Splitting methods with boundary corrections

Transcription

1 Splitting methods with boundary corrections Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017

2 Strang s paper, SIAM J. Numer. Anal., 1968 S (5) k f = e k 2 A x e kb y e k 2 A x f or rather second-order approximations to the exponentials

3 Neumann boundary conditions t u = 1 (a 1 u) + 2 (a 2 u), Ω = (0, 1) Neumann Experiment 2 a(x 1, x 2 ) = 16x 1 (1 x 1 )x 2 (1 x 2 ) + 1 ( ) u 0 (x 1, x 2 ) = c exp 1 x 1 1(1 x 1) x 2(1 x 2) error Ψ τ (1/3) Φ τ Φ τ (1,2) Φ τ (1,3) τ T /τ Ψ τ Φ τ Φ τ (1, 2) Φ τ (1, 3)

4 Dirichlet and periodic boundary conditions 10 4 Dirichlet Experiment 10 6 Periodic Experiment error error Ψ τ (1/3) Φ τ Φ τ (1,2) Φ τ (1,3) τ Ψ τ (1/3) Φ τ Φ τ (1,2) Φ τ (1,3) τ E. Hansen, A.O., High order splitting methods for analytic semigroups exist. BIT Numer. Math. 49, (2009)

5 Error is concentrated on the boundary x x Dirichlet problem Neumann problem Pointwise errors at time T = 0.1 for the splitting Φ h (1, 3), step size τ = T /512. Large errors are located along a thin strip around the boundary.

6 Outline Diffusion-reaction splitting Oblique boundary conditions References

7 Diffusion-reaction equations: Dirichlet problem Diffusion-reaction initial-boundary value problem u t = Du + f (u) u Ω = b u(0) = u 0 where u = u(t, x) for 0 < t T and x Ω R d ; D is an elliptic differential operator (e.g., the Laplacian); f : R R is the reaction term (usually f (0) = 0); b : [0, T ] Ω R is allowed to depend on time.

8 Diffusion-reaction splitting The system u t = Du + f (u), u Ω = b is split up into v t = Dv, v Ω = b w t = f (w) Numerical example in Ω = (0, 1) with u 0 (x) = 1 + sin 2 πx, f (u) = u 2, 500 grid points, b 0 = b 1 = 1. Error at t = 0.1 Strang Strang (modified) step size l 2 error order l 2 error order 2.000e e e e e e e e e e e e e e e

9 Error analysis for Lie splitting Reduction to homogeneous Dirichlet boundary conditions: let Dz = 0, z Ω = b and consider U = u z which satisfies U t = DU + f (U + z) z t, U Ω = 0, U(0) = u 0 z 0. Write PDE as an abstract parabolic problem U t = AU + k(t) + g(t, U), U(0) = u 0 z 0, where D(A) = H 2 (Ω) H0 1 (Ω), e.g., and split. The leading term in the local error is then τ 2 A g(t k, U(t k )).

10 Abstract convergence, classic Lie splitting Theorem. (L. Einkemmer, AO, SIAM J. Sci. Comput., 2015) The classic Lie splitting is convergent of order τ log τ, i.e. u n u(t n ) Cτ(1 + log τ ), 0 nτ T, where C depends on T but is independent of τ and n. Proof. We employ the parabolic smoothing property to bound e ta ( ta) α C, α 0 e (n k 1)τA Ag(t k, U(t k )) τ 2 n 1 k=0 which is the leading error term.

11 Numerical results for Lie splitting Numerical example in Ω = (0, 1) with u 0 (x) = 1 + sin 2 πx, f (u) = u 2, 500 grid points, b 0 = b 1 = 1. Error at t = 0.1 Lie Lie (modified) step size l error order l error order 2.000e e e e e e e e e e e e e e e e e e e e e

12 Essential modification step The critical error term is: e (n k 1)τA Ag(t k, U(t k )) Satisfy a compatibility condition for the nonlinearity. (E. Hansen, F. Kramer, AO, Appl. Numer. Math., 2012) Split the nonlinearity f (U + z) into a term g(t, U) such that g(t, 0) = 0 and a second term that does not depend on U. Obvious choice: v t = Dv + f (z) z t, v Ω = 0 Modified nonlinearity w t = f (w + z) f (z) satisfies g(t, 0) = 0 as required. g(t, U) = f (U + z(t)) f (z(t))

13 Abstract convergence, Strang splitting Theorem. (L. Einkemmer, AO, SIAM J. Sci. Comput., 2015) The classic Strang splitting is first-order convergent. The modified Strang splitting is second-order convergent. Proof. The leading error term is e (n k 1)τA A 2 g(t k, U(t k )). One power of A is bounded by parabolic smoothing; Another power of A is bounded by the modification. Remark. Spatially smooth functions lie in D(( ) 1/4 ε ); therefore one observes order 1.25 in L 2 for uncorrected splitting (order p in Lp ).

