Implicit-explicit exponential integrators

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1 Implicit-explicit exponential integrators Bawfeh Kingsley Kometa joint work with Elena Celledoni MaGIC 2011 Finse, March 1-4

2 1 Introduction Motivation 2 semi-lagrangian Runge-Kutta exponential integrators SL exponential Runge-Kutta integrators semi-lagrangian methods Krylov methods 3 Semi-Lagrangian multistep exponential integrators Operator-integrating-factor (OIF) SL multistep exponential integrators 4 Remarks 5 Conclusion

3 We consider implicit-explicit (IMEX) commutator-free semi-lagrangian exponential integrators for solving convection-diffusion problems of the form t u =N(u)+Lu (1) with given boundary and initial data; wheren is a nonlinear convection operator and L is a (spatial) linear operator. Examples: Burgers, KdV, Navier-Stokes. Semi-discretization in space leads to an ODE of the form or DAE ẏ = C(y)y + Ay (2) ẏ = f(y, z) (3) 0 = g(y) (4)

4 Motivation Methods presented here are shown to work better than standard IMEX methods for problems with small diffusion coefficients. One considers exact flows of the convection term in a semi-lagrangian fashion via exponential integrators Better linear stability

5 Method SL exponential Runge-Kutta[Celledoni,Kometa;2009] ẏ = C(y)y + Ay (5) Change of variables y = Wz (y z) with the matrix W chosen such that Ẇ C(Wz)W = 0 W(0) = I (6) to obtain ż = W 1 AWz z(0) = y 0 (7) where y(0) = y 0 represents the initial data.

6 Method SL exponential Runge-Kutta[Celledoni,Kometa;2009] Starting with additive s-stage IMEX PRK pair I :={A, b, c} and E :={Â,ˆb, c} respectively, where I represents a DIRK method while E is the explicit RK; Construct coefficients of an explicit commutator-free (CF) Lie-group exponential integrator [Celledoni,Martinsen,Owren;2003], [Owren;2006] C :={α j il,βj }, l = 1,..., J, i, j = 1,..., s, l where J = J(i) is the maximum number of exponentials allowed at each stage i, and J J α j = â il ij, β j = ˆbj l l=0 l=0 C has the same classical order as E

7 Method SL exponential Runge-Kutta[Celledoni,Kometa;2009] Apply the explicit CF method C to solve the convection part (6) and the implicit RK method I to solve the diffusion part (7).

8 Method SL exponential Runge-Kutta[Celledoni,Kometa;2009] Apply the explicit CF method C to solve the convection part (6) and the implicit RK method I to solve the diffusion part (7). The resulting method in the variable y has is given by the algorithm: for i = 1 to s Y i =ϕ i y n + h i a ij ϕ i ϕ 1 j A Y j j=1 ϕ i = ( exp h kα k ij C(Y k) ) (...exp h kα k i1 C(Y k) ) end y n+1 =ϕ s+1 y n + h s b i ϕ s+1 ϕ 1 i A Y i i=1 ϕ s+1 = ( exp h kβ k J C(Y k) ) (...exp h kβ k 1 C(Y k) ) We refer to methods constructed this way as DIRK-CF.

9 SL semi-lagrangian methods To approximate y 1 = exp(hc(ỹ))y 0 (8) we compute the flow of the pure convection problem t u+v u = 0 u(x, 0) = u 0 (9) where V(x i ) = ỹ i and u 0 (x i ) = y i 0 for each grid point x i. This is computed accurately using a semi-lagrangian method. See for example [Pironneau; ], [Giraldo;1998], [Xiu, Karniadakis; 2001] etc.

10 SL Krylov methods In this case y 1 = exp(hc(ỹ))y 0 (10) can also be approximated using a Krylov subspace method. See for example [Celledoni, Moret; 1996], [Hochbruck, Lubich; 1997], [Sidje; 1998], [Lopez,Simoncini; 2006] etc.

