Three approaches for the design of adaptive time-splitting methods

Transcription

1 Three approaches for the design of adaptive time-splitting methods Mechthild Thalhammer Leopold Franzens Universität Innsbruck, Austria Workshop on Geometric Integration and Computational Mechanics Organisers: Fernando Casas, Elena Celledoni, David Martin de Diego FoCM 27, Barcelona, July 27

2 Theme Splitting methods Splitting methods. Time integration of nonlinear evolution equations by exponential operator splitting methods { u (t) = F ( u(t) ) = A ( u(t) ) + B ( u(t) ), t (,T ), u() given. Linear case. For linear evolution equations, exponential operator splitting methods can be cast into general form (time grid points = t < < t N T and stepsizes τ n = t n t n ) u (t) = A u(t) + B u(t), t (,T ), s u n = e b j τ n B a e j τ n A un u(t n ) = e τ n (A+B) u(tn ), n {,..., N }. j = Areas of application. Schrödinger equations (Quantum mechanics) Damped wave equations (Nonlinear acoustics) Parabolic equations (Pattern formation) Kinetic equations (Plasma physics)

3 Main theme Splitting methods Local error control. Use of local error control to adjust time stepsize ( τ optimal = τ current min α max,max (α min, p+ α )) tol err local in general enhances reliability and efficiency of time integration. Question. How to construct estimators for local error in the context of splitting methods? Approches. Different approaches rely on embedded splitting methods (with OTHMAR KOCH), defect-based a posteriori local error estimators (with HARALD HOFSTÄTTER, OTHMAR KOCH, WINFRIED AUZINGER), associated approximations with negligible additional cost (with SERGIO BLANES, FERNANDO CASAS).

4 Outline Splitting methods Main theme. Design and theoretical analysis of local error estimators for exponential operator splitting methods. Outline. Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Numerical illustrations Nonlinear Schrödinger equations Diffusion-reaction systems

5 General form Fundamental tools, Examples Exponential operator splitting methods for nonlinear evolution equations Calculus of Lie-derivatives and Gröbner Alekseev formula

6 General form Fundamental tools, Examples Exponential operator splitting methods Splitting methods. For nonlinear evolution equations of form { u (t) = F ( u(t) ) = A ( u(t) ) + B ( u(t) ), t (,T ), u() given, determine approximations at time grid points = t < < t N T with associated stepsizes τ n = t n t n for n {,..., N } by recurrence u n = S F (τ n,u n ) u(t n ) = E F ( τn,u(t n ) ). Splitting methods rely on presumption that corresponding subproblems are solvable in accurate and efficient manner v (t) = A ( v(t) ), w (t) = B ( w(t) ), v(t) = E A ( t, v() ), w(t) = EB ( t, w() ). High-order splitting methods are cast into following format with suitably chosen real (or complex) coefficients S F (τ, ) = E B (b s τ, ) E A (a s τ, ) E B (b τ, ) E A (a τ, ) E F (τ, ).

7 Compact formulation Splitting methods General form Fundamental tools, Examples Compact formulation. Calculus of Lie-derivatives permits compact formulation and reveals analogies to significantly simpler linear case e a τd A e b τdb e a sτd A e b sτd B = E B (b s τ, ) E A (a s τ, ) E B (b τ, ) E A (a τ, ). Recipe. In order to extend result for linear case to nonlinear case, replace operator A,B by Lie-derivatives D A,D B and reverse order of evolution operators.