14 1D example, time dependent boundary conditions 1D example, Ω = (0, 1), and f (u) = u 2, initial value u 0 (x) = 1 + sin 2 πx, 500 grid points, boundary values b 0 (t) = b 1 (t) = 1 + sin 5t. Strang Strang (modified) step size l error order l error order 2.000e e e e e e e e e e e e e e e e e e e e e

15 2D example, time invariant boundary conditions u Strang Strang (modified) step size l error order l error order e e e e e e e e

16 Outline Diffusion-reaction splitting Oblique boundary conditions References

17 Model problem with oblique b.c. Diffusion-reaction problem on domain Ω R d where u t = Du + f (u) Bu Ω = b u(0) = u 0 D = d i,j=1 d ij(x) ij + d i=1 d i(x) i + d 0 (x)i is an elliptic operator with positive definite (d ij (x)); B = d i=1 β i(x) i + α(x)i is a first-order operator; B satisfies the uniform non tangentiality condition inf d x Ω i=1 β i(x)n i (x) > 0. Neumann problem: α = 0, β i (x) = d j=1 d ij(x)n j (x) for all i.

18 Condition on the correction Choose as correction a smooth function q that satisfies the boundary conditions of f (u) Since Bq n (0) Ω = Bf (u(t n )) Ω + O(τ). Bf (u) Ω = αf (u) Ω + f (u) d i=1 β i(x) i u Ω = αf (u) Ω + f (u)(b αu) Ω and our numerical methods converge at least with order one, we can simply take Bq n Ω = αf (u n ) Ω + f (u n ) ( b n αu n ) Ω. Dirichlet case: α = 1, β 1 =... = β s = 0 and q Ω = f (b). (previously called f (z))

19 Modified splitting With the correction q n satisfying Bq n Ω = αf (u n ) Ω + f (u n ) ( b n αu n ) Ω. at hand, we consider the boundary-corrected splitting t v n = Dv n + q n, t w n = f (w n ) q n, Bv n Ω = b n and solve it on the time interval [t n, t n+1 ] by the standard Lie or Strang approach.

20 Modified Strang splitting For a given initial value u n, first solve t v n = Dv n + q n, Bv n Ω = b n with initial value v n (0) = u n to obtain v n ( τ ). 2 Next, integrate t w n = f (w n ) q n w n (0) = v n ( τ ) to obtain w 2 n(τ). with initial value Finally, integrate once more t ṽ n = Dṽ n + q n, Bṽ n Ω = b n but this time with initial value ṽ n (0) = w n (τ), and set u n+1 = S τ u n = ṽ n ( τ ). 2

21 Convergence results Theorem. (L. Einkemmer, AO, arxiv: , 2016) The modified Strang splitting scheme is second-order convergent. More precisely, the global error satisfies u n u(t n ) Cτ 2 (1 + log τ ), 0 nτ T, where C depends on T but is independent of τ and n. Orders of convergence for classic Strang splitting in various norms: boundary type L 1 L 2 L β 1 =... = β d = j with β j

22 Inhomogeneous Neumann boundary conditions We take f (u) = u 2, Ω = (0, 1) with b 0 = 0 and b 1 = 1, 500 points. Admissible correction q n (s, x) = x 2 u n (1). Strang Strang step size l error order l 2 error order 3.125e e e e e e e e e e e e e e e Strang (modified) Strang (modified) step size l error order l 2 error order 3.125e e e e e e e e e e e e e e e

23 Mixed Dirichlet/Neumann boundary conditions Dirichlet b.c. (with b 0 = 1) and Neumann b.c. (with b 1 = 1). Correction is given by q n (s, x) = 1 + 2xu n (1). Strang Strang step size l error order l 2 error order 1.250e e e e e e e e e e e e e e e Strang (modified) Strang (modified) step size l error order l 2 error order 1.250e e e e e e e e e e e e e e e

24 Various possible corrections Problem with Dirichlet b.c. (b 0 = 1, b 1 = 2) and 500 grid points. The error is given at t = 0.1 with a step size τ = method correction l error Lie none 2.31e-03 Lie (mod.) harmonic: q = 1 + x 1.20e-03 Lie (mod.) q = 1 + x + sin πx 2.43e-03 Lie (mod.) q = 1 + x + sin 10πx 3.21e-03 method correction l error Strang none 1.84e-03 Strang (mod.) harmonic: q = 1 + x 1.20e-06 Strang (mod.) q = 1 + x + sin πx 1.27e-06 Strang (mod.) q = 1 + x + sin 10πx 7.02e-05

25 References L. Einkemmer, A. Ostermann. A comparison of boundary correction methods for Strang splitting. Preprint ( ) L. Einkemmer, A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 2: oblique boundary conditions. SIAM J. Sci. Comput. 38, A3741 A3757 (2016) L. Einkemmer, A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions. SIAM J. Sci. Comput. 37, A1577 A1592 (2015)