Numerical analysis for the BCF method in complex fluids simulations

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1 for the BCF method in complex fluids simulations Tiejun Li School of Mathematical Sciences, Peking University, Beijing Joint work with Weinan E and Pingwen Zhang CSCAMM conference, April 16-19, Maryland

2 Outline Introduction

Polymeric fluids Figure 1. Experimental images of a collapsing liquid bridge of polymer solution in a viscous solvent (plate radius R 0 = 3mm, distance 13.8 mm).

8Pas, respectively, the polymer timescale is λ = 8.1s. Relative to the capillary timescale τ = ρr0 3 /γ this results in a Deborah number of De = 296.

3 Polymeric fluids Figure 1. Experimental images of a collapsing liquid bridge of polymer solution in a viscous solvent (plate radius R 0 = 3mm, distance 13.8 mm). The surface tension is γ = 37mN/m, the density ρ = 1026kg/m 3. The solvent and polymeric contributions to the viscosity are η s = 65.2Pas and η p = 9.8Pas, respectively, the polymer timescale is λ = 8.1s. Relative to the capillary timescale τ = ρr0 3 /γ this results in a Deborah number of De = 296. Complex fluids: viscoelastic fluids containing many macromolecules. Figure 2. High speed videoimage of a jet of dilute (0.01 wt%) aqueous polyacryamide solution (surface Anomalous tensionviscoelastic γ = 62mN/m) properties: undergoing capillary shear shinning, thinning. The shear sharp-edged thicking, jet orifice tubeless is at the left of the image (radius R 0 = 0.30 mm) and the free jet velocity is 30 cm/s. The polymeric contribution siphon, Rod to climbing, the viscosityetc. η p = Pas, and the polymer timescale is found to be λ = 0.012s. This corresponds to a Deborah number of De = Many industrial usage: liquid crystal displayer, synthesized products (CD,...), plastics,... convected from left to right, fluid is forced by capillarity from the thinning ligaments into the spherical droplets Most analytical studies of this structure have been performed using simplifying assumptions about the slenderness of the liquid jet, see Yarin (1993) for a review. However, despite a considerable number of studies (see e.g. Goren & Gottlieb (1982), Entov & Yarin (1984), Bousfield et al. (1986), Forest et al. (1990), Shipman et al. (1991), Larson (1992), Renardy (1994), Renardy (1995), Chang at al. (1999)) a full analytical description of the beads-on-string phenomenon is still open, even in the context of one-dimensional models. In this paper we seek a self-similar solution that encompasses the elasto-capillary balance documented in experimental observations in liquid bridges, pinching drops and thin-

4 Mathematical models (solution case) Newtonian fluid (Linear constitutive relation) { ut + (u )u + p = τ, u = 0, where τ = τ s = η s( u + u T ). Non-Newtonian fluid (Nonlinear constitutive relation) { ut + (u )u + p = τ, u = 0, where τ = τ s + τ p, τ p is due to the polymer s contribution.

5 Mathematical models (solution case) Newtonian fluid (Linear constitutive relation) { ut + (u )u + p = τ, u = 0, where τ = τ s = η s( u + u T ). Non-Newtonian fluid (Nonlinear constitutive relation) { ut + (u )u + p = τ, u = 0, where τ = τ s + τ p, τ p is due to the polymer s contribution.

6 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

7 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

8 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

9 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

10 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

11 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

12 Coarse graining of polymers Molecular dynamics: limited space and time scales Macroscopic modelling: neglecting all of the molecular details Compromise: kinetic theory Bending persistence length: l p Length of one single polymer: l Flexible: l l p Rigid: l l p Semi-flexible: l O(l p) Additional angle potential Random flight Flexible Rigid Semiflexible

13 Simplest flexible polymers Dumbbell model z Q = r 1 r 2 r 1 O r 2 x y

14 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

15 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

16 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

17 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

18 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

19 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

20 Dumbbell model (dilute limit) Stochastic differential equations (homogeneous approximation, mean position) dx = udt dq = (κq 2 ζ F (Q))dt + 4k B T dw ζ Kramers Expression: τ p = nk BT I + n F (Q) Q Spring force Hookean Dumbbell: F (Q) = HQ FENE Dumbbell: F (Q) = HQ 1 Q 2 /Q 2 0 General nonlinear: F (Q) = γ(q 2 )Q, γ 0 A coupled deterministic-stochastic system (u and Q).

