Finite Element Decompositions for Stable Time Integration of Flow Equations

Transcription

1 MAX PLANCK INSTITUT August 13, 2015 Finite Element Decompositions for Stable Time Integration of Flow Equations Jan Heiland, Robert Altmann (TU Berlin) ICIAM 2015 Beijing DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 1/31

2 Problem Statement We want to integrate the semi-discrete flow equations, to solve for the velocity v and the pressure p: M v + K(v) + B T p = f Bv = g Why not simply use semi-implicit Euler scheme, which is of first order for v and p? M v+ v c + K(v c ) + B T p c = f c Bv + = g + Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 2/31

3 Numerical Example The error in the pressure variable p for varying p,tol p L Index if we iteratively solve the resulting linear equation systems up to a residual of tol = Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 3/31

4 Introduction Flow equations u + Ku + π = f, u = g, u(0) = u 0. Stokes equation: Ku = ν u Euler equation: Ku = (u )u Navier-Stokes equations: Ku = (u )u ν u Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 4/31

5 Minimal Extension for the PDE u + Ku + π = f u = g We can decompose the solution u = u 1 + u 2 such that u 1 = 0 and u 2 = g. We add the derivative of the constrained u = g to the system u + Ku + π = f, u = g, u = ġ. Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 5/31

6 Minimal Extension for the PDE u + Ku + π = f u = g u = ġ With u = u 1 + u 2, u 1 = 0, and ũ 2 := u 2, one obtains u 1 + ũ 2 + K ( u 1 + u 2 ) π = f ũ 2 u 2 = g = ġ which under reasonable conditions is equivalent to the original system [Altmann 15, Altmann/JH 15] is an index reduced on the PDE level applying Minimal Extension [Kunkel/Mehrmann 04] Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 6/31

7 Spatial Discretization u + Ku + π = f u = g Velocity: u V := [ H 1 0(Ω) ] d v Vh Pressure: π Q := L 2 (Ω)/R p Q h V h and Q h satisfy an inf-sup or the LBB condition like the Taylor-Hood or Crouzeix-Raviart scheme V h Q h Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 7/31

8 Spatial Discretization u + Ku + π = f u = g M v + K(v) + B T p = f Bv = g ( ) Assumptions on the DAE (met by LBB-stable schemes) M R nv,nv, B R np,nv, n v > n p M symmetric positive definite and B is of full rank Proposed reformulation Add the time-derivative of the constraint ( ) Add a variable that makes the system square again Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 8/31

9 Reformulation Find orthogonal matrix Q so that BQ = [ ] B 1 B 2 with invertible B2 [ ] q1 and define := Q T v q 2 M v + K(v) + B T p = f Bv = g ( ) Add constraint B q = ġ and the new variable q 2 := q 2 Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 9/31

10 Reformulation Find orthogonal matrix Q so that BQ = [ ] B 1 B 2 with invertible B2 [ ] q1 and define := Q T v q 2 M v + K(v) + B T p = f Bv = g ( ) Add constraint B q = ġ and the new variable q 2 := q 2 Reformulated index-1 system: [ ] q1 MQ + K(q q 1, q 2 ) + B T p = f 2 B 1 q 1 + B 2 q 2 B 1 q 1 + B 2 q 2 = g = ġ Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 9/31

11 Finite Element Decomposition Problem: how to find Q Computation via QR decomposition is not feasible We propose: decomposition of velocity ansatz space Splitting V h = V h,1 V h,2 Basis functions V h,1 = span{ϕ 1,..., ϕ nv m}, V h,2 = span{ϕ nv m+1,..., ϕ nv } Q = permutation matrix B [ ] B 1 B 2 through column swapping Aim: find a subspace V h,2, or ϕ nv m+1,..., ϕ nv, such that B 2 is invertible Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 10/31

12 Finite Element Decomposition Algorithm to find V h,2 are available for Crouzeix-Raviart elements, Taylor-Hood elements, Rannacher-Turek elements for quads, and their 3D counterparts. see [Altmann/JH 15]. Illustration for the Crouzeix-Raviart scheme The splitting is based on a mapping ι: T \ {T 0 } E Range(ι) = {E 1,..., E m } subset of the set of edges E of the triangulation Edges correspond to ϕ n m+1,..., ϕ n... The algorithm terminates in a triangular B 2 Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 11/31

