Convergence and error estimates for the compressible Navier-Stokes - PowerPoint PPT Presentation

Convergence and error estimates for the compressible Navier-Stokes equations Antonin Novotny ( 1 ) , IMATH ( 2 ) , Universit´ e du Sud Toulon Var ( 1 ) http://myweb.labscinet.com/novotny ( 2 ) http://imath.univ-tln.fr 13.09-18.09, 2015 Part 1 : Concept and stability analysis in the continuous case based on joint work with E. Feireisl Part 2 : Error estimates : joint work with T. Gallouet, R. Herbin, D. Maltese and applications with E. Feireisl, R. Hosek, D. Maltese Antonin Novotny Relative energy method

Compressible Navier-Stokes equations We consider in [ 0 , T ) × Ω , Ω ⊂ R 3 (a bounded Lipschitz domain) the following system of equations Continuity equation ∂ t ̺ + div x ( ̺ u ) = 0 (1) Momentum equation ∂ t ( ̺ u ) + div x ( ̺ u ⊗ u ) + ∇ x p ( ̺ ) = µ ∆ u + ( µ/ 3 ) ∇ div x u (2) Boundary conditions � � ( 0 , T ) × ∂ Ω = 0 u � (3) Initial conditions ̺ ( 0 , x ) = ̺ 0 ( x ) , ̺ u ( 0 , x ) = ̺ 0 u 0 ( x ) . (4) Antonin Novotny Relative energy method

Viscosity coefficients µ > 0 , η = 0 (5) Pressure p ∈ C 1 [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , p ′ ( ̺ ) > 0 , p ( 0 ) = 0 , (6) ̺ →∞ p ′ ( ̺ ) /̺ γ − 1 = p ∞ > 0 , γ ≥ 1 . lim Helmholtz function H � ̺ p ( s ) ̺ H ′ ( ̺ ) − H ( ̺ ) = p ( ̺ ) , H ( ̺ ) = ̺ s 2 d s 1 Relative (potential) energy function E E ( ̺, r ) = H ( ̺ ) − H ′ ( r )( ̺ − r ) − H ( r ) E ( ̺, r ) ≥ 0 , E ( ̺, r ) = 0 ⇔ ̺ = r

Weak solutions Functional spaces ̺ ( t , x ) ≥ 0 for a.a. ( t , x ) ∈ ( 0 , T ) × Ω , ̺ ∈ L ∞ ( 0 , T ; L 1 (Ω)) , ̺ u ∈ L ∞ ( 0 , T ; L 1 (Ω; R 3 )) , ̺ u 2 ∈ L ∞ ( 0 , T ; L 1 (Ω)) , u ∈ L 2 ( 0 , T ; W 1 , 2 0 (Ω; R 3 )) , p ( ̺ ) ∈ L ∞ ( 0 , T ; L 1 (Ω)) . Continuity equation ̺ ∈ C weak ([ 0 , T ]; L 1 (Ω)) and equation (1) is replaced by the family of integral identities � � τ � � � � τ � ̺ϕ d x 0 = ̺∂ t ϕ + ̺ u · ∇ x ϕ (7) � d x d t Ω Ω 0 for all τ ∈ [ 0 , T ] and for any ϕ ∈ C 1 ([ 0 , T ] × Ω) ;

Momentum equation ̺ u ∈ C weak ([ 0 , T ]; L 1 (Ω; R 3 )) and momentum equation (2) is satisfied in the sense of distributions, specifically, � � τ � � � � τ � ̺ u · ϕ d x 0 = ̺ u · ∂ t ϕ + ̺ u ⊗ u : ∇ x ϕ (8) � d x dt Ω 0 Ω � τ � � � p ( ̺ ) div x ϕ − S ( ϑ, ∇ x u ) : ∇ x ϕ + d x d t 0 Ω for all τ ∈ [ 0 , T ] and for any ϕ ∈ C 1 c ([ 0 , T ] × Ω; R 3 ) ; Energy inequality � � τ � � 1 � � τ � 2 ̺ u 2 + E ( ̺, ̺ ) 0 + S ( ∇ x u ) : ∇ x u d x d t ≤ 0 , (9) d x � Ω Ω 0 for a.a. τ ∈ ( 0 , T ) , where ̺ > 0 . Antonin Novotny Relative energy method

Relative entropy (relative energy) inequality � � 1 � � τ � 2 ̺ | u − U | 2 + E ( ̺ | r ) � (10) d x 0 Ω � τ � � � � � + ∇ x ( u − U ) : ∇ x u − U S d x d t 0 Ω � τ � � � � ≤ S ( ∇ x U ) : ∇ x U − u d x d t 0 Ω � τ � � � + ̺∂ t U + ̺ u · ∇ x U · ( U − u ) d x d t Ω 0 � τ � − p ( ̺ ) div x U d x d t 0 Ω � τ � � r − ̺ � ∂ t p ( r ) − ̺ + r u · ∇ x p ( r ) d x d t r Ω 0 for all r ∈ C 1 c ([ 0 , T ] × Ω) , r > 0 , U ∈ C 1 c ([ 0 , T ] × Ω) .