8 Calculus of Lie-derivatives General form Fundamental tools, Examples Formal calculus. Calculus of Lie-derivatives is suggestive of less involved linear case, see for instance HAIRER, LUBICH, WANNER (22) and SANZ-SERNA, CALVO (994). Problem. Consider nonlinear evolution equation on Banach space involving (unbounded) nonlinear operator F : D(F ) X X Employ formal notation for exact solution u (t) = F ( u(t) ), t (,T ). u(t) = E F ( t,u() ) = e td F u(), t [,T ]. Evolution operator, Lie-derivative. For (unbounded) nonlinear operator G : D(G) X X define evolution operator and Lie-derivative by e td F G v = G ( E F (t, v) ), D F G v = G (v)f (v). Remark. Definition of Lie-derivative is natural extension of identity L = d dt e tl t= d dt e td F G v = d t= dt G ( E F (t, v) ) = G ( E F (t, v) ) F ( E F (t, v) ) t= = G (v)f (v) = D F G v. t=

9 Example methods (p =, 2) General form Fundamental tools, Examples Low-order methods. First-order Lie Trotter splitting method a = = b, S F (τ, ) = e τd B e τd A. Second-order Strang splitting method a = 2 = a 2, b =, b 2 =, S F (τ, ) = e 2 τd A e τd B e 2 τd A.

10 Example methods (p = 4) General form Fundamental tools, Examples Higher-order methods. Symmetric fourth-order splitting method by BLANES, MOAN (22) a =, a 2 = = a 7, a 3 = = a 6, a 4 = 2 (a 2 + a 3 ) = a 5, b = = b 7, b 2 = = b 6, b 3 = = b 5, b 4 = 2(b + b 2 + b 3 ). Stability ensured for evolution equations of Schrödinger type. Symmetric fourth-order splitting method by YOSHIDA (s = 4, complex variant of famous scheme) α = i, β = i, a = 2 α, a 2 = 2 (α + β) = a 3, a 4 = a, b = α = b 3, b 2 = β, b 4 =. Stability ensured for evolution equations of parabolic type, since R(a j ),R(b j ) for j {,...,4}.

11 General form Fundamental tools, Examples Nonlinear variation-of-constants formula Theorem (Gröbner Alekseev formula) The solutions to the autonomous problem { d dt E G (t t, v) = G ( E G (t t, v) ), t t T, E G (, v) = v, and the related non-autonomous problem { d dt E G+R (t, t, v) = G ( E G+R (t, t, v) ) + R(t), t t T, E G+R (t, t, v) = v, satisfy the integral relation t ( E G+R (t, t, v) E G (t t, v) = 2 E G t τ,eg+r (τ, t, v) ) R(τ) dτ. t

12 Proof Splitting methods General form Fundamental tools, Examples The fundamental identity ( 2 E G (t, v)g(v) = 2 E G t,eg (, v) ) G ( E G (, v) ) = d ( ds E G t,eg (s, v) ) s= E G (t + s, v) = G ( E G (t, v) ) s= = d ds implies 2 E G (t τ, w)g(w) = G(E G (t τ, w)) for w = E G+R (τ, t, v). A brief calculation shows the stated relation E G+R (t, t, v) E G (t t, v) = E G ( t τ,eg+r (τ, t, v) ) t = = t t t t d dτ E G( t τ,eg+r (τ, t, v) ) dτ τ=t ( 2 E G ( t τ,eg+r (τ, t, v) )( G ( E G+R (τ, t, v) ) + R(τ) ) G ( ( E G t τ,eg+r (τ, t, v) ))) dτ t ( = 2 E G t τ,eg+r (τ, t, v) ) R(τ) dτ. t

13 Embedded splitting methods Defect-based local error estimators Associated approximations Approaches for design and analysis of local error estimators

14 Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Approaches. Study different approaches for design and theoretical analysis of local error estimators for splitting methods. Embedded splitting methods O. KOCH, CH. NEUHAUSER, M. TH. Embedded exponential operator splitting methods for the time integration of nonlinear evolution equations (23). A posteriori local error estimators W. AUZINGER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, with application to Schrödinger equations. Part I. The linear case (22). W. AUZINGER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, with application to Schrödinger equations. Part II. Higher-order methods for linear problems (24). W. AUZINGER, H. HOFSTÄTTER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, with application to Schrödinger equations. Part III. The nonlinear case (25). Approximations with negligible additional cost (recent work with SERGIO and FERNANDO) Simplification. Specify local error estimators for first time step (τ > ).