21 SDE for liquid crystal (high concentration case) Rod model for liquid crystals: an interacting particle system. L m m L Excluded volume 2b The non-dimensionalized form for pdf ψ: (U is usually taken as mean field potential, R = m m.) tψ = 1 R (Rψ + ψru) R (m κ mψ). De The corresponding stochastic version: nonlinear SDEs in the sense of McKean. m ( t +(u x)m = (I m m) 1 2 ). De RU +m κ m+ De Ẇ (t) where means the Stratonovich integral because the Brownian motion is on the sphere S 2.

22 SDE for liquid crystal (high concentration case) Rod model for liquid crystals: an interacting particle system. L m m L Excluded volume 2b The non-dimensionalized form for pdf ψ: (U is usually taken as mean field potential, R = m m.) tψ = 1 R (Rψ + ψru) R (m κ mψ). De The corresponding stochastic version: nonlinear SDEs in the sense of McKean. m ( t +(u x)m = (I m m) 1 2 ). De RU +m κ m+ De Ẇ (t) where means the Stratonovich integral because the Brownian motion is on the sphere S 2.

23 SDE for liquid crystal (high concentration case) Rod model for liquid crystals: an interacting particle system. L m m L Excluded volume 2b The non-dimensionalized form for pdf ψ: (U is usually taken as mean field potential, R = m m.) tψ = 1 R (Rψ + ψru) R (m κ mψ). De The corresponding stochastic version: nonlinear SDEs in the sense of McKean. m ( t +(u x)m = (I m m) 1 2 ). De RU +m κ m+ De Ẇ (t) where means the Stratonovich integral because the Brownian motion is on the sphere S 2.

24 SDE for liquid crystal (high concentration case) Rod model for liquid crystals: an interacting particle system. L m m L Excluded volume 2b The non-dimensionalized form for pdf ψ: (U is usually taken as mean field potential, R = m m.) tψ = 1 R (Rψ + ψru) R (m κ mψ). De The corresponding stochastic version: nonlinear SDEs in the sense of McKean. m ( t +(u x)m = (I m m) 1 2 ). De RU +m κ m+ De Ẇ (t) where means the Stratonovich integral because the Brownian motion is on the sphere S 2.

25 Outline Introduction

26 Numerical methods for the dumbbell model CONNFFESSIT: Calculation Of Non-Newtonian Flows: Finite Elements and Stochastic SImulation Technique (Laso-Öttinger) Fields of Q Sparse element Concentrated element CONNFFESSIT BCF Brownian Configuration Fields (BCF) dq + u Q = (κq 2 ζ F(Q))dt + 4k BT dw ζ Replacing Lagrangian description with Eulerian description Replacing random particles with random fields

27 Numerical methods for the dumbbell model CONNFFESSIT: Calculation Of Non-Newtonian Flows: Finite Elements and Stochastic SImulation Technique (Laso-Öttinger) Fields of Q Sparse element Concentrated element CONNFFESSIT BCF Brownian Configuration Fields (BCF) dq + u Q = (κq 2 ζ F(Q))dt + 4k BT dw ζ Replacing Lagrangian description with Eulerian description Replacing random particles with random fields

28 Numerical methods for the dumbbell model CONNFFESSIT: Calculation Of Non-Newtonian Flows: Finite Elements and Stochastic SImulation Technique (Laso-Öttinger) Fields of Q Sparse element Concentrated element CONNFFESSIT BCF Brownian Configuration Fields (BCF) dq + u Q = (κq 2 ζ F(Q))dt + 4k BT dw ζ Replacing Lagrangian description with Eulerian description Replacing random particles with random fields

29 Numerical methods for the dumbbell model CONNFFESSIT: Calculation Of Non-Newtonian Flows: Finite Elements and Stochastic SImulation Technique (Laso-Öttinger) Fields of Q Sparse element Concentrated element CONNFFESSIT BCF Brownian Configuration Fields (BCF) dq + u Q = (κq 2 ζ F(Q))dt + 4k BT dw ζ Replacing Lagrangian description with Eulerian description Replacing random particles with random fields

