Entropy-based artificial viscosity Jean-Luc Guermond Department of Mathematics Texas A&M University SMAI 2001 23-27 Mai Guidel Jean-Luc Guermond High-Order Hydrodynamics
Acknowledgments SSP collaborators: Jim Morel (PI), Bojan Popov, Valentin Zingan (Grad Student) Other collaborators: Andrea Bonito, Texas A&M Murtazo Nazarov (Grad student) KTH, sweden Richard Pasquetti, Univ. Nice Guglielmo Scovazzi, Sandia Natl. Lab., NM Other Support: NSF (0811041), AFSOR Jean-Luc Guermond High-Order Hydrodynamics
Outline Part 1 INTRODUCTION 1 Jean-Luc Guermond High-Order Hydrodynamics
Outline Part 1 INTRODUCTION 1 LINEAR TRANSPORT EQUATION 2 Jean-Luc Guermond High-Order Hydrodynamics
Outline Part 1 INTRODUCTION 1 LINEAR TRANSPORT EQUATION 2 NONLINEAR SCALAR CONSERVATION 3 Jean-Luc Guermond High-Order Hydrodynamics
Outline Part 2 COMPRESSIBLE EULER EQUATIONS 4 Jean-Luc Guermond High-Order Hydrodynamics
Outline Part 2 COMPRESSIBLE EULER EQUATIONS 4 LAGRANGIAN HYDRODYNAMICS 5 Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION NONLINEAR SCALAR CONSERVATION EQUATIONS INTRODUCTION 1 LINEAR TRANSPORT EQUATION 2 NONLINEAR SCALAR CONSERVATION 3 Introduction Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? Solve 1D eikonal | u ′ ( x ) | = 1 , u (0) = 0 , u (1) = 0 Exists infinitely many weak solutions Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? Exists a unique (positive) viscosity solution, u | u ′ ǫ | − ǫ u ′′ ǫ = 1 , u ǫ (0) = 0 , u ǫ (1) = 0 . 1 2 , � u − u ǫ � H 1 ≤ c ǫ Sloppy approximation. Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? One can do better with L 1 (of course ) Define mesh T h = ∪ N i =0 [ x i , x i +1 ], h = x i +1 − x i . Use continuous finite elements of degree 1. V = { v ∈ C 0 [0 , 1]; v | [ x i , x i +1 ] ∈ P 1 , v (0) = v (1) = 0 } . Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? Consider p > 1 and set � 1 N � � � � | v ′ | − 1 � d x ( v ′ ( x + i ) − v ′ ( x − i )) p + h 2 − p J ( v ) = + 0 � �� � 1 � �� � L 1 -norm of residual Entropy Define u h ∈ V u h = arg min v ∈ V J ( v ) Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? Implementation: use mid-point quadrature � � N � � � � | v ′ ( x i + 1 J h ( v ) = h 2 ) | − 1 +Entropy . � i =0 � �� � ℓ 1 -norm of residual Define � u h = arg min v ∈ V J h ( v ) Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Why L1 for PDEs? Theorem (J.-L. G.&B. Popov (2008)) u h → u strongly in W 1 , 1 (0 , 1) ∩ C 0 [0 , 1] . u h → u and � Fast solution in 1D (JLG&BP 2010) and in higher dimension (fast-marching/fast sweeping, Osher/Sethian) to compute � u h . Similar results in 2D for convex Hamiltonians (JLG&BP 2008). Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION A new idea based on L 1 minimization Some provable properties of minimizer ˜ u h (JLG&BP 2008, 2009, 2010). Minimizer ˜ u h is such that: Residual is SPARSE: ∀ i such that 1 u ′ | ˜ h ( x i + 1 2 ) | − 1 = 0 , 2 �∈ [ x i , x i +1 ] . u ′ Entropy makes it so that graph of ˜ h ( x ) is concave down in [ x i , x i +1 ] ∋ 1 2 . Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION A new idea based on L 1 minimization Conclusion: Residual is SPARSE: PDE solved almost everywhere. Entropy does not play role in those cells. Entropy plays a key role only in cell where PDE is not solved. Jean-Luc Guermond High-Order Hydrodynamics
INTRODUCTION Why L1 for PDEs? LINEAR TRANSPORT EQUATION A new idea based on L 1 minimization NONLINEAR SCALAR CONSERVATION Can L1 help anyway? New idea: Go back to the notion of viscosity solution Add smart viscosity to the PDE: | u ′ ǫ | − ∂ x ( ǫ ( u ǫ ) ∂ x u ǫ ) = 1 Make ǫ depend on the entropy production Viscosity large (order h) where entropy production is large 1 Viscosity vanish when no entropy production 2 Entropy plays a key role in cell where PDE is not solved. Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests NONLINEAR SCALAR CONSERVATION EQUATIONS INTRODUCTION 1 LINEAR TRANSPORT EQUATION 2 NONLINEAR SCALAR CONSERVATION 3 Transport, mixing Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The PDE Solve the transport equation ∂ t u + β ·∇ u = 0 , u | t =0 = u 0 , +BCs Use standard discretizations (ex: continuous finite elements) Deviate as little possible from Galerkin. Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The idea Entropy for linear transport? Notion of renormalized solution (DiPerna/Lions (1989)) Good framework for non-smooth transport. ∀ E ∈ C 1 ( R ; R ) is an entropy If solution is smooth ⇒ E ( u ) solves PDE, ∀ E ∈ C 1 ( R ; R ) (multiply PDE by E ′ ( u ) and apply chain rule) ∂ t E ( u ) + β ·∇ E ( u ) = 0 � �� � Entropy residual Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The idea Key idea 1: Use entropy residual to construct viscosity Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The idea viscosity ∼ entropy residual Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The idea viscosity ∼ entropy residual Viscosity ∼ residual (Hughes-Mallet (1986) Johnson-Szepessy (1990)) Entropy Residual ∼ a posteriori estimator (Puppo (2003)) Add entropy to formulation (For Hamilton-Jacobi equations Guermond-Popov (2007)) Application to nonlinear conservation equations (Guermond-Pasquetti (2008)) Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The algorithm + time discretization Numerical analysis 101: Up-winding=centered approx + 1 2 | β | h viscosity Proof: u i − u i − 1 u i +1 − u i − 1 − 1 u i +1 − 2 u i + u i − 1 β i = β i 2 β i h i h i 2 h i h i Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The algorithm + time discretization Key idea 2: Entropy viscosity should not exceed 1 2 | β | h Jean-Luc Guermond High-Order Hydrodynamics
Linear transport INTRODUCTION The idea LINEAR TRANSPORT EQUATION The algorithm NONLINEAR SCALAR CONSERVATION A little bit of theory Numerical tests The algorithm Choose one entropy functional. EX1: E ( u ) = | u − u 0 | , EX2: E ( u ) = ( u − u 0 ) 2 , etc. Define entropy residual D h := ∂ t E ( u h ) + β ·∇ E ( u h ), Define local mesh size of cell K : h K = diam( K ) / p 2 Construct a wave speed associated with this residual on each mesh cell K : v K := h K � D h � ∞ , K / E ( u h ) Define entropy viscosity on each mesh cell K : ν K := h K min(1 2 � β � ∞ , K , v K ) Jean-Luc Guermond High-Order Hydrodynamics
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