15 Embedded splitting methods Defect-based local error estimators Associated approximations Embedded splitting methods Examples and theoretical basis

16 Embedded splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Heuristic approach. Consider splitting method of nonstiff order p u = s e a s+ j τd A e b s+ j τd B u. j = Design related splitting method of nonstiff order p such that certain coefficients coincide û = ŝ τd A e b s+ j τd B eâs+ j u. j = Use difference between two approximations as local error estimator err local = u û X. Remark. Approach in spirit of embedded Runge-Kutta methods (but with higher cost).

17 Embedded splitting methods Defect-based local error estimators Associated approximations Example (Schrödinger equations) Example. Favourable scheme (p = 4, BLANES & MOAN) and embedded scheme ( p = 3, KOCH & TH.). j a j 2, , ,5 /2 (a 2 + a 3 ) j â j a 2 a 2 3 a 3 4 a j b j, , , (b + b 2 + b 3 ) j b j b 2 b 2 3 b 3 4 b

18 Example (Parabolic equations) Embedded splitting methods Defect-based local error estimators Associated approximations Example. Complex scheme (p = 4, YOSHIDA) and embedded scheme ( p = 3, KOCH & TH.). j 2, i i j a i i i a j â j j, i 2, i j b i i 4 b j b j

19 Theoretical justification Embedded splitting methods Defect-based local error estimators Associated approximations Theoretical justification. Consider high-order splitting methods and employ local error representations that are suitable for nonlinear evolution equations involving unbounded operators. Theorem (Th. 28, Th. 22, Koch & Neuhauser & Th. 23) A splitting method of nonstiff order p admits the (formal) expansion L F (t, v) = p k= µ N k µ p k C kµ = k µ! t k+ µ C kµ α λ λ Λ k l= l= k b λl c µ l ad µ l D A (D B ) e td A v + R p+ (t, v), k λ l µ l + +µ k +k l+. l= Main tools. Calculus of Lie derivatives, Gröbner Alekseev formula. Remark. Further considerations show that resulting (redundant) stiff order conditions C kµ = coincide with nonstiff order conditions.

20 Theoretical justification Embedded splitting methods Defect-based local error estimators Associated approximations Remarks. Application of local error representation to different classes of (non)linear evolution equations such as Schrödinger equations or diffusion-reaction systems requires characterisation of domains of iterated Lie-commutators (regularity and consistency requirements). In connection with Schrödinger equations, it is often justified to assume that exact solution is regular. For linear equations versus nonlinear equations, the regularity requirements are D = H p (Ω), D = H 2p (Ω). For sufficiently regular solutions (bounded in D), above local error representation implies L F (τ, v) = S F (τ, v) E F (τ, v) = O ( τ p+). Provided that p > p, this justifies use of local error estimator err local = u û X = O ( τ p+).

21 Embedded splitting methods Defect-based local error estimators Associated approximations Global error estimate (Full discretisations) Discretisation. Full discretisation of nonlinear Schrödinger equations (GPE) by high-order variable stepsize time-splitting methods combined with pseudo-spectral methods (Fourier, Sine, Hermite). Theorem (Th. 22) Provided that exact solution remains bounded in fractional power space X β defined by principal linear part for β p, global error estimate holds un M u(t N ) ( u X C u() X + τ p max + M q). Extensions. Time-dependent Gross Pitaevskii equations with additional rotation term, see HOFSTÄTTER, KOCH, TH. (24). Multi-revolution composition time-splitting pseudo-spectral methods for highly oscillatory problems (with CHARTIER, MÉHATS).