30 Numerical methods for the dumbbell model CONNFFESSIT: Calculation Of Non-Newtonian Flows: Finite Elements and Stochastic SImulation Technique (Laso-Öttinger) Fields of Q Sparse element Concentrated element CONNFFESSIT BCF Brownian Configuration Fields (BCF) dq + u Q = (κq 2 ζ F(Q))dt + 4k BT dw ζ Replacing Lagrangian description with Eulerian description Replacing random particles with random fields

31 An overview of the results for the BCF methods Well-posedness analysis Hookean dumbbell under shear flow: Jourdain-Lelievre-Le Bris (2002) FENE dumbbell under shear flow: Jourdain-Lelievre-Le Bris (2004) General dumbbell model, high dim with polynomial growth: E-Li-Zhang (2004) Convergence analysis Hookean dumbbell under shear flow: Jourdain-Lelievre-Le Bris, E-Li-Zhang (2002) Rod-like model under shear flow: Li-Zhang-Zhou (2004) High dimensional Hookean dumbbell model: Li-Zhang (2005) Nonlinear dumbbell: Only partial results.

32 Simple SDE and discretization Simple SDE dx t = b(x t)dt + dw t Euler-Maruyama scheme X n+1 = X n + tb(x n) + W n where W n are i.i.d. N(0, t) Gaussian R.V.s. Convergence result: Strong order 1. (Lipschitz continuity assumption) sup(een) C t n

33 Simple SDE and discretization Simple SDE dx t = b(x t)dt + dw t Euler-Maruyama scheme X n+1 = X n + tb(x n) + W n where W n are i.i.d. N(0, t) Gaussian R.V.s. Convergence result: Strong order 1. (Lipschitz continuity assumption) sup(een) C t n

34 Simple SDE and discretization Simple SDE dx t = b(x t)dt + dw t Euler-Maruyama scheme X n+1 = X n + tb(x n) + W n where W n are i.i.d. N(0, t) Gaussian R.V.s. Convergence result: Strong order 1. (Lipschitz continuity assumption) sup(een) C t n

35 BCF scheme for Hookean dumbbell under shear flow Discretized equations (temporal and stochastic) ( u n+1 u n = u n+1 yy t + y P n Q n N ) t, ( P n+1 i = Pi n + u n+1 y Q n i P i n ) t + dvi n, 2 where Q n+1 i = Q n i Qn i 2 t + dw n i, P n Q n N = 1 N Remark: u n are random variables, too! N Pi n Q n i. i=1

36 BCF scheme for Hookean dumbbell under shear flow Discretized equations (temporal and stochastic) ( u n+1 u n = u n+1 yy t + y P n Q n N ) t, ( P n+1 i = Pi n + u n+1 y Q n i P i n ) t + dvi n, 2 where Q n+1 i = Q n i Qn i 2 t + dw n i, P n Q n N = 1 N Remark: u n are random variables, too! N Pi n Q n i. i=1

37 A new type of convergence Coupling issue The errors e n are random variables! E e n+1 where ˆQ n satisfies y 2 L 2 ( ˆQ n ) 2 N E e n+1 y 2 y L 2 E ( ˆQ n ) 2 N y Large deviation estimate ˆQ n+1 i = ˆQ n i ˆQ n i 2 t + dw n i. t ( ˆQ n ) 2 N 1 2 after excluding a set A of exponentially small probability. Mean square convergence after excluding a set of exponentially small probability ) E ( e n 2L 2y 1Ac t N.

38 A new type of convergence Coupling issue The errors e n are random variables! E e n+1 where ˆQ n satisfies y 2 L 2 ( ˆQ n ) 2 N E e n+1 y 2 y L 2 E ( ˆQ n ) 2 N y Large deviation estimate ˆQ n+1 i = ˆQ n i ˆQ n i 2 t + dw n i. t ( ˆQ n ) 2 N 1 2 after excluding a set A of exponentially small probability. Mean square convergence after excluding a set of exponentially small probability ) E ( e n 2L 2y 1Ac t N.

