Numerical discretization of tangent vectors of hyperbolic - PowerPoint PPT Presentation

Numerical discretization of tangent vectors of hyperbolic conservation laws. Michael Herty IGPM, RWTH Aachen www.sites.google.com/michaelherty joint work with Benedetto Piccoli MNCFF 2014, Bejing, 22.5.2014

Contents ◮ Originally on multi-phase fluid flow on networks (extension of coupling conditions - NumHyp 2013 ) ◮ Recent results on tangent vectors in collaboration with B. Piccoli (Rutgers) ◮ Tangent vectors are used for optimization problems where the dynamics is governed by 1–d hyperbolic systems ◮ Tangent vectors are a first–order sensitivity calculus for ∂ solutions ∂ u 0 u ( t , x ) ◮ Examples are control of compressor stations in gas networks, control of gates in open canals, parameter identification problems subject to transport problems, . . .

This presentation ◮ References and theoretical problem ◮ Example of tangent vectors for Burgers ◮ Numerical challenges ◮ Approximation of evolution of tangent vectors ◮ Numerical examples

References Concept of tangent vectors present one possibility to compute and analyze sensitivity of systems on 1-d hyperbolic balance laws ◮ Existing calculus for spatially one–dimensional systems of conservation laws ◮ Introduced by Bressan/Marson (1995), extended by Bressan and co–workers (1997,2007), Bianchini (2000), Piccoli and co–workers (2000–) ◮ Lipschitz continuous dependence for 1–d system Crasta/Bressan/Piccoli (2001) ◮ Related for scalar convex case: Ulbrich (2001), Giles (1996), Ulbrich/Giles (2010), Zuazua/Castro et al (2008–). ◮ Application of sensitivity equations for general inverse problems, optimal control problems, and control / controllability questions, . . .

Theoretical challenge ∂ t u + ∂ x f ( u ) = 0 , u ( t = 0 , x ) = u 0 ( x ) ◮ Solution operator for nonlinear conservation laws u ( t , · ) = S t u 0 generically non–differentiable on L 1 ◮ No classical calculus for first–order variations of S t u 0 with respect to u 0 ◮ No characterizing conditions for parameter identification problems or optimal control problems � min J ( u ( T , x )) dx subject to ∂ t u + ∂ x f ( u ) = 0 , u ( t = 0 , x ) = u 0 ( x ) u 0

Example for Burger’s equation u 2 2 = 0 , u ( t = 0 , x ) = u ǫ ∂ t u + ∂ x 0 = ǫ · x · χ [0 . 1] ◮ u ǫ 0 variations of initial data and interest in the behavior of the solution S t u ǫ 0 with respect to h at e.g. ǫ S t u ǫ + h ≈ S t u ǫ 0 + hv + O ( h ) 0 ◮ Exact solution u ǫ ( t , x ) = (1+ ǫ ) x 1+ ǫ t χ [0 , √ 1+ ǫ t ] is Lipschitz in L 1 wrt ǫ ◮ Shock position is 1 + ǫ t ◮ t = 0: a first–order approximation in L 1 exists and is u ǫ + h ( x ) − u ǫ 0 ( x ) v ( t , x ) = lim 0 = ǫ x · χ [0 , 1] ǫ h → 0 ◮ But for any t > 0 the jump depends on ǫ + h and the previous limit does not define any function in L 1

Idea of tangent vectors for Burger’s equation � √ � u ǫ + h ( t , x ) − u ǫ ( t , x ) 1+( ǫ + h ) t dx = O (1)+1 ( ǫ + h ) x lim 1 + ( ǫ + h ) t dx √ 1+ ǫ t h h h → 0 = O ( 1 √ ) h Use weak formulation to compute the limit h → 0 for arbitrary value of ǫ � u ǫ + h ( t , x ) − u ǫ ( t , x ) lim φ ( x ) dx = h h → 0 � x lim (1 + ( ǫ + h ) t )(1 + ǫ t ) χ [0 , √ 1+ ǫ t ] ( x ) φ ( x ) dx + h → 0 √ ( ǫ + h ) y t � lim φ ( y ) 1 + ( ǫ + h ) t , y ∈ [ 1 + ǫ t , 1 + ( ǫ + h ) t ] � h → 0 2 1 + ( ǫ + h ) t