22 Embedded splitting methods Defect-based local error estimators Associated approximations Defect-based a posteriori local error estimators Examples and theoretical basis

23 Alternative approach Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Approach. Consider nonlinear evolution equation and deduce evolution equation of similar form for splitting operator d dt E F (t, ) = (D A + D B )E F (t, ), s S F (t, ) = e a s+ j td A e b s+ j td B. j = Employ Gröbner Alekseev formula and suitable further expansion. For low-order methods, obtain compact local error representation t τ L F (t, ) = e τ D A e τ 2 D [ ] B D A,D B e (τ τ 2 )D B e (t τ )D F dτ2 dτ t τ ( = 2 E F t τ,s F (τ, ) ) ( 2 E B τ τ 2,E A (τ, ) )[ B, A ]( ( E B τ2,e A (τ, ) )) dτ 2 dτ. As rigorous extension to higher-order splitting methods becomes highly involved, study linear case and use formal extension. Remark. Related approach studied in context of (non)linear Schrödinger equations in semi-classical regime, see DESCOMBES, TH. (2, 22).

24 Embedded splitting methods Defect-based local error estimators Associated approximations A posteriori local error estimators (Linear case) Approach. Consider linear evolution equation { E F (t) = ( A + B ) E F (t), t (,T ), E F () = I. For splitting method, deduce evolution equation of similar form { S F (t) = ( A + B ) S F (t) + D F (t), t (,T ), S F () = I. Local error L F = S F E F satisfies evolution equation { L F (t) = ( A + B ) L F (t) + D F (t), t (,T ), L F () =. Employ variation-of-constants formula to obtain integral representation for local error involving defect L F (t) = t e (t τ)(a+b) D F (τ) dτ.

25 Embedded splitting methods Defect-based local error estimators Associated approximations A posteriori local error estimators (Linear case) Approach. Recall integral representation for local error L F (t) = t e (t τ)(a+b) D F (τ) }{{} =f (τ) Apply Hermite quadrature approximation and use that in present situationvalidity of order conditions implies f () = = f (p ) () = p l= ω l t l+ f (l) () } {{ } = + p+ t f (t) t dτ. f (τ) dτ = O ( t p+2). For any splitting method of order p, obtain asymptotically correct defect-based a posteriori local error estimator P F (t) = p+ t D F (t), P F (t) L F (t) = O ( t p+2).

26 A posteriori local error estimators Embedded splitting methods Defect-based local error estimators Associated approximations Result. Asymptotically correct a posteriori local error estimator associated with splitting method of order p given by D F = s k= S m k m (t) = e b j tb e a j t A, j =k S s k a k A S k S F (t) E F (t) = O ( t p+), s + k= S F (t) = S s (t), S s k+ b kb S k ( A + ( b s )B ) S F (t), P F = p+ t D F, P F (t) L F (t) = O ( t p+2). Extension to nonlinear evolution equations by calculus of Lie-derivatives. Theoretical analysis. In context of linear Schrödinger equations, rigorous analysis given in AUZINGER, KOCH, TH. (22, 24). Corresponding result for nonlinear case deduced in AUZINGER, HOFSTÄTTER, KOCH, TH. (25) for second-order Strang splitting method.

27 Embedded splitting methods Defect-based local error estimators Associated approximations Special case (Lie Trotter splitting method) Special case. A posteriori local error estimator for Lie Trotter splitting method applied to linear evolution equation given by P F (t, v) = 2 t D F (t, v), D F (t, v) = ( e tb e t A A A e tb e t A) v. Extension to nonlinear case yields D F (t, v) = 2 E B ( t,e A (t, v) ) 2 E A (t, v) A v A E B ( t,e A (t, v) ). Explanation. Extension by formal calculus of Lie-derivatives implies D F (t, v) = D A e td A e td B v e td A e td B D A v and G(v) = e td A e td B v = E B ( t,e A (t, v) ), G (v) = 2 E B ( t,e A (t, v) ) 2 E A (t, v), e td A e td B D A v = A E B ( t,e A (t, v) ), D A e td A e td B v = G (v) A v = 2 E B ( t,e A (t, v) ) 2 E A (t, v) A v. Remark. Improved approximation S F (t, ) P F (t, ) = E F (t, ) + O ( t p+2). Realisation and computational effort. Realisation for nonlinear Schrödinger equations (Gross Pitaevskii equation) straightforward. Computational effort comparable with splitting pair Lie/Strang (two additional applications of A required, FFT) P (t, v) = e it (U+ϑ w 2 ) ( A w iϑt ( A w w 2 + A w w 2)) A e it (U+ϑ w 2 ) w, w = e t A v. Explanation. With G(v) = e it (U+ϑ w 2 ) w, G (v) = e it (U+ϑ w 2 ) ( e t A ( ) iϑt ( w e t A ( ) + w e t A ( ) ) ) w obtain e td A e td B D A v = A e it (U+ϑ w 2 ) w, D A e td A e td B v = e it (U+ϑ w 2 ) ( A w iϑ t ( Aw w 2 + A w w 2)).