39 A new type of convergence Coupling issue The errors e n are random variables! E e n+1 where ˆQ n satisfies y 2 L 2 ( ˆQ n ) 2 N E e n+1 y 2 y L 2 E ( ˆQ n ) 2 N y Large deviation estimate ˆQ n+1 i = ˆQ n i ˆQ n i 2 t + dw n i. t ( ˆQ n ) 2 N 1 2 after excluding a set A of exponentially small probability. Mean square convergence after excluding a set of exponentially small probability ) E ( e n 2L 2y 1Ac t N.

40 BCF for Liquid Crystal Polymers SDE form for Doi-Edwards model(shear flow, 1D) ) dθ t = (a(θ t, L(Θ t)) + yu sin 2 Θ t dt + dw t, τ(y, t) = E [ sin 2Θ t + a(θ t, L(Θ t)) cos 2 Θ t + yu sin 2 2Θ t ], where a(θ t, L(Θ t)) = + sin 2(Θt θ )L(Θ t)(dθ ). Weakly interacting particle system after discretization. dθ t = a(θ t, L(Θ t))dt + dw t, becomes dθ i t = 1 N N sin 2(Θ i t Θ j t)dt + dwt i, j=1 {Θ i t} coupled each other!

41 BCF for Liquid Crystal Polymers SDE form for Doi-Edwards model(shear flow, 1D) ) dθ t = (a(θ t, L(Θ t)) + yu sin 2 Θ t dt + dw t, τ(y, t) = E [ sin 2Θ t + a(θ t, L(Θ t)) cos 2 Θ t + yu sin 2 2Θ t ], where a(θ t, L(Θ t)) = + sin 2(Θt θ )L(Θ t)(dθ ). Weakly interacting particle system after discretization. dθ t = a(θ t, L(Θ t))dt + dw t, becomes dθ i t = 1 N N sin 2(Θ i t Θ j t)dt + dwt i, j=1 {Θ i t} coupled each other!

42 Convergence analysis for Liquid Crystal Polymers Convergence result (without excluding the small set) E Ef(Θ t) 1 N N f(θ j t) 2 C N j=1 This mean square convergence is due to the boundedness of sin function, which is quite special for this problem. Techniques (A.-S. Sznitman, LNM 1464 (1991)) Wasserstein metric for iterative convergence. Centering for mean square convergence.

43 Convergence analysis for Liquid Crystal Polymers Convergence result (without excluding the small set) E Ef(Θ t) 1 N N f(θ j t) 2 C N j=1 This mean square convergence is due to the boundedness of sin function, which is quite special for this problem. Techniques (A.-S. Sznitman, LNM 1464 (1991)) Wasserstein metric for iterative convergence. Centering for mean square convergence.

44 High-D convergence analysis for dumbbell model Define the error E n := Q n ˆQ n only with time discretization, then E n satisfies 1 t (En+1 E n ) + u n E n+1 + e n ˆQ n+1 = κ n E n+1 + e n ˆQn+1 E n. Identity u n E n+1 E n+1 dx = 0 D In order to estimate D en ˆQ n+1 E n+1 dx, one needs ˆQ n+1 2 N L Const. Difficult! L 2 -type estimate ˆQ n+1 2 N L 2 Const. is easy. But it can t be transfered back to L norm.

45 High-D convergence analysis for dumbbell model Define the error E n := Q n ˆQ n only with time discretization, then E n satisfies 1 t (En+1 E n ) + u n E n+1 + e n ˆQ n+1 = κ n E n+1 + e n ˆQn+1 E n. Identity u n E n+1 E n+1 dx = 0 D In order to estimate D en ˆQ n+1 E n+1 dx, one needs ˆQ n+1 2 N L Const. Difficult! L 2 -type estimate ˆQ n+1 2 N L 2 Const. is easy. But it can t be transfered back to L norm.

