Resonance based schemes for dispersive equations via decorated trees - PowerPoint PPT Presentation

Resonance based schemes for dispersive equations via decorated trees Yvain Bruned University of Edinburgh (joint work with Katharina Schratz) "Higher Structures Emerging from Renormalisation", ESI Vienna, 15 October 2020 1/15

Dispersive PDEs We consider nonlinear dispersive equations of the form i ∂ t u ( t , x ) + L u ( t , x ) = p ( u ( t , x ) , u ( t , x )) ( t , x ) ∈ R + × T d u ( 0 , x ) = v ( x ) , where L is a differential operator and p is a polynomial nonlinearity. Assume local wellposedness of the problem on the finite time interval ] 0 , T ] , T < ∞ for v ∈ H n . Aim: give a numerical approximation of u at low regularity when n is small. NLS: L = ∆ and p ( u , u ) = | u | 2 u . KdV: L = i ∂ 3 x and p ( u , u ) = i ∂ x ( u 2 ) . 2/15

Decorated trees approach Mild solution given by Duhamel’s formula: � t u ( t ) = e it L v + e it L e − i ξ L ( − i p ( u ( ξ ) , u ( ξ )) d ξ ) ���� ���� 0 edge edge � �� � edge Definition of a character Π : Decorated trees → Iterated integrals 1 e it L v = (Π T 0 )( t , v ) , T 0 = 2 − ie it L � t � � 0 e − i ξ L p e i ξ L v , e − i ξ L v d ξ = (Π T 1 )( t , v ) , T 1 = 3/15

B-series type expansion Solution U r up to order r can be represented by a series: Υ p ( T ) � U r ( t , v ) = S ( T ) (Π T )( t , v ) , T ∈V r V r : decorated trees of order r . S ( T ) : symmetry factor. Υ p : elementary differentials. Error of order � � t r + 1 q ( v ) O for some polynomial q . 4/15

Treatment of oscillations The principal oscillatory integral takes the form � t I 1 ( t , L , v , p ) = Osc ( ξ, L , v , p ) d ξ 0 with the central oscillations Osc given by � � Osc ( ξ, L , v , p ) = e − i ξ L p e i ξ L v , e − i ξ L v . In general it will be � � Osc ( ξ, L , v , p ) = e − i ξ L p e i ξ ( L + L 1 ) q 1 ( v ) , e − i ξ ( L + L 2 ) q 2 ( v ) . 5/15

Various approaches Classical Methods: Osc ( ξ, L , v , p ) ≈ e − i ξ L p ( v , v ) exponential method: splitting method: Osc ( ξ, L , v , p ) ≈ p ( v , v ) Resonance as a computational tool: � � � � e i ξ L dom p dom ( v , v ) Osc ( ξ, L , v , p ) = p low ( v , v ) + O ξ L low v . Here, L dom denotes a suitable dominant part of the high frequency interactions and L low = L − L dom 6/15

Experiments Comparison of classical and resonance based schemes for the Schrödinger equation for smooth ( C ∞ data) and non-smooth ( H 2 data) solutions. 10 -1 10 -5 10 -2 10 -3 10 -4 10 -5 10 -10 10 -6 10 -7 10 -3 10 -2 10 -3 10 -2 7/15

Experiments Comparison of classical and resonance based schemes for the KdV equation with smooth data in C ∞ . 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -3 10 -2 8/15

Fourier iterated integrals Mild solution given by Duhamel’s formula in Fourier ( P ( k ) ↔ L ): � t u k ( t ) = e itP ( k ) v k + e itP ( k ) e − i ξ P ( k ) ˆ ˆ ( − i p k ( u ( ξ ) , u ( ξ )) d ξ ) � �� � � �� � 0 edge edge � �� � edge Definition of a character ˆ Π : Decorated trees → Iterated integrals 1 e itP ( k ) = (ˆ k Π T 0 )( t ) , T 0 = 2 − ie itP ( k ) � t 0 e − i ξ ( P ( k ) − P ( − k 1 )+ P ( k 2 )+ P ( k 3 )) d ξ = (ˆ Π T 1 )( t ) , k 3 k 1 k 2 T 1 = 9/15

Fourier B-series type expansion Resonance scheme U r k of order r with regularity n (initial data): Υ p ( T )( v ) � � � ˆ U r Π r k ( τ, v ) = n T ( τ ) S ( T ) T ∈V r k V r k : decorated trees of order r with frenquency k . Character ˆ n resonance approximation of ˆ Π r Π . Examples of decorated trees for NLS ( r = 2): k 3 k 3 k 1 k 2 k 1 k 2 k 3 k 1 k 2 k 4 k 5 k 4 k 5 k 10/15

A practical example k 2 k 1 k 3 The iterated integral associated to T = is given by: � t e i ξ ( − k 2 − k 2 1 + k 2 2 + k 2 3 ) d ξ, (ˆ Π T )( t ) = k = − k 1 + k 2 + k 3 0 One has − k 2 − k 2 1 + k 2 2 + k 2 3 = L dom + L low ���� ���� − 2 k 2 order one 1 Taylor expansion of L low : Π T )( t ) = e − 2 itk 2 1 − 1 � � (ˆ + O t L low − 2 ik 2 1 � �� � (ˆ Π r n T )( t ) 11/15

Local Error Analysis Main idea is to single out oscillations: � t e i ξ P ( k ) d ξ = e itP ( k ) − 1 iP ( k ) 0 Butcher-Connes-Kreimer coproduct ∆ k 3 k 1 k 2 ℓ k 3 k 4 k 5 k 4 k 5 k 1 k 2 ∆ = ⊗ + · · · , ℓ = − k 1 + k 2 + k 3 � t 0 ξ ℓ e i ξ P ( k ) d ξ → deformed BCK coproduct ˆ Integrals ∆ (SPDEs). 12/15

Main result Birkhoff type factoriation: � � ˜ ˆ n ⊗ ( Q ◦ ˆ ˆ Π r Π r Π r n = n A· )( 0 ) ∆ . where A is an antipode and Q is a projector. Theorem (B., Schratz 2020) For every T ∈ V r k � � � � Π T − ˆ ˆ τ r + 1 L r Π r n T ( τ ) = O low ( T , n ) . where L r low ( T , n ) involves all lower order frequency interactions. 13/15

Family of schemes ˆ Π r n deg( L r low ( T , n )) n r exp ( T ) n r low ( T ) Minimum Regularity Low Regularity Classical Exponential Resonance scheme Resonance scheme Integrator type scheme n n r n r low ( T ) exp ( T ) 14/15

Perspectives B-series → Regularity Structures → PDEs Numerical Schemes New example of a deformation of the BCK coproduct. Birkhoff type factorisation as in SPDEs see (B., Ebrahimi-Fard 2020). Backward error analysis for the scheme. Structure preservation. Generalisation to more general domains not only T d and wave equations. Potential connection with the study of dispersive (S)PDEs. 15/15