11 Burgers equation We consider five (2nd, 3rd & 4th order) DIRK-CF methods contructed from additive IMEX RK methods taken from [Ascher et al.; 1997] and [Kennedy & Capernter; 2003]. Apply the methods to viscous Burgers equation t u+ u x u =ν xx u, x (0, 1), t (0, T] (11) with initial data u(x, 0) = u 0 (x) = sin(πx), and homogeneous Dirichlet boundary conditions.

12 Burgers equation Methods considered here are Methods order IMEX DIRK property DIRK-CF2 2 IMEX(1,2,2)[ARS] impl. midpoint DIRK-CF2L 2 IMEX(2,2,2)[ARS] stiff. acc., L-stable DIRK-CF3 3 IMEX(2,3,3)[ARS] non-stiff. acc. DIRK-CF-3L 3 IMEX(3,4,3)[ARS] stiff. acc., L-stable DIRK-CF4 4 ARK4(3)6L[2]SA[KC] stiff. acc., L-stable [ARS] = [Ascher et al.; 1997]; [KC] = [Kennedy & Capernter; 2003]

13 Burgers equation In the experiments we use Centered finite difference schemes (FDM) for spatial discretization for both the convection and diffusion operators Semi-Lagrangian schemes with cubic spline interpolations; solving the characteristics via 4th order explicit RK method. Krylov projections are used for some of the experiments. The global error are measured in the L 2 -norm Reference or exact solution are obtained via MATLAB s ode45

14 Burgers equation: Temporal error Error (L 2 ) 10 6 DIRK CF DIRK CF2L DIRK CF DIRK CF3L DIRK CF h Figure: DIRK-CF methods; Centered FDM are used in space with N = 32 nodes; x (0, 1), u(x, 0) = sin(πx), T = 1; stepsize h = T/2 k, k = 3 : 9; Temporal order [2.0076, , , , ] respectively. Krylov projections

15 Burgers equation: The three classes of methods compared We now compare the three classes of methods IMEX Runge-Kutta DIRK-CF using Krylov methods to compute the exponentials DIRK-CF using semi-lagrangian methods to compute the exponentials exponentials = flows of convecting vector fields (exp(c(ỹ)y 0 ) We measure the relative error in the discrete L -norm at time T = 2, as a function of viscosity. We use MATLAB sode15s function to compute a reference solution.

16 Burgers equation: The three classes of methods compared 10 0 Rel. Error ( L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90; We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.01

17 Burgers equation: The three classes of methods compared 10 0 Rel. Error ( L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90; We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.01

18 Burgers equation: The three classes of methods compared 10 0 Rel. Error ( L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90; We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.01

19 Burgers equation: The three classes of methods compared 10 0 Rel. Error (L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90. We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.001

20 Burgers equation: The three classes of methods compared 10 0 Rel. Error (L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90. We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.001

21 Burgers equation: The three classes of methods compared 10 0 Rel. Error (L ) ν Figure: Centered FDM are used in space with N = 81 nodes; x (0, 1), u(x, 0) = sin(πx), T = 2; stepsize h = T/90. We have SL DIRK-CF, IMEX, DIRK-CF with Krylov projections in blue, red, black resp ly. = order 2, =order 3, =order 4. Viscosity range ν 0.001

22 Operator-integrating-factor (OIF) Operator-integrating-factor (OIF)[Maday et al.; 1990] Given ẏ = C(y)y + Ay, y(t n ) = y n ; (12) Find an integrating factor I c via the convection operator C such that di c dt = I c C(ỹ(t)), I c (t n+1 ) = I = (identity matrix). (13) Differentiate the product I c y and substitute (12) and (15) to obtain d dt (I cy) = I c Ay. (14)