28 Embedded splitting methods Defect-based local error estimators Associated approximations Associated approximations with negligible additional computational cost Examples and theoretical basis

29 Novel approach Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Approach. Consider nonlinear evolution equation { u (t) = A ( u(t) ) + B ( u(t) ), t (,T ), u() = u. Realise higher-order splitting method in straightforward manner u = u n for j = : s end u = E A (a j τ n,u) Solution of subproblem u (t) = A ( u(t) ) u = E B (b j τ n,u) Solution of subproblem u (t) = B ( u(t) ) u n+ = u Use suitable linear combination of intermediate values to compute associated approximation that serves as local error estimator.

30 Novel approach Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Schrödinger equations. Consider splitting method by BLANES, MOAN p = 4, s = 7. Associated third-order approximation obtained by certain linear combination of intermediate values yields local error estimator u = u n for j = : s u = E A (a j τ n,u) u = E B (b j τ n,u) end u n+ = u u Estimator = α u u Estimator = u Estimator + α 2j u u Estimator = u Estimator + α 2j u Local error estimator = u u Estimator Parabolic equations. Consider instead splitting method by YOSHIDA with complex coefficients and melt two subsequent time steps (p = 4, s = 7).

31 Novel approach Splitting methods Embedded splitting methods Defect-based local error estimators Associated approximations Benefit. Compared to approaches based on embedded splitting methods or defect-based local error estimators, novel approach leads to local error estimators with negligible additional computational cost. Open questions. Provide coefficients for favourable higher-order splitting methods. Numerical tests confirm stability of associated approximations. Rigorous argument?

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33 Local error control Splitting methods Local error control. Use of local error control to adjust time stepsize ( τ optimal = τ current min α max,max (α min, p+ α α max =.5, α min =.2, α =.25, enhances reliability and efficiency of time integration. tol err local )),

34 Illustration (Schrödinger equation) Test equation (see BAO ET AL.). Consider nonlinear Schrödinger equation under harmonic potential (d =, ω = 2, ϑ = ) ( i t ψ(x, t) = 2 ε + ε U (x) + ε ϑ ) ψ(x, t) 2 ψ(x, t). Small value of (semi-classical) parameter ε > causes high oscillations in initial condition and solution ψ(x,) = ϱ (x)e i ε σ (x), ϱ (x) = e x2, σ (x) = ln ( e x + e x). Use Fourier spectral space discretisation combined with fourth-order time-splitting method by BLANES & MOAN (x [ 8,8], M = 892, t [,3]).

35 Illustration (Solution behaviour for ε = 2 ) First observation. Even a simple local error control for second-order Strang splitting method based on first-order Lie Trotter splitting method is useful to enhance reliability! See Movie. Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator "embedded" (p = 2, Strang / Lie Trotter) Current time t = 3 (Tol =., dt = e 5, N = 763, FFTs = 3528, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator "embedded" (p = 2, Strang / Lie Trotter) Current time t = 3 (Tol =., dt = e 5, N = 563, FFTs = 262, M = 892).8.8 ψ(x,t) 2.6 ψ(x,t) x.5.5 x Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator "embedded" (p = 2, Strang / Lie Trotter) Current time t = 3 (Tol =., dt = e 5, N = 768, FFTs = 526, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator "embedded" (p = 2, Strang / Lie Trotter) Current time t = 3 (Tol =., dt = e 5, N = 223, FFTs = 2446, M = 892).8.8 ψ(x,t) 2.6 ψ(x,t) x.5.5 x