Dissipative solutions Relative entropy � 1 � � � � 2 ̺ | u − U | 2 + E ( ̺ | r ) E ( ̺, u � r , U ) = d x Ω Functional spaces ̺ ( t , x ) ≥ 0 for a.a. ( t , x ) ∈ ( 0 , T ) × Ω , ̺ ∈ L ∞ ( 0 , T ; L 1 (Ω)) , ̺ u ∈ L ∞ ( 0 , T ; L 1 (Ω; R 3 )) , ̺ u 2 ∈ L ∞ ( 0 , T ; L 1 (Ω)) , u ∈ L 2 ( 0 , T ; W 1 , 2 0 (Ω; R 3 )) , p ( ̺ ) ∈ L ∞ ( 0 , T ; L 1 (Ω)) . Relative energy inequality � τ � � � � � � � E ( ̺, u ∇ x ( u − U ) : ∇ x u − U � r , U )( τ ) + S d x d t 0 Ω � τ � � � � � � ≤ E ( ̺ 0 , u 0 � r ( 0 ) , U ( 0 )) + R ̺, u � r , U d t 0 where the remainder R is given by the r.h.s. of formula (12) and the test functions are the same as in formula (12). Antonin Novotny Relative energy method

Weak and dissipative solutions Finite energy initial data � 1 2 ̺ 0 u 2 0 � = ̺ 0 ≥ 0 , 0 + E ( ̺ 0 | ̺ ) d x < ∞ . (11) Ω Weak solutions : Lions,98 ( γ ≥ 9 5 ), Feireisl, Petzeltova, N., 02 ( γ > 3 2 ) Under assumptions on the initial data (11) and pressure with γ > 3 / 2 , the compressible Navier-Stokes system (1–5) admits at least one weak solution. Weak solutions are dissipative : Feireisl, Jin, N., 2012 Under assumptions on initial data(11) and pressure with γ ≥ 1 , any weak solution of the compressible Navier-Stokes system (1–5) is a dissipative one.

Relative entropy with ( r , U ) strong solution of CNSE � � 1 � � τ � 2 ̺ | u − U | 2 + E ( ̺ | r ) d x � (12) 0 Ω � τ � � � � � + ∇ x ( u − U ) : ∇ x u − U S d x d t 0 Ω � τ � � � ≤ ( ̺ − r ) ∂ t U + U · ∇ x U · ( U − u ) d x d t 0 Ω � τ � � − ̺ ( u − U ) · ∇ x U · ( U − u ) d x d t 0 Ω � τ � � � p ( ̺ ) − p ′ ( r )( ̺ − r ) − p ( r ) − div x U d x d t Ω 0 � τ � r − ̺ + ( u − U ) · ∇ x p ( r ) d x d t . r 0 Ω

Relative entropy and stability Weak strong uniqueness, stability Feireisl, Jin, N. 2012 Let the pressure verifies (6) γ ≥ 1 . Let ( ̺, u ) be a weak solution to the compressible Navier-Stokes equations (1-5) emanating from the initial data ( ̺ 0 , u 0 ) , and let ( r , U ) be a classical solution of the same system emanating from the initial data ( 0 < r 0 , U 0 ) . Then there exists c = c (Ω , T , � r − 1 � 0 , ∞ , � r � 1 , ∞ , � U � 1 , ∞ ) such that � � � � � � � � E ̺, u � r , U ≤ c E ̺ 0 , u 0 � r 0 , U 0 . Goal To get � � � � � � � � � + h a + ∆ t b ) , E ̺, u � r , U ≤ c E ̺ 0 , u 0 � r 0 , U 0 a > 0 , b > 0 for a numerical solution corresponding to the size ( h , ∆ t ) of the space-time discretization.

Approximating system of PDEs for weak solutions Approximating system ∂ t ̺ + div ( ̺ u ) − ε ∆ ̺ = 0 , ∂ n ̺ | ‘ ∂ Ω = 0 , ∂ t ( ̺ u ) + div ( ̺ u ⊗ u ) + ∇ x ( p ( ̺ )+ δ̺ 4 )+ ε ∇ x ̺ · ∇ x u = µ ∆ u + ( µ 3 + η ) ∇ x div u u | ∂ Ω = 0 Weak solutions are obtained letting first ε → 0 and then δ → 0 . This is not exploitable in the numerics !