23 Operator-integrating-factor (OIF) Operator-integrating-factor (OIF)[Maday et al.; 1990] We have two equations and di c dt = I c C(ỹ(t)), I c (t n+1 ) = I = (identity matrix) (15) d dt (I cy) = I c Ay. (16) We apply a k-step method of order p to (16) and a one-step (multi-step also allowed) to (15). The approximation ỹ(t) := ỹ n (t) is constructed in terms of the initial values of y at time t n+k i, i = 1,...,k, such that y(t n+1 ) ỹ n (t) =O(h p+1 )

24 Operator-integrating-factor (OIF) Operator-integrating-factor (OIF)[Maday et al.; 1990] The resulting method is of the form k k α i (I c ) i y n+1 k i = h β i (I c ) i A y n+1 k i, (17) i=0 where for i = 0,...,k 1, (I C ) i w, represents the flow of the linearized ODE dψ ds = C(ỹn (t))ψ, t n+1+k i < s t n+1, ψ(t n+1+k i ) = w, and (I C ) k w = w. Meanwhileα i,β i withα k 0 are coefficients of the multistep method. i=0 (18)

25 Method SL multistep exponential[celledoni,kometa;2011] Analogous to the OIF, a general k-step multistep method is written as k k α i (I c ) i y n+1 k i = h β i (I c ) i A y n+1 k i, (19) where i=0 ϕ k = I, ϕ i = exp(hc(ỹ n i )), ỹ n i = i=0 k 1 j=0 e ij y n+1 k j, i = 0,...,k 1. Here e ij are the extrapolation coefficients, used to approximate C(y(t n+1 )) by frosen vector fields C(ỹ n i ). The coefficients e ij are be chosen so as to achieve the appropriate order of the overall method.

26 Method SL multistep exponential[celledoni,kometa;2011] Analogous to the OIF, a general k-step multistep method is written as k k α i ϕ i y n+1 k i = h β i ϕ i A y n+1 k i, (20) where i=0 ϕ k = I, ϕ i = exp(hc(ỹ n i )), ỹ n i = i=0 k 1 j=0 e ij y n+1 k j, i = 0,...,k 1. Here e ij are the extrapolation coefficients, used to approximate C(y(t n+1 )) by frosen vector fields C(ỹ n i ). The coefficients e ij are be chosen so as to achieve the appropriate order of the overall method.

27 Method SL multistep exponential[celledoni,kometa;2011] An example is the Crank-Nicolson exponential integrator (2-step, order 2): y n+1 = exp(hc(ỹ n 0 ))y n + h 2 (exp(hc(ỹn 0 ))Ay n + Ay n+1 ) (21) where ỹ n 0 := 3 2 y n 1 2 y n 1. Here we have ( α 0 = 0, α 1 =α 2 = 1;β 0 = 0,β 1 =β 2 = 1 2 while (e ij) = Using BDF schemes, methods of the type (20) upto order 4 have been derived. We refer to them as BDF-CF. We compute flows exp(hc(ỹ))y 0 in a semi-lagrangian fashion as mentioned earlier. 3 2 ).

28 Navier-Stokes, BDF-CF methods We consider the Navier-Stokes problem t u+(u )u = u p (22) u = 0 (23) with Dirichlet boundary conditions on Ω = [0, 1] 2 and initial data determined from the exact solution u 1 = y(1 x 2 )e t (24) u 2 = x(1 y 2 )e t (25) p = = t 3 x + y 1 2 (1+t 3 ) (26)

29 Navier-Stokes, BDF-CF methods We consider integrate in space, using (P N P N 2 ) spectral method, with N = 8 [Fischer,2002] in time, on interval t [0, 1] using the BDF-CF methods [Celledoni & Kometa,2011], choosing time steps h = 1/2 k, k = 3,...,8 Measuring the global error at time T = 1 we demonstrate the temporal convergence of the BDF-CF methods

30 Navier-Stokes, BDF-CF methods Error (L 2 ) BDF1 CF BDF2 CF BDF3 CF BDF4 CF h Figure: Temporal order (velocity): BDF1-CF (1.0029), BDF2-CF (1.9282), BDF3-CF (3.2982), BDF4-CF (4.3509)