36 Illustration (Global error) Expectation. Use of higher-order methods will enhance efficiency. 5 Nonlinear Schrödinger equation (ε =, T = ) Estimator "embedded" (p =, Strang / Lie Trotter) Integrator "embedded" (p = 2, Strang / Lie Trotter) Estimator novel approach (p = 3, Blanes & Moan) Estimator "embedded" (p = 3, Blanes & Moan / unrelated) Estimator embedded (p = 3, Blanes & Moan / Koch & Th.) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Integrator novel approach (p = 4, Blanes & Moan) Global error 5 2 Time stepsize

37 Illustration (ε = ) Splitting methods Comparison. Compare approach based on embedded splitting methods with novel approach. Obtain expected results for ε =, T =, Tol = 4. Higher-order method superior to low-order method (e.g. with respect to number of FFT transforms). Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 2, Strang / Lie Trotter) Current time t = 2 (Tol =., dt =., N = 483, FFTs = 966, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 2 (Tol =., dt =., N = 258, FFTs = 548, M = 5).8.8 ψ(x,t) 2.6 ψ(x,t) x.5.5 x

38 Illustration (ε = ) Splitting methods Comparison. Compare approach based on embedded splitting methods with novel approach. Obtain expected results for ε =, T =, Tol = 4. Good performance of novel approach in comparison with embedded method (e.g. with respect to number of FFT transforms). Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 2 (Tol =., dt =., N = 267, FFTs = 243, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 2 (Tol =., dt =., N = 258, FFTs = 548, M = 5).8.8 ψ(x,t) 2.6 ψ(x,t) x.5.5 x

39 Illustration (ε = ) Splitting methods Best performance. Observe best performance for novel approach (FFTs). Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) Current time t = 2 (Tol =., dt =., N = 96, FFTs = 864, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) Current time t = 2 (Tol =., dt =., N = 67, FFTs = 53, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) Current time t = 2 (Tol =., dt =., N = 293, FFTs = 2637, M = 5) ψ(x,t) 2 ψ(x,t) 2 ψ(x,t) x x x Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 2 (Tol =., dt =., N = 78, FFTs = 72, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 2 (Tol =., dt =., N = 45, FFTs = 35, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 2 (Tol =., dt =., N = 267, FFTs = 243, M = 5) ψ(x,t) 2 ψ(x,t) 2 ψ(x,t) x x x Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 2 (Tol =., dt =., N = 9, FFTs = 54, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 2 (Tol =., dt =., N = 49, FFTs = 894, M = 5) Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 2 (Tol =., dt =., N = 258, FFTs = 548, M = 5) ψ(x,t) 2 ψ(x,t) 2 ψ(x,t) x x x

40 Illustration (ε = ) Splitting methods Time stepsizes. Compare sequences of time stepsizes. Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 96, FFTs = 864, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 67, FFTs = 53, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 293, FFTs = 2637, M = 5 Time stepsize Time stepsize Time stepsize Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 78, FFTs = 72, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 45, FFTs = 35, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 267, FFTs = 243, M = 5 Time stepsize Time stepsize Time stepsize Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 9, FFTs = 54, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 49, FFTs = 894, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 258, FFTs = 548, M = 5 Time stepsize Time stepsize Time stepsize

41 Illustration (ε = ) Splitting methods Local errors. Compare local errors. Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 96, FFTs = 864, M = 5 x 3 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 67, FFTs = 53, M = 5 x 4 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator "embedded" (p = 4, Blanes & Moan / unrelated) T = 2, Tol =., dt =., N = 293, FFTs = 2637, M = 5 x 5 Local errors Local errors Local errors Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 78, FFTs = 72, M = 5 9 x 3 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 45, FFTs = 35, M = 5 9 x 4 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 2, Tol =., dt =., N = 267, FFTs = 243, M = 5 x 5 Local errors Local errors Local errors Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 9, FFTs = 54, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 49, FFTs = 894, M = 5 Nonlinear Schrödinger equation (ε =, ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 2, Tol =., dt =., N = 258, FFTs = 548, M = 5 5 x 3 x 4 x 5 Local errors Local errors Local errors

