Order of convergence of splitting schemes for both deterministic and - PowerPoint PPT Presentation

1 Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schr¨ odinger equations Jie Liu National University of Singapore, Singapore Numerics for Stochastic Partial Differential Equations and their Applications, RICAM, Linz, Dec. 2016.

2 Outline • The second order convergence of Strang-type splitting scheme for nonlinear Schr¨ odinger equation. • Mass preserving splitting scheme for stochastic nonlinear Schr¨ odinger equation with multiplicative noise ⋆ Explicit formula for the nonlinear step ⋆ first order strong convergence

3 Strang-type splitting scheme for nonlinear Schr¨ odinger equation Let i = √− 1 and V be a real-valued function. Consider the following nonlinear Schr¨ odinger equation in R d . idu = ∆ udt + V ( x, | u | ) udt (1) Note three things: • If a ∈ R , i du dt = au ⇒ u ( t ) = e − iat u (0) . Hence | u ( t ) | = | u (0) | . • The exact solution of idu = V ( x, | u | ) udt , u (0) = u 0 satisfies | u ( t ) | = | u (0) | and therefore u ( x, t ) = exp {− itV ( x, | u 0 ( x ) | ) } u 0 ( x ) . • Recall the Strang splitting for du = ( Au + Bu ) dt : δt δt u ( δt ) = e δt ( A + B ) u (0) ≈ e 2 B e δtA e 2 B u (0) .

4 We will study the following scheme 1 u n − 1 → ˜ 2 → u n → ˜ 2 → u n +1 → · · · : u n − 1 u n − 1 u n + 1 u n + 1 ◦ ◦ 2 → 2 → � � − iδt u n − 1 2 = exp 2 V ( x, | u n − 1 | ) u n − 1 , (2) ˜ u n − 1 u n − 1 ◦ 2 = exp {− iδt ∆ } ˜ 2 , (3) � − iδt � u n = exp u n − 1 u n − 1 ◦ ◦ 2 V ( x, | 2 | ) 2 . (4) u n − 1 u n + 1 u n + 1 ◦ ◦ 2 → ˜ 2 → 2 (skip u n ) because It can be implemented as � � − iδt u n = exp � � u n + 1 u n − 1 u n − 1 ◦ ◦ 2 = exp 2 V ( x, | u n | ) ˜ − iδtV ( x, | 2 | ) 2 . The second order convergence in L 2 norm has been proved by Lubich 2 . 1 Hardin & Tappert 1973, Taha & Ablowitz 1984. 2 C. Lubich, On splitting methods for Schr¨ odinger-Poisson and cubic nonlinear Schr¨ odinger equations. Math. Comp., 77 (2008) 2141–2153.

5 Second order convergence Consider (1) and its numerical scheme (2)–(4) in Theorem 1. R 3 . Assume V ( x, | u | ) = | u | 2 , and take any γ > 3 / 2 and any If � u 0 � γ = M γ < ∞ , then there are constants T σ ∈ { 0 , 1 , 2 , 3 , ... } . and C 1 depending on M γ such that for any δt > 0 , n =0 , 1 ,..., [ T /δt ] � u n � γ ≤ C 1 . (5) max If � u 0 � σ +4 = M σ +4 < ∞ , then there are constants T and C 2 depending on M σ +4 such that for any δt > 0 , n =0 , 1 ,..., [ T /δt ] � u ( t n ) − u n � σ ≤ C 2 δt 2 . max (6) ¯ ¯ Assume in addition that there are constants T and M σ +4 so that ¯ sup 0 ≤ t ≤ ¯ T � u ( s ) � σ +4 = M σ +4 < ∞ , then there is a constant δt 0 > 0 so that when δt ≤ δt 0 , the T in (6) can be taken as ¯ T .

6 Integral formulation of the scheme Fix T , δt , and N = [ T/δt ] . Define a right continuous function φ N ( x, t ) with • φ N ( x, 0) = u 0 ( x ) . • φ N ( t ) = e {− i [ tV ( x, | φ N (0) | )] } φ N (0) when t ∈ [0 , δt 2 ) • On any interval [ t n − 1 2 , t n + 1 2 ) ( n = 1 , 2 , ... ),  e {− iδt ∆ } lim when t = t n − 1 2 φ N ( t ) 2 , t ↑ t n − 1   φ N ( t ) = � ( t − t n − 1 2 ) V ( x, | φ N ( t n − 1 � {− i 2 ) | ) } φ N ( t n − 1 when t ∈ [ t n − 1 2 , t n + 1  e 2 ) 2 ) .  Then φ N ( t n − 1 u n − 1 ◦ φ N ( t n ) = u n . 2 ) = and 2

7 Integral formulation of the scheme  when t = tn − 1 e {− iδt ∆ } lim 2 ,  φN ( t )  t ↑ tn − 1    2  Recall φN ( t ) = � � ( t − tn − 1 2) V ( x, | φN ( tn − 1 2) | ) {− i }   φN ( tn − 1 when t ∈ [ tn − 1 2 , tn +1   2) 2) .  e  • When t ∈ [ t n − 1 2 , t n + 1 2 ) , dφ N ( t ) = − iV ( x, | φ N ( t ) | ) φ N ( t ) dt and therefore � t φ N ( t ) = φ N ( t n − 1 (7) 2 ) − i V ( x, | φ N ( s ) | ) φ N ( s ) ds. t n − 1 2 • When t = t n + 1 2 , � t n +1   2 φ N ( t n + 1  φ N ( t n − 1  . 2 ) = e {− iδt ∆ } 2 ) − i V ( x, | φ N ( s ) | ) φ N ( s ) ds t n − 1 2 (8)