31 Navier-Stokes, BDF-CF methods Error (L 2 ) BDF1 CF BDF2 CF BDF3 CF BDF4 CF Figure: Temporal order (pressure): BDF1-CF (1.0257), BDF2-CF (2.0578), BDF3-CF (3.1176), BDF4-CF (4.3446) h

32 Burgers eq: BDF-CF vs SBDF methods We compare the performance of some BDF-CF methods and the SBDF IMEX multistep methods of [Ascher et al.; 1995] for the viscous Burgers equation t u+ u x u =ν xx u, x ( 1, 1), t (0, 2] (27) with initial data u(x, 0) = u 0 (x) = sin(πx), and homogeneous Dirichlet boundary conditions.

33 Burgers eq: BDF-CF vs SBDF methods Spatial discretization: Gauss-Lobatto-Chebyshev spectral collocation method with N = 40 nodes. Relative error in L grid-norm is measured at time T = 2 as a function of viscosity, ν 0.1. The reference or exact solution is computed for N = 80 spatial nodes using MATLABs build-in ode45 function For different stepsizes h = 1/10, 1/20, 1/40, 1/80, compare the performance of SBDF and BDF-CF.

34 Burgers eq: BDF-CF vs SBDF methods Rel. Error (L ) SBDF1 SBDF2 SBDF3 BDF1 CF BDF2 CF BDF3 CF ν Figure: h = 1/80

35 Burgers eq: BDF-CF vs SBDF methods Rel. Error (L ) SBDF1 SBDF2 SBDF3 BDF1 CF BDF2 CF BDF3 CF ν Figure: h = 1/40

36 Burgers eq: BDF-CF vs SBDF methods Rel. Error (L ) SBDF1 SBDF2 SBDF3 BDF1 CF BDF2 CF BDF3 CF ν Figure: h = 1/20

37 Burgers eq: BDF-CF vs SBDF methods Rel. Error (L ) 10 3 SBDF SBDF2 SBDF3 BDF1 CF 10 5 BDF2 CF BDF3 CF ν Figure: h = 1/10

38 DIRK-CF methods on Navier-Stokes Direct applications of DIRK-CF methods on the discrete equations leads to a loss in order of convergence. Possible reasons include low stage order of the constituent DIRK methods [Hairer & Wanner,2ed.; 1996] insufficient treatment of the vector fields on the constraint manifold [O. Verdier]

39 Can we fix this? Replace the discrete problem ẏ = C(y)y + Ay D T z, Dy = 0 (28) by its restriction on the constraint manifold M :={y : Dy = 0}, namely: ẏ = ΠC(y)y +ΠAy, (29) where Π represents the orthogonal projection on M. Apply the DIRK-CF method on (29) instead of (28).

40 Can we fix this? slope = Error ( L 2 ) 10 6 slope = NS Projected Figure: Toy Navier-Stokes: dim(y) = 20, dim(z) = 10; DIRK-CF3L; Error in y : T = 1, h = T/2 l, l = 2 : 8 h

41 Can we fix this? slope = Error (L 2 ) slope = NS Projected h Figure: Toy Navier-Stokes: dim(y) = 20, dim(z) = 10; DIRK-CF4; Error in y : T = 1, h = T/2 l, l = 2 : 8

42 Conclusion Classical order theory required Efficient implementation of the semi-lagrangian interpolations required Work still in progress. Thanks for your attention!

Index. higher order methods, 52 nonlinear, 36 with variable coefficients, 34 Burgers equation, 234 BVP, see boundary value problems

Index. higher order methods, 52 nonlinear, 36 with variable coefficients, 34 Burgers equation, 234 BVP, see boundary value problems Index A-conjugate directions, 83 A-stability, 171 A( )-stability, 171 absolute error, 243 absolute stability, 149 for systems of equations, 154 absorbing boundary conditions, 228 Adams Bashforth methods,