ICFD25 Workshop p. 1/39 Fast Solvers for Unsteady Incompressible Flow David Silvester University of Manchester http://www.maths.manchester.ac.uk/~djs/
ICFD25 Workshop p. 2/39 Outline Semi-Discrete Navier-Stokes Equations: DAE theory 25 year overview: Decoupled Solution Methods Today: A black-box "Smart Integrator (SI)
ICFD25 Workshop p. 3/39 Joint work with Phil Gresho (ex-llnl) David Griffiths (University of Dundee) David Kay (Oxford University Computing Laboratory) For further details, see 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.
Semi-Discrete Navier-Stokes Equations: DAE theory ICFD25 Workshop p. 4/39
ICFD25 Workshop p. 5/39 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 D on Γ D [0,T]; ν u n p n = 0 u( x, 0) = u 0 ( x) in Ω. on Γ N [0,T];
ICFD25 Workshop p. 6/39 Spatial Discretization I In R 2 the method-of-lines discretized system is a semi-explicit system of DAEs: M v 0 0 F v 0 Bx T 0 M v 0 + 0 F v B T y 0 0 0 B x B y 0 α x t α y t α p t α x α y α p = f x f y f p The DAEs have index equal to two The discrete problem is nonlinear F v := νa v + N v (α) The vector f is constructed from the boundary data g D. To reduce the index we differentiate the constraint...
ICFD25 Workshop p. 7/39 Spatial Discretization II... to give an index one DAE system: α x t Mv 1 F v 0 Mv 1 α y t + 0 Mv 1 F v Mv 1 0 B x Mv 1 F v B y Mv 1 F v A p Bx T By T α x α y α p = f x f y f p The matrix A p := B x Mv 1 Bx T + B y Mv 1 By T 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!
ICFD25 Workshop p. 8/39 Semi-Discrete Navier-Stokes Equations: DAE theory 25 year overview: Decoupled Solution Methods
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. ICFD25 Workshop p. 9/39
ICFD25 Workshop p. 10/39 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.
ICFD25 Workshop p. 11/39 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.
ICFD25 Workshop p. 12/39 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.
ICFD25 Workshop p. 13/39 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.
ICFD25 Workshop p. 14/39 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.
ICFD25 Workshop p. 15/39 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.
ICFD25 Workshop p. 16/39 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.
ICFD25 Workshop p. 17/39 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).
ICFD25 Workshop p. 18/39 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.
ICFD25 Workshop p. 19/39 Semi-Discrete Navier-Stokes Equations: DAE theory Decoupled Solution Methods Gresho s black-box "Smart Integrator
ICFD25 Workshop p. 20/39 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+1/2 u n+1 ν 2 u n+1 + p n+1 = f n+1, with linearization u n+1 = 0 u n+1 = g n+1 D in Ω on Γ D ν u n+1 n p n+1 n = 0 on Γ N, n+1 f = 2 k n+1 u n + u n t + ( un u n w n+1/2 u n ) w n+1/2 = (1 + k n+1 k n ) u n k n+1 k n u n 1
ICFD25 Workshop p. 21/39 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
ICFD25 Workshop p. 22/39 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
ICFD25 Workshop p. 23/39 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
ICFD25 Workshop p. 24/39 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 bounded independently of the discrete problem parameters.
ICFD25 Workshop p. 25/39 Saddle-point system The Oseen system (*) is: Fv n+1 0 Bx T B T y 0 Fv n+1 B x B y 0 α x,n+1 α y,n+1 α p,n+1 = f x,n+1 f y,n+1 f p,n+1 F n+1 v := 2 k n+1 M v + νa v + N v ( w n+1/2 h ) The timestep k n+1 is computed adaptively The vector f is constructed from the boundary data, the computed velocity un h at the previous time g n+1 D level and the acceleration un h t The system can be efficiently solved using appropriately preconditioned GMRES...
ICFD25 Workshop p. 26/39 Preconditioned system ( F B 0 B T ) 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.
ICFD25 Workshop p. 27/39 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...
ICFD25 Workshop p. 28/39 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 ) 1 (BM }{{} 1 amg FM 1 B T )(BM 1 B T ) 1 } {{ } amg
ICFD25 Workshop p. 29/39 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 Q 1 F p A 1 p }{{} amg
ICFD25 Workshop p. 30/39 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 D, we model the impulse with a time-dependent boundary condition: u( x,t) = g D (1 e 5t ) on Γ D [0,T]. We specify the frequency of averaging, typically n = 10. We also choose a very small initial timestep, typically, k 1 = 10 8.
ICFD25 Workshop p. 31/39 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. Otherwise, accept the step and continue with n = n + 1 and k n+2 based on the LTE estimate and the accuracy tolerance tol.
ICFD25 Workshop p. 32/39 Example Flow Problem I (ν = 1/1000) -1.2-1.4-1.6 1 0-1 -1 0 1
Time step evolution ICFD25 Workshop p. 33/39
Linear solver performance ICFD25 Workshop p. 34/39
Example Flow Problem II (ν = 1/100) ICFD25 Workshop p. 35/39
Lift Coefficient ICFD25 Workshop p. 36/39
Time step evolution ICFD25 Workshop p. 37/39
Linear solver performance ICFD25 Workshop p. 38/39
ICFD25 Workshop p. 39/39 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. Optimal convergence: rate is bounded independently of h. Given an appropriate time accuracy tolerance convergence is also robust with respect to ν