42 Illustration (ε = 2 ) Splitting methods Observation. Time integration of test equation for ε = 2 is delicate task. Number of time steps does not necessarily increase for smaller tolerances. Influence of choice of initial time stepsize? Improvement of local error control in this situation? Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 3 (Tol =., dt = e 5, N = 237, FFTs = 2384, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 3 (Tol =., dt = e 5, N = 2468, FFTs = 22239, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) Current time t = 3 (Tol =., dt = e 5, N = 2287, FFTs = 26, M = 892) ψ(x,t) 2.6 ψ(x,t) 2.6 ψ(x,t) x.5.5 x.5.5 x Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 3 (Tol =., dt = e 5, N = 5788, FFTs = 34752, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 3 (Tol =., dt = e 5, N = 569, FFTs = 342, M = 892) Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) Current time t = 3 (Tol =., dt = e 5, N = 8825, FFTs = 5295, M = 892) ψ(x,t) 2.6 ψ(x,t) 2.6 ψ(x,t) x.5.5 x.5.5 x

43 Illustration (ε = 2 ) Splitting methods Time stepsizes. Compare sequences of time stepsizes. Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 237, FFTs = 2384, M = 892 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 2468, FFTs = 22239, M = 892 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 2287, FFTs = 26, M = Time stepsize Time stepsize Time stepsize Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 5788, FFTs = 34752, M = 892 x Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 569, FFTs = 342, M = 892 x Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 8825, FFTs = 5295, M = 892 x Time stepsize Time stepsize 2.5 Time stepsize

44 Illustration (ε = 2 ) Splitting methods Local errors. Compare local errors. Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 237, FFTs = 2384, M = 892 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 2468, FFTs = 22239, M = 892 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator embedded (p = 4, Blanes & Moan / Koch & Th) T = 3, Tol =., dt = e 5, N = 2287, FFTs = 26, M = 892 x 3 8 x 4 9 x Local errors 5 4 Local errors 5 4 Local errors Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 5788, FFTs = 34752, M = 892 x 3 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 569, FFTs = 342, M = 892 x 4 Nonlinear Schrödinger equation (ε =., ω = 2, θ = ) Integrator novel approach (p = 4, Blanes & Moan) T = 3, Tol =., dt = e 5, N = 8825, FFTs = 5295, M = x Local errors 3 2 Local errors 4 3 Local errors

45 Illustration (Gray Scott equations) Test equation. Consider system of diffusion reaction equations (d = 2) { t u(x, t) = ( D u α ) u(x, t) u(x, t) ( v(x, t) ) 2 + α, t v(x, t) = ( D v β ) v(x, t) + u(x, t) ( v(x, t) ) 2. Special choice of parameters causes formation of patterns over long times D u = 2D v =.6, f =.35, k =.6, α = f, β = f + k, see also NICOLAS P. ROUGIER. Use Fourier spectral space discretisation combined with fourth-order complex time-splitting method by YOSHIDA (x [ 75,75], M = 5 5). Movie Formation of patterns

46 Illustration (Global error) 5 Gray Scott equations (T = ) Estimator "embedded" (p =, Strang / Lie Trotter) Integrator "embedded" (p = 2, Strang / Lie Trotter) Estimator novel approach (p = 3, complex Yoshida) Estimator novel approach (p = 3, Blanes & Moan) Integrator novel approach (p = 4, Blanes & Moan) Integrator novel approach (p = 4, complex Yoshida) Global error 5 2 Time stepsize

47 Conclusions and future work Conclusions. Adaptivity in time essential for reliable and efficient numerical simulations. Novel approach for time-splitting methods provides local error estimators with negligible additional cost. Open questions. Theoretical understanding of novel local error estimators (favourable stability behaviour). Design of higher-order schemes. Detailed study of local error control for semi-classical Schrödinger equation to understand unexpected behaviour. Thank you!