Convection and total variation flow

Transcription

1 Convection and total variation flow R. Eymard joint work wit F. Boucut and D. Doyen LAMA, Université Paris-Est marc, 2nd, 2016

2 Flow wit cange-of-state simplified model for Bingam fluid + Navier-Stokes equations u : Q T R tu + div F (x, t, u) div u u = 0 in Q T u(x, 0) = u ini(x) on R d Q T := R d (0, T ) u ini L 1 (R d ) L (R d ) wit a 0 u ini b 0 a.e. F C 1 (Q T R, R d ) locally Lipscitz continuous suc tat d F i div xf (x, t, u) := (x, t, u) = 0 Example : F (x, t, u) = 1 x i 2 u2 v i=1 First properties of tis model u u not regularized if constant, ence entropy sense needed div u total variation flow [see Mazón] u TV flow = 1-laplacian = nonlocal problem

3 Entropy sense BV (R d ) = {u L 1 { (R d ), TV (u) < } } wit TV (u) = sup u divφ dx; φ Cc 1 (R d ; R d ) wit φ L (R d ) 1 R d entropy solution : u L (Q T ) L 1 (0, T ; BV (R d )) if tere exists λ L (Q T ) d, wit λ 1 a.e. suc tat for all ϕ Cc (Q T ) wit ϕ 0 for all η C (R) convex and Φ u (x, t, u) = η (u) F (x, t, u) u ( η(u) tϕ + ( Φ(x, t, u) η (u)λ ) ) ϕ dxdt Q T ϕ D[η (u)] dt + η(u ini(x))ϕ(x, 0) dx 0 Q T R d obtained by passing to te limit in tu ɛ + div F (x, t, u ɛ) div uɛ u ɛ ɛ uɛ = 0 multiplying by η (u ɛ)ϕ and using uɛ u ɛ η (u ɛ) = η (u ɛ) and uɛ u ɛ λ

4 Particular case F = 0 and analytical solutions ( ) η(u) tϕ η (u)λ ϕ dxdt ϕ D[η (u)] dt + η(u ini(x))ϕ(x, 0) dx 0 Q T Q T R d η(s) = s and η(s) = s yield ( ) u tϕ λ ϕ dxdt + u ini(x)ϕ(x, 0) dx = 0 R d Q T implies divλ = tu on Q T 1D example u ini(x) = 1 ] 1,1[ (x) 1 if x < 1 solution for t < 1 : u(x, t) = (1 t)1 ] 1,1[ (x) and λ(x, t) = x if 1 < x < 1 1 if 1 < x 2D example u ini(x) = 1 B(0,1) (x) { x if x < 1 solution for t < 1 : u(x, t) = (1 2t)1 2 B(0,1)(x) and λ(x, t) = x if x > 1 x 2

5 Uniqueness of te entropy solution 1 Kruskov doubling variable tecnique 2 start wit regular entropy η( κ) 3 sign of terms issued from TV flow allows to get rid of tem 4 let η tend to Kruskov entropies 5 same conclusion as in te pure convection case

6 Numerical sceme quasi-uniform simplicial mes of R d wit nonobtuse angles (exists in any space dimension [J. Brandts et al, 2009]) v R N v C 0 (R d ) ; v K is affine for eac K T, v (x p) = v p, p N v L 1 loc(r d ) ; v Qp = v p, p N û (x) v (x)dx K = Tpq(u K p u q)(v p v q) p V K q V K wit Tpq K = 1 φ p(x) φ q(x)dx 0 2 K and u û L 1 (R d ) û L 1 (R d ) d

7 Numerical sceme : Initialization and finite volume step Initialization of (u 0 p) n N suc tat u 0 L (R d ) L 1 (R d ) : u 0 p = 1 m p Q p u ini(x)dx, p N Finite volume step Letting (u n p) p N suc tat u n L (R d ) L 1 (R d ) seek (u n+ 2 1 p ) p N suc tat u n+ 2 1 L (R d ) L 1 (R d ) and u n+ 1 2 p up n m p δt + q N p F n p,q(u n p, u n q) = 0, were (F n p,q) p,q,n N is admissible and consistent : p N F n p,q C 0 ([a 0, b 0] 2, R) Lipscitz continuous wit constant m p,ql F n p,q(u, v) wit u wit v Fp,q(u, n v) = Fq,p(v, n u) Fp,q(u, n u) = 1 t n+1 F (x, t, u) ν p,q ds(x)dt δt t n σ p,q under CFL condition δt 1 2L inf (p,q) E m p m p,q