The mesh The physical space is a polyhedral domain Ω ⊂ R 3 coinciding with the numerical domain Ω h . K ∈ T - regular partition of Ω h into (closed) tetrahedra of size h : Ω = ∪ K ∈T h K . If K ∩ L � = ∅ , K � = L , then K ∩ L is either a common face, or a common edge, or a common vortex. Furthermore, we suppose that each K is a tetrahedron such that ξ [ K ] diam [ K ] ≥ θ 0 > 0 min (13) K where ξ [ K ] is the radius of the largest ball contained in K . Notation σ = K | L ∈ E int - set of internal faces, E - set of all faces. 0 < t 1 < . . . < t n < . . . < T - time discretisation of step ∆ t . Antonin Novotny Relative energy method

Discretisation CR space : V h , 0 (Ω h ) = { v | K ∈ P 1 ( R 3 ) | if σ = K | L then [ v | K ] σ = [ v | L ] σ , v σ = 0 if σ ∈ ∂ Ω h } ̺ ( t n , x ) ≈ � K ∈T ̺ n K 1 K ( x ) ∈ Q h (Ω h ) - space of piecewise constants. u ( t n , x ) ≈ � σ ∈E int u n σ φ σ ( x ) ∈ V h , 0 (Ω h ) - the CR space. Upwind :   ̺ K if u σ · n σ, K > 0   ̺ up σ =  , where σ = K | L .  ̺ L otherwise Mean values : � � � V K = 1 V σ = 1 ˆ V d x , V = v K 1 K ( x ) , V d S | K | | σ | K σ K ∈T Antonin Novotny Relative energy method

Numerical scheme ̺ n ∈ Q h (Ω h ) , ̺ n ≥ 0 , u n ∈ V h , 0 (Ω h ; R 3 ) , n = 0 , 1 , . . . , N , (14) � � � K − ̺ n − 1 | K | ̺ n K | σ | ̺ n , up ( u n σ · n σ, K ) φ K = 0 φ K + (15) σ ∆ t K ∈T K ∈T σ ∈E ( K ) for any φ ∈ Q h (Ω h ) and n = 1 , . . . , N , � � � � � | K | | σ | ̺ n , up u n , up ̺ n K u n K − ̺ n − 1 u n − 1 [ u n · v K + ˆ σ · n σ, K ] · v K σ σ K K ∆ t K ∈T K ∈T σ ∈E ( K ) (16) � � � � ∇ u n : ∇ v d x p ( ̺ n − K ) | σ | v σ · n σ, K + µ K K ∈T σ ∈E ( K ) K ∈T � � + µ div u n div v d x = 0 , for any v ∈ V h , 0 (Ω; R 3 ) and n = 1 , . . . , N . 3 K K ∈T

� ( r , V ) � X T ( R 3 ) ≡ � r � C 1 ([ 0 , T ] × R 3 ) + � ∂ t ∇ x r � C ([ 0 , T ]; L 6 ( R 3 ; R 3 )) + � ∂ 2 t , t r � C ([ 0 , T ]; L 6 ( R 3 )) � V � C 1 ([ 0 , T ] × R 3 ; R 3 ) + � V � C ([ 0 , T ]; C 2 ( R 3 ; R 3 )) + � ∂ t ∇ x V � C ([ 0 , T ]; L 6 ( R 3 ; R 3 × 3 )) + � ∂ 2 t , t V � L 2 ( 0 , T ; L 6 ( R 3 )) and r ≥ r > 0 . Case Ω = Ω h , Gallouet, Herbin, Maltese, N., 2014 Let ( ̺ n h , u n h ) = ( ̺, u ) be a family of numerical solutions of the numerical scheme (14–15) with γ ≥ 3 / 2 . Let ( r , V ) be a classical solution of the compressible Navier-Stokes equations (1–6) in the class X T ( R 3 ) . Then there exists c > 0 independent of h , ∆ t , ̺ , u , such that E ( ̺ n , u n � � � � √ � � � r 0 , U 0 ) + h α + � r , U ) ≤ c E ( ̺ 0 , u 0 ∆ t , where α = 2 γ − 3 if 3 / 2 ≤ γ < 2 , α = 1 2 if γ ≥ 2 , 2 � E ( ̺ n , u n � � 1 � � � K ( V n − ˆ K ) 2 + E ( ̺ n 2 ̺ n u n K | r n ) � r , V ) = K K ∈T