46 Essential ingredient 1: Explicit solution for Q SPDE for Q dq + u Q = (κq Q)dt + dw Introduce flow map dx(α, t) dt and deformation tensor = u(x(α, t), t), x(α, 0) = α. F (α, t) = x α, Explicit solution in Lagrangian coordinates xi i.e. Fij = α j Q(α, t) = e t F (α, t)q 0 (α) + F (α, t) t 0 e s t F 1 (α, s) dw s. Q is a Gaussian process with S.P.D. covariance matrix. This allows large deviation estimates.

47 Essential ingredient 1: Explicit solution for Q SPDE for Q dq + u Q = (κq Q)dt + dw Introduce flow map dx(α, t) dt and deformation tensor = u(x(α, t), t), x(α, 0) = α. F (α, t) = x α, Explicit solution in Lagrangian coordinates xi i.e. Fij = α j Q(α, t) = e t F (α, t)q 0 (α) + F (α, t) t 0 e s t F 1 (α, s) dw s. Q is a Gaussian process with S.P.D. covariance matrix. This allows large deviation estimates.

48 Essential ingredient 1: Explicit solution for Q SPDE for Q dq + u Q = (κq Q)dt + dw Introduce flow map dx(α, t) dt and deformation tensor = u(x(α, t), t), x(α, 0) = α. F (α, t) = x α, Explicit solution in Lagrangian coordinates xi i.e. Fij = α j Q(α, t) = e t F (α, t)q 0 (α) + F (α, t) t 0 e s t F 1 (α, s) dw s. Q is a Gaussian process with S.P.D. covariance matrix. This allows large deviation estimates.

49 Essential ingredient 1: Explicit solution for Q SPDE for Q dq + u Q = (κq Q)dt + dw Introduce flow map dx(α, t) dt and deformation tensor = u(x(α, t), t), x(α, 0) = α. F (α, t) = x α, Explicit solution in Lagrangian coordinates xi i.e. Fij = α j Q(α, t) = e t F (α, t)q 0 (α) + F (α, t) t 0 e s t F 1 (α, s) dw s. Q is a Gaussian process with S.P.D. covariance matrix. This allows large deviation estimates.

50 Essential ingredient 2: Strang s trick for estimating L h norm Inverse inequality under spatial discretization u n L h h d 2 u n L 2 h, Q n L h h d 2 Qn L 2 h Continuation technique u n h L h u n h u n L h + u n L h e n L h + u C 0 (D [0,T ]) h d 2 e n L 2 + u h C 0 (D [0,T ]) Ch p d 2 + u C 0 (D [0,T ]) If p > d 2, un h L h is bounded.

51 Essential ingredient 2: Strang s trick for estimating L h norm Inverse inequality under spatial discretization u n L h h d 2 u n L 2 h, Q n L h h d 2 Qn L 2 h Continuation technique u n h L h u n h u n L h + u n L h e n L h + u C 0 (D [0,T ]) h d 2 e n L 2 + u h C 0 (D [0,T ]) Ch p d 2 + u C 0 (D [0,T ]) If p > d 2, un h L h is bounded.

52 A stronger convergence result for high dimensional Hookean dumbbell model Final convergence result e,n+1 2 L 2 + h and E,,n+1 2 L 2 h h e,n 2 L 2 τ L2 h N C(δt 2 + h N 1 ɛ ) C(δt 2 + h ). N 1 ɛ where means after excluding a set of exponentially small probability. This type of convergence is considered by D.-G. Long (1988) for analysis of random vortex method.

53 A stronger convergence result for high dimensional Hookean dumbbell model Final convergence result e,n+1 2 L 2 + h and E,,n+1 2 L 2 h h e,n 2 L 2 τ L2 h N C(δt 2 + h N 1 ɛ ) C(δt 2 + h ). N 1 ɛ where means after excluding a set of exponentially small probability. This type of convergence is considered by D.-G. Long (1988) for analysis of random vortex method.

54 Nonlinear dumbbells: shear flow (decoupled case) Difficulties: non-lipschitz property of the force F ( ) dq t = κ Q t F (Q) dt + dw t One sided Lipschitz condition! (F (a) F (b), a b) 0 a, b R m. For FENE dumbbell model (implicit method) ( P n+1 = P n + u y(t n+1)q n 1 2 Q n+1 P n+1 1 (Q n+1 ) 2 /b Q n+1 = Q n (Q n+1 ) 2 /b t + dw n, ) t + dv n, A cubic equation for Q n+1, explicit unique solution in [0, b). Convergence for FENE is OK now.