8 Numerical sceme : Finite element step Seek (u n+1, λ n+1 ) X Λ suc tat n+1 u u n+ 2 1 v dx + + θ() û n+1 ) v dx = 0, v X δt R d R d (λ n+1 λ n+1 1 if û n+1 = 0, oterwise λ n+1 = ûn+1 û n+1 X = {v R N ; v L 1 (R d ) L 2 (R d ), v L 2 (R d )} wit Λ := {µ L (R d ) d ; µ K is constant for eac K T } θ() > 0 and lim θ() = lim 0 0 θ() = 0 Remark : approximate finite element step used in practice Seek u n+1 X suc tat n+1 u u n+ 1 ( ) 2 1 v dx + + θ() û n+1 R d δt R d (ε(x) 2 + û n+1 2 ) 1/2 v dx = 0, v X

9 Estimates on finite volume step numerical entropy flux Φ n p,q(x, y) := 1 2 b0 a 0 η (κ) ( F n p,q(x κ, y κ) F n p,q(x κ, y κ) ) dκ + η (a 0) + η (b 0) Fp,q(x, n y) 2 entropy inequality satisfied by te finite volume step using CFL condition η(u n+ 1 2 p ) η(up) n m p + Φ n p,q(up, n uq) n 0 δt q N p (proof using Kruskov entropies) implies for η(0) = 0, η (0) = 0 η(u n+ 1 2 R d (x))dx η(u n (x))dx R d (multiply by exp( ε x p ), sum over te mes and let ε 0) η(s) s b 0 b 0 implies u n+ 1 2 p η(s) a 0 s a 0 implies u n+ 1 2 p a 0 b 0 for all p N η(s) s implies u n+ 1 2 L 1 (R d ) η(s) = s 2 implies u n+ 1 2 L 2 (R d ) u n L 2 (R d )

10 Estimates on finite element step u n+1 X minimizer of J n+1 (v ) := 1 2δt R d ( v u n+ 1 2 ) 2 dx + ( v + 1 R d R d 2 θ() v 2 ) dx tanks to saddle point teorem, deduce tat exists (u n+1, λ n+1 ) X Λ suc tat n+1 u u n+ 2 1 v dx + + θ() û n+1 ) v dx = 0, v X δt R d (λ n+1 λ n+1 1 if û n+1 = 0, oterwise λ n+1 take v p = η (u n+1 p = ûn+1 û n+1 ) wit η(0) = 0, η (0) = 0 and η bounded gives η(u n+1 (x))dx η(u n+ 1 2 R d R d (x))dx as finite volume step conclusion on L and L 1 bounds Remark : approximate finite element step u n+1 X minimizer of Jn+1 (v ) := 1 ( ) 2 v u n+ 1 2 dx + 2δt R d R d ((ε(x) 2 + v 2 ) 1/ ) θ() v 2 dx

11 L 1 (0, T ; BV (R d )) estimate and N 1 2 u,δt 2 L (0,T ;L 2 (R d )) + δt û n dx 1 R d 2 uini 2 L 2 (R d ) n=1 N 1 n=0 δt θ() û n+1 R d N 1 2 dx = θ() n=0 δt Tpq(u K p n+1 uq n+1 ) uini 2 L 2 (R d ) K T q V K p V K proof : use L 2 control on finite volume step and v = u n+1 leads to L 1 loc time and space translates estimates for u,δt and terefore û,δt converges as well Used for control of numerical fluxes in te finite volume step only for n 1, no need of more regularity for u ini

12 Entropy estimate wit η(ū,δt ) tϕ dxdt + Φ(x, t, ū,δt ) ϕ dxdt Q T Q T η (û,δt )λ,δt ϕ dxdt ϕ η (û,δt ) dxdt + e,δt 0 Q T Q T lim e,δt = 0 0,δt 0 proof : use entropy inequality of finite volume step v n+1 p = η (up n+1 )ϕ(x p, (n + 1)δt) in finite element step difficulty : error between δt λ n+1 v n+1 dxdt and R d n term in θ() wit /θ() 0 necessary θ() 0 necessary for tis term to vanis Q T ϕ η (û,δt ) dxdt conclusion by passing to te limit 0, δt 0 using lower semi-continuity of te BV norm

13 A 1D numerical example tu + 1 xu 2 x(u2 ) g x = 0 xu wit g = : = 0.002, δt = 0.001, initial data u x

14 A 1D numerical example tu + 1 xu 2 x(u2 ) g x = 0 xu wit g = , t = u x

15 A 1D numerical example wit convection tu + 1 xu 2 x(u2 ) g x = 0 xu wit g = , t = u x

16 A 2D numerical example wit convection tu + c x xu + c y y u g div u u = 0 u ini(x, y) = 1 D0 (x, y) Analytical solution u : (x, y, t) ( 1 2gt ) + 1 D0 (x cxt, y cy t) r 0

17 2D initial data

18 2D solution at t = 0.5

19 2D solution at t = 1

20 2D solution at t = 1.5

21 Compararison wit analytical solution convection and TV flow convection witout TV flow

22 Conclusions 1 a first step to Navier-Stokes equations wit solid-fluid transition 2 full finite volume sceme not expected to converge in te TV flow part 3 need of viscous regularization for te analysis of convergence, not needed in te numerical tests