8 Integral formulation of the scheme φ N satisfies � t φ N ( t ) = e − it n ∆ φ N (0) − i � � S N,n,t ( s ) V ( x, | φ N ( s ) | ) φ N ( s ) ds 0 when t ∈ [ t n − 1 2 , t n + 1 2 ) , where φ N (0) = u (0) , 2 − j ] ( s ) e − i ( jδt )∆ + I 2 ] ( s ) + � n − 1 S N,n,t ( s ) = e − inδt ∆ I j =1 I 2 ,t ] ( s ) . 1 [ t n − 1 2 − j ,t n +1 [ t n − 1 [0 ,t Let S ( t − s ) = e − i ( t − s )∆ . The exact solution u ( t ) of NLS satisfies � t u ( t ) = e − it ∆ u (0) − i S ( t − s ) ( V ( x, | u ( s ) | ) u ( s )) ds. 0 � t n Note that 0 S N,n,t n ( s ) ds is the midpoint rule approximation of 1 � t n +1 � t n � t 2 − j � t 0 S ( t n − s ) ds on the partition 0 + � n − 1 2 2 − j + t n − 1 t n − 1 j =1 2

9 Error equation Let r ( t ) = u ( t ) − φ N ( t ) . We have r ( t ) =( e − it ∆ − e − it n ∆ ) u 0 � t − i ( S ( t − s ) − S N,n,t ( s )) V ( x, | φ N ( s ) | ) φ N ( s ) ds 0 � t − i S ( t − s ) [ V ( x, | u ( s ) | ) u ( s ) − V ( x, | φ N ( s ) | ) φ N ( s )] ds 0 =: J 1 ( t ) + J 2 ( t ) + J 3 ( t ) (9) for t ∈ [ t n − 1 2 , t n + 1 2 ) (but n is arbitrary). Note that only at t = t n , J 1 ( t n ) vanishes instead of being of size O ( δt ) .

10 Error estimate • Step 1: (First order error estimate over a short time interval) Since � fg � γ ≤ c � f � γ � g � γ when γ > d/ 2 and � e − it ∆ w � σ = � w � σ , � ( e − it ∆ − I ) w � α = t � ∆ w � α , one immediately get sup t ≤ T � φ N ( t ) � 4 < C and sup � r ( t ) � 2 ≤ Cδt t ≤ T for some T > 0 . Here r ( t ) = u ( t ) − φ N ( t ) , � r � α = � r � H α • Step 2: (Error estimate over a short time interval) There is a constant T which depends on the initial data so that n − 1 � � r ( t n ) � 0 ≤ C 3 δt � r ( t j ) � 0 + C 5 δt 2 j =1 for all n ≤ [ T/δt ] . So, second order convergence follows from the standard discrete Gronwall inequality.

11 Step 3: (Error estimate up to the blowing up time of the exact solution) Note that to prove the 2nd order convergence in L 2 , one has to prove the stability in H 4 : By Calculus inequality in the Sobolev space, �| φ N | 2 φ N � 4 ≤ C � φ N � 2 (10) 2 � φ N � 4 . Hence � φ N ( t ) � 4 ≤� φ N (0) � 4 � t ( � u ( s ) � 2 + � φ N ( s ) − u ( s ) � 2 ) 2 � φ N ( s ) � 4 ds. + C 0

12 Stochastic Schr¨ odinger equation Consider the stochastic Schr¨ odinger equation with multiplicative noise: in R d , idv = ∆ vdt + V ( x, | v | ) vdt + v ◦ dW (11) where v is a complex valued function, i = √− 1 , W is a real valued Wiener process, ◦ means Stratonovich product, V is also real valued. Let { β k : k ∈ N } be a sequence of independent Brownian motions that are associated with {F t : t ≥ 0 } . Let { e k : k ∈ N } be an orthonormal basis of L 2 ( R d , R ) . Then ˆ � W = β k ( t, ω ) e k ( x ) k and W = Φ ˆ � W = β k ( t, ω )(Φ e k ( x )) . k Φ is a Hilbert-Schmidt operator from L 2 to H α 3 . 3 To present our scheme, we need α = 0 . To prove stability and convergence, we will need larger α .

13 Applications of Stochastic NLS • • It can be used to model the wave propagation in nonhomogeneous or random media 4 . (Prof. Solna’s talk on Tuesday) • It has been introduced by Bang etc 5 as a model for molecular monolayers arranged in Scheibe aggregates with thermal fluctuations of the phonons. • It has also been widely used in quantum trajectory theory 6 , which determines the evolution of the state of a continuously measured quantum system (e.g. the continuous monitoring of an atom by the detection of its fluorescence light). 4 V. Konotop, and L. V´ azquez, Nonlinear random waves, World Scientific, NJ, 1994. 5 O. Bang, P.L. Christiansen, F. If, K. Ø. Rasmussen and Y. B. Gaididei, Temperature effects in a nonlinear model of monolayer Scheibe aggregates, Phys. Rev. E, 49 (1994) 4627–4636. 6 A. Barchielli and M. Gregoratti, Quantum trajectories and measurements in continuous time: the diffusive case, Lecture Notes in Physics 782, Springer, Berlin, 2009.