55 Nonlinear dumbbells: shear flow (decoupled case) Difficulties: non-lipschitz property of the force F ( ) dq t = κ Q t F (Q) dt + dw t One sided Lipschitz condition! (F (a) F (b), a b) 0 a, b R m. For FENE dumbbell model (implicit method) ( P n+1 = P n + u y(t n+1)q n 1 2 Q n+1 P n+1 1 (Q n+1 ) 2 /b Q n+1 = Q n (Q n+1 ) 2 /b t + dw n, ) t + dv n, A cubic equation for Q n+1, explicit unique solution in [0, b). Convergence for FENE is OK now.

56 Nonlinear dumbbells: shear flow (decoupled case) Difficulties: non-lipschitz property of the force F ( ) dq t = κ Q t F (Q) dt + dw t One sided Lipschitz condition! (F (a) F (b), a b) 0 a, b R m. For FENE dumbbell model (implicit method) ( P n+1 = P n + u y(t n+1)q n 1 2 Q n+1 P n+1 1 (Q n+1 ) 2 /b Q n+1 = Q n (Q n+1 ) 2 /b t + dw n, ) t + dv n, A cubic equation for Q n+1, explicit unique solution in [0, b). Convergence for FENE is OK now.

57 Nonlinear dumbbells: shear flow (decoupled case) Difficulties: non-lipschitz property of the force F ( ) dq t = κ Q t F (Q) dt + dw t One sided Lipschitz condition! (F (a) F (b), a b) 0 a, b R m. For FENE dumbbell model (implicit method) ( P n+1 = P n + u y(t n+1)q n 1 2 Q n+1 P n+1 1 (Q n+1 ) 2 /b Q n+1 = Q n (Q n+1 ) 2 /b t + dw n, ) t + dv n, A cubic equation for Q n+1, explicit unique solution in [0, b). Convergence for FENE is OK now.

58 Nonlinear dumbbells: shear flow (decoupled case) For general nonlinear force, implicit equations are difficult to be solved. How to construct explicit schemes? Direct discretization P n+1 = P n tγ( Q n 2 )P n + u y(t n+1)q n t + dv n, Q n+1 = Q n tγ( Q n 2 )Q n + dw n. Difficult to obtain the L p bound of Q n, which is inevitable for analysis! New scheme P n+1 = Q n+1 = Convergence is OK! tγ( Q n 2 ) P n + u y(t n+1)q n t + dv n, tγ( Q n 2 ) Qn + dw n.

59 Nonlinear dumbbells: shear flow (decoupled case) For general nonlinear force, implicit equations are difficult to be solved. How to construct explicit schemes? Direct discretization P n+1 = P n tγ( Q n 2 )P n + u y(t n+1)q n t + dv n, Q n+1 = Q n tγ( Q n 2 )Q n + dw n. Difficult to obtain the L p bound of Q n, which is inevitable for analysis! New scheme P n+1 = Q n+1 = Convergence is OK! tγ( Q n 2 ) P n + u y(t n+1)q n t + dv n, tγ( Q n 2 ) Qn + dw n.

60 Nonlinear dumbbells: shear flow (decoupled case) For general nonlinear force, implicit equations are difficult to be solved. How to construct explicit schemes? Direct discretization P n+1 = P n tγ( Q n 2 )P n + u y(t n+1)q n t + dv n, Q n+1 = Q n tγ( Q n 2 )Q n + dw n. Difficult to obtain the L p bound of Q n, which is inevitable for analysis! New scheme P n+1 = Q n+1 = Convergence is OK! tγ( Q n 2 ) P n + u y(t n+1)q n t + dv n, tγ( Q n 2 ) Qn + dw n.

61 Nonlinear dumbbells: shear flow (coupled case) Not available now! Reference: review paper by Li-Zhang, Comm. Math. Sci. 5 (2007), 1-51.

62 Thank you!

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