Idea of tangent vectors (cont’d) � x lim (1 + ( ǫ + h ) t )(1 + ǫ t ) χ [0 , √ 1+ ǫ t ] ( x ) φ ( x ) dx + h → 0 √ t ( ǫ + h ) y � lim φ ( y ) 1 + ( ǫ + h ) t , y ∈ [ 1 + ǫ t , 1 + ( ǫ + h ) t ] � h → 0 2 1 + ( ǫ + h ) t A suitable differential of u ǫ ( t , x ) at ǫ may therefore consist of two components: an L 1 part and a measure located at the jump of u ǫ S t u ǫ + h = u ǫ + h ( t , · ) = S t u ǫ v ( t , · ) + δ s ǫ ( t ) ( · )[ u ǫ ] ξ ( t ) � � 0 + h 0

Idea of tangent vectors for Burger’s equation (2/2) � x lim (1 + ( ǫ + h ) t )(1 + ǫ t ) χ [0 , √ 1+ ǫ t ] ( x ) φ ( x ) dx + h → 0 √ ( ǫ + h ) y t � lim φ ( y ) 1 + ( ǫ + h ) t , y ∈ [ 1 + ǫ t , 1 + ( ǫ + h ) t ] � h → 0 2 1 + ( ǫ + h ) t ◮ v ( t , x ) is the L 1 − part consists of the a.e. pointwise limit u ǫ + h ( t , x ) − u ǫ ( t , x ) x v ( t , x ) = lim = (1 + ǫ t ) 2 h h → 0 ◮ ξ ( t ) is a real number and is the variation in the shift of the shock position s ǫ s ǫ + h ( t ) − s ǫ ( t ) t ξ ( t ) = lim = 2 √ 1 + ǫ t , h h → 0 ◮ u ǫ + h ( t , · ) = u ǫ ( t , · ) + h v ( t , · ) + δ s ǫ ( t ) ( · )[ u ǫ ] ξ ( t ) � � ◮ ( v , ξ ) is called generalized tangent vector

Generalized tangent vectors ( v , ξ ) ◮ v ( t , x ) is the L 1 − part consists of the a.e. pointwise limit u ǫ + h ( t , x ) − u ǫ ( t , x ) v ( t , x ) = lim h h → 0 ◮ ξ ( t ) is a real number and is the variation in the shift of the s ǫ + h ( t ) − s ǫ ( t ) shock position s ǫ ξ ( t ) = lim h h → 0 ◮ u ǫ + h ( t , · ) = u ǫ ( t , · ) + h � v ( t , · ) + δ s ǫ ( t ) ( · )[ u ǫ ] ξ ( t ) � Question ◮ Which variations ǫ → u ǫ 0 lead to well–defined tangent vectors? Different settings possible. Here, we follow B/M with u ǫ 0 piecewise Lipschitz with a finite number of isolated discontinuities ◮ A tangent vector a t = 0 may lead to a tangent vector at time t by solving suitable evolution equations (first–order variations of hyperbolic system). For hyperbolic balance laws and suitable variations this is proven up to shock interaction (B/M) and extended in case of conservation laws (C/B/P).

Suitable variations of initial data leading to well–defined tangent vectors ( v , ξ ) A suitable variation u h of u 0 ( x ) is within an L 1 equivalence class to � � u h ( x ) = u 0 ( x )+ hv ( x )+ [ u 0 ] χ [ x i + h ξ, x i ] ( x ) − [ u 0 ] χ [ x i , x i + h ξ ] ( x ) ξ i < 0 ξ i > 0 for some v ∈ L 1 and ξ ∈ R N where N is the number of isolated discontinuities located at x i in u 0 ( x ) .