13 Time Integration Consider the semi-explicit Euler scheme... time-step length v c, v +... current, next time instance of velocity For the index-2 system: M v+ v c + K(v c ) + B T p c = f c Bv + = g + For the index-1 system with (q 1, q 2 ) Q T v: [ 1 ] MQ (q+ 1 q1) c + K(q1, c q2) c + B T p c = f c q c 2 B 1 q B 2 q + 2 = g + B 1 1 (q+ 1 q c 1) + B 2 q c 2 = ġ c Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 12/31

14 What s the Gain Consider the original system M v+ v c The inherent equation for the pressure reads + K(v c ) + B T p c = f c Bv + = g + BM 1 B T p c = Bvc Bv + + BM 1 [f c K(v c )] If we solve the algebraic up to an residual of size ε, then Bv c Bv + = gc g + + εc ε +, i.e. in the pressure, the algebraic error is amplified by 1. Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 13/31

15 What s the Gain While for the index-1 system, BM 1 B T p c = Bvc Bv + + BM 1 [f c K(v c )] Bv c Bv + = gc g + + εc ε + the inherent equation for the pressure reads BM 1 B T p c = 1 [ ] q BQ c 1 q 1 + q 2 c + BM 1 [f c K(q1, c q2)], c which is affected by the algebraic error via [ ] 1 q BQ c 1 q 1 + q 2 c = ġ + + ε c. Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 14/31

16 The Setup Navier-Stokes equations for the cylinder wake at Re = 60 Nonuniform spatial discretization Crouzeix-Raviart finite elements using [FEniCS] about velocity and 5000 pressure nodes Uniform time-discretization Numerically computed reference solution Semi-Explicit Euler scheme GMRes with [KryPy] for the linear systems Fair balancing of the residuals Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 15/31

17 Velocity Approximation q,tol q L 2 Index Index tol = tol = The error in the velocity v versus time step length for varying tolerances tol in the linear solver the + and in the plots denote exact solves (Index-2) and very rough solves (Index-1) Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 16/31

18 Pressure Approximation p,tol p L 2 Index Index tol = tol = The error in the pressure p versus time step length for varying tolerances tol in the linear solver the + and in the plots denote exact solves (Index-2) and very rough solves (Index-1) Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 17/31

19 Residual in the Divergence Constraint Bq,tol g L 2 Index Index tol = tol = The residual in the divergence free constraint versus time step length for varying tolerances tol in the linear solver Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 18/31

20 Conclusion & Outlook Summary: the time integration of semi-discretized flow equations requires some index reduction (well known) Minimal extension can be formulated on the PDE level and numerically realized in FEM by analytical splittings of velocity ansatz spaces (Necessary) future work: Efficiency for the solution of the index-1 systems Formulation for higher order time integration Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 19/31

21 Thank you for your attention Github: github.com/highlando Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 20/31

22 Literature R. Altmann and J. Heiland, Finite element decomposition and minimal extension for flow equations, accepted in M2AN (2015), also available as Preprint from TU Berlin, Germany. A. Gaul, Krypy, Public Git Repository, Commit: 110a1fb756fb, Iterative Solvers for Linear Systems, P. Kunkel and V. Mehrmann, Index reduction for differential-algebraic equations by minimal extension, ZAMM, Z. Angew. Math. Mech. 84 (2004), no. 9, Anders Logg, Kristian B. Ølgaard, Marie E. Rognes, and Garth N. Wells, FFC: the FEniCS form compiler, Automated Solution of Differential Equations by the Finite Element Method, Springer-Verlag, Berlin, Germany, 2012, pp Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 21/31

23 Penalty Method A perturbation in the constraints gives the strangeness-free approximation to the NSEs [ ] [ ] [ ] [ ] [ ] M 0 vε A B T vε f = 0 0 B εi g ṗ ε In Shen 1995 it was shown that in continous space the modelling error behaves like p ε Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 22/31

