2)257 283, 1989. Zeist 2009 p. 48/57

Zeist 2009 p. 49/57 Semi-Implicit, Fractional-Step, Pressure Projection II P.M. Gresho, S.T. Chan, R.L. Lee & C.D. Upson A modified finite element method for solving the time dependent, incompressible Navier Stokes equations. Part 1: Theory, Int. J. Numer. Meth. Fluids, 4:557 598, 1984. P.M. Gresho & S.T. Chan On the theory of semi implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation, Int. J. Numer. Meth. Fluids, 11(5):621-660, 1990. J.B. Perot An analysis of the fractional step method, J. Comput. Phys., 108:51 58, 1993.

Zeist 2009 p. 50/57 Semi-Implicit, Fractional-Step, Pressure Projection III J. L. Guermond Sur l approximation des équations de Navier Stokes instationnaires par une méthode de projection, C.R. Acad. Sci. Paris, 319:887 892, 1994. Serie I. R. Rannacher The Navier Stokes Equations II: Theory and Numerical Methods, Springer Verlag, Berlin, Germany, 1992. Chap.On Chorin s projection method for the incompressible Navier Stokes equations; pp. 167 183; Lecture Notes in Mathematics, Vol. 1530.

Zeist 2009 p. 51/57 Semi-Implicit, Fractional-Step, Pressure Projection IV W. E & J. G. Liu Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32(4):1017 1057, 1995. W. E & J. G. Liu Vorticity boundary conditions and related issues for finite difference schemes, J. Comput. Phys., 124:368 382, 1996.

Zeist 2009 p. 52/57 Semi-Implicit, Fractional-Step, Pressure Projection V J. Shen On error estimates of some higher order projection and penalty projection methods for Navier Stokes equations, Numer. Math., 62:49 73, 1992. J. Shen On error estimates of the projection methods for the Navier Stokes equations: Second order schemes, Math. Comput., 65, 215:1039 1065, 1996. A. Prohl Analysis of Chorin s projection method for solving the incompressible Navier Stokes equations, Universität Heidelberg, Institut für Angewandte Mathematik, INF 294, D 69120 Heidelberg, Germany, 1996.

Zeist 2009 p. 53/57 Transport Diffusion, Lagrange-Galerkin, Backward Method of Characteristics I P. Hansbo The characteristic streamline diffusion method for the time dependent incompressible Navier Stokes equations, Comput. Meth. Appl. Mech. Eng., 99:171 186, 1992. O. Pironneau On the transport diffusion algorithm and its application to the Navier Stokes equations, Numer. Math., 38:309 332, 1982. K.W. Morton, A. Priestley & E. Süli Stability of the Lagrange Galerkin method with non exact integration, Math. Model. Numer. Anal., 22:(4)625 653, 1988.

Zeist 2009 p. 54/57 Transport Diffusion, Lagrange-Galerkin, Backward Method of Characteristics II A. Priestley Exact projections and the Lagrange Galerkin method: A realistic alternative to quadrature, J. Comput. Phys., 112:316 333, 1994 R. Bermejo Analysis of an algorithm for the Galerkin characteristic method, Numer. Math., 60:163 194, 1991.

Zeist 2009 p. 55/57 Fractional Step, le Θ scheme I M.O. Bristeau, R. Glowinski, B. Mantel, J. Periaux & P. Perrier, Numerical methods for incompressible and compressible Navier Stokes problems, Finite Elements in Fluids 6 (R.H. Gallagher, G. Carey, J.T. Oden & O.C. Zienkiewicz, eds), Chichester: Wiley, 1985. pp. 1 40. E.J. Dean & R. Glowinski On some finite element methods for the numerical simulation of incompressible viscous flow, Incompressible Computational Fluid Dynamics., (M.D. Gunzburger & R.A. Nicolaides, eds), Cambridge University Press, 1993. pp. 17 65.

Zeist 2009 p. 56/57 Fractional Step, le Θ scheme II R. Rannacher Applications of mathematics in industry and technology, B.G. Teubner, Stuttgart, Germany, 1989. Chap. Numerical analysis of nonstationary fluid flow (a survey); pp. 34 53; (V.C. Boffi & H. Neunzert, eds). R. Rannacher Navier Stokes equations: theory and numerical methods, Springer Verlag, Berlin, Germany, 1990. Chap. On the numerical analysis of the non stationary Navier Stokes equations; pp. 180 193; (J. Heywood et al., eds).

Zeist 2009 p. 57/57 Fractional Step, le Θ scheme III S. Turek A comparative study of some time stepping techniques for the incompressible Navier Stokes equations: From fully implicit nonlinear schemes to semi implicit projection methods, Int. J. Numer. Meth. Fluids, 22(10):987 1012, 1996. A. Smith & D.J. Silvester Implicit algorithms and their linearization for the transient incompressible Navier Stokes equations, IMA J. Numer. Anal. 17: 527 545, 1997.