Theoretical results (Theorem 2.2 (B/M) ◮ For previous variations the evolution equation for the tangent vector is well–defined up to any point in time when two shocks coincide ◮ Result holds for system of balance laws u t + f ( u ) x = h ( t , x , u ) , A ( u ) = Df ( u ) .

Numerical implementation of tangent vectors ◮ Requires knowledge on exact shock positions for evaluation of the evolution of ξ ( t ) ◮ Requires solution of compatibility condition between waves of different families ◮ Evolution of the L 1 part v influences evolution of shift ξ ( t ) ◮ Non–conservative system for v

Approximation to the problem for numerical implementation ◮ Presentation on the simplest case of a 1–d scalar conservation law y (1) + f ( y (1) ) x = 0 , y (1) ( t = 0 , x ) = u 0 ( x ) t ◮ Replace equation by Jin–Xin relaxation approximation for 0 < µ << 1 � 0 � y x = 1 � � 1 0 y t + a 2 f ( y (1) ) − y (2) 0 µ y (1) (0 , x ) = u 0 ( x ) , y (2) (0 , x ) = f ( u 0 ( x )) ◮ Relaxation is a viscous approximation �� a 2 − f ′ ( y (1) ) 2 � � y (1) y (1) + f ( y (1) ) x = µ x . x t ◮ System is linear hyperbolic with stiff source term

Tangent vectors for relaxation system � 0 � y x = 1 � 0 � 1 y t + a 2 f ( y (1) ) − y (2) 0 µ ◮ L 1 − part of tangent vector does not depend on y (1) and is linear system � 0 � � � 1 v x = 1 0 v (0 , · ) = ¯ v ( · ) , v t + , f ′ ( y (1) ) v (1) − v (2) a 2 0 µ v = ( v (1) , v (2) ) outside the discontinuities of y ◮ N discontinuities and l j j th eigenvector and ∆ i v = v ( x i ( t )+ , t ) − v ( x i ( t ) − , t ) jump across i the discontinuities ξ i ( t ) = ¯ ξ i and l j · (∆ i v + ∆ i y x ξ i ) = 0 j � = k .

Variation of general cost functional wrt initial data ◮ Given y d ∈ L 1 ( R ), T > 0 given and y = y (1) solution to relaxation system � χ I ( x ) ( y ( T , x ) − y d ( x )) 2 dx J ( y ( · , · ) , y d ( · )) = ◮ Variation of J with respect to variations in u 0 generating the tangent vectors ( v , ξ ) and a fixed number N of discontinuities � � � y (1) ( T , x ) − y d ( x ) v (1) ( T , x ) dx + ∇ u 0 J ( y , y d ) = 2 χ I ( x ) N ( u 0 ) � � � � y (1) ( T , x i +) − y d ( x i +) + i =1 � �� y (1) ( T , x i − ) − y d ( x i − ) ∆ i y (1) ( T , · ) ξ i ( T )

Variation of cost and tangent vectors used for maximal descent � � y (1) ( T , x ) − y d ( x ) � v (1) ( T , x ) dx + ∇ u 0 J ( y , y d ) = 2 χ I ( x ) N ( u 0) � � y (1) ( T , x i +) − y d ( x i +) � � y (1) ( T , x i − ) − y d ( x i − ) �� ∆ i y (1) ( T , · ) ξ i ( T ) � + i =1 ◮ Update of control u 0 using gradient based information with stepsize ρ > 0   N ( u 0 ) � u 0 ( x ) = u 0 ( x ) − ˜  ρ v (0 , x ) − χ [ x i + ρξ i (0) , x i +1 + ρξ i +1 (0)] ( x ) u 0 ( x i +)  i =1 ◮ Choice of v for maximal descent in J � � v (1) ( T , x ) = y (1) ( T , x ) − y d ( x ) , v (2) ( T , x ) = 0 ◮ Choice of ξ for maximal descent in J �� y (1) ( T , x i +) − y d ( x i +) � � y (1) ( T , x i − ) − y d ( x i − ) �� ∆ i y (1) ( T ) ξ i ( T ) = +