24 Penalty Method Backdraw Problem: For fixed time step a smaller ε makes the problem ill conditioned: For illustration we observe that with Bv + εp = g the linear case reads M v Av B T p = f or p = 1 [Bv g] ε or M v [A 1 ε BT B]v = f 1 ε Bg and the application of Backward Euler requires solves with the coefficient matrix [ M [A 1 ε BT B] ] which is ill-conditioned if ε. Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 23/31

25 Pressure Poisson Formulations [ ] [ ] [ M 0 v A B T 0 0 ṗ B 0 Differentiation of the constraint, ] [ ] v = p [ ] f g Bv = g becomes B v = ġ, and the application of BM 1 to the differential part leads to the system [ ] [ ] [ ] [ ] [ ] M 0 v A B T v f 0 0 ṗ BM 1 A BM 1 B T = p BM 1, f + ġ which is strangeness-free, since BM 1 B T is invertible. Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 24/31

26 Numerical Treatment of the PPE Formulation [ ] [ M 0 v 0 0 ṗ ] [ A BM 1 A B T BM 1 B T ] [ ] [ ] v f = p BM 1 f + ġ One can use standard time integration methods, e.g. Backward Euler global error is O() in v and p Trapezoidal rule O( 2 ) see Gresho/Sani 2000 for a concise discussion But the solution drifts off the original constraints unless one uses modified discretization schemes Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 25/31

27 Splitting/Projection/PressureCorrection Algorithm Algorithm for one time step according to M un+1 u n Au n B T p n+1 = f n+1 (2) Bu n+1 = g n+1 (3) 1 guess a pressure gradient B T p and solve M ũ un Au n B T p = f n+1 (1 ) 2 Having substracted (1) (1 ) one finds M un+1 ũ B T (p n+1 p) = 0 (4) 3 Use Bu n+1 = g n+1, cf. (2), to solve (3) for (p n+1 p) 4 get u n+1 from (3) and p n+1 = p + (p n+1 p) Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 26/31

28 Splitting/Projection/PressureCorrection Analysis Straight forward implementation for nonlinear formulations but it amplifies the error by 1 correction when solving for the and it also explicitly requires boundary conditions for the pressure It is nothing else than Stabilization by Projection, cf. Hairer/Wanner 2002 for convergence results Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 27/31

29 It is actually a time integration scheme for the strangeness-free system ṽ 0 M 1 A 0 M 1 B T ṽ 0 0 I I M 1 B T 0 v 0 B 0 0 φ = 0 0 BM 1 A 0 BM 1 B T p f 0 g BM 1 f + ġ Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 28/31

30 Divergence Free Bases [ ] [ M 0 v 0 0 ṗ ] [ A BM 1 A B T BM 1 B T ] [ ] [ ] v f = p BM 1 f + ġ Find an orthogonal transformation matrix Q, such that BQ = [R 0] and transform the equations to get [ ] [ ] [ ] R Q T MQ 0 d Q T T [ ] [ ] v QT AQ 0 Q T v Q 0 0 dt p [ ] = T f, p g R 0 0 which can be decomposed into algebraic in differential equations The computation of Q is expensive to unfeasible and if it is done numerically, it may cause instabilities But the system is reduced less computational effort in the time integration Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 29/31

31 Common Solution Approaches M v + K(v) + B T p = f Bv + ɛp = g ( ) (a) Splitting, projection, or pressure correction schemes (b) Pressure penalization scheme (c) Divergence-free methods with common difficulties Need for pressure boundary conditions (a) Parameter dependency (b) Ill-conditioned resulting linear system (b) Instability in the pressure approximation (a-c) Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 30/31

32 References E. Hairer and G. Wanner, Solving ordinary differential equations. II: Stiff and differential-algebraic problems. Springer, R. Rannacher, On the numerical solution of the incompressible Navier-Stokes equations. Z. Angew. Math. Mech. (ZAMM), 73 (1993), no. 9, P. M. Gresho and R. L. Sani, Incompressible flow and the finite element method. Vol. 2: Isothermal laminar flow. Wiley, Max-Planck-Institut Magdeburg J. Heiland, FEM Decomposition for Time Integration for Flows 31/31