A fractional step θ-method approximation of time-dependent viscoelastic fluid flow

Transcription

1 A fractional step θ-metod approximation of time-dependent viscoelastic fluid flow J.C. Crispell V.J. Ervin E.W. Jenkins July 5, 008 Abstract A fractional step θ-metod for te approximation of time dependent viscoelastic fluid flow equations, is described and analyzed in tis article. Te θ-metod implementation allows te velocity and pressure updates to be resolved separately from te stress, reducing te number of unknowns resolved at eac step of te metod. A streamline upwinded Petrov-Galerkin SUPG-metod is used to stabilize te constitutive equation. A priori error estimates are establised for te approximation sceme. Numerical computations supporting te teoretical results and demonstrating te θ-metod are also presented. Key words. θ-metod; splitting metod; viscoelastic flow AMS Matematics subject classifications. 65N30 Introduction Modeling viscoelastic fluid flow is computationally difficult for a variety of reasons. Assuming slow flow te modeling equations represent a Stokes system for te conservation of mass and momentum equations, coupled wit a non-linear yperbolic equation describing te constitutive relationsip between te fluid s extra stress and velocity. Te numerical approximation requires te determination of te fluid s velocity, pressure and stress a symmetric tensor. A direct approximation tecnique requires te solution of a very large nonlinear system of equations at eac time step. Te fractional step θ-metod [33, 34, 35] is an appealing numerical approximation tecnique for several reasons. Te θ-metod separates te updates for velocity/pressure and stress into several substeps. Variables are alternately lagged in te updates to reduce te size of te algebraic systems wic ave to be solved at eac substep. In addition, te splitting allows for te application of Department of Matematical Sciences, Clemson University, Clemson, SC , USA. Tis work was supported in part by te National Science Foundation under Award Number DMS-04079, and U.S. Army Researc Office Grant W9NF Corresponding autor. addresses: jcrisp@clemson.edu J.C. Crispell Department of Matematical Sciences, Clemson University, Clemson, SC , USA. Tis work was supported in part by te National Science Foundation under Award Number DMS-04079, and U.S. Army Researc Office Grant W9NF address: vjervin@clemson.edu V.J. Ervin, lea@clemson.edu E.W. Jenkins.

2 appropriate approximation tecniques to be applied wen resolving te resulting parabolic model for velocity and pressure, and yperbolic constitutive relation for stress. An additional benefit of te θ-metod [35] is tat by separating te update of velocity and pressure from te stress update, te algebraic systems to be solved at eac step in te metod are linear. Researc on viscoelastic materials can be traced back to Maxwell, Boltzman, and Volterra, in te late eigteen undreds, but it was te work of Oldroyd in 950 tat produced a constitutive model tat worked well wen modeling fluids wit large deformations [7, 38]. Since Oldroyd s original work, many constitutive equations ave been formulated to describe te motion of viscoelastic fluids. Tese include te models of Giesekus [5], Oldroyd [8], and Pan-Tien and Tanner [9], as well as te Jonson and Segalman [] constitutive model used in tis work. Error analysis of finite element approximations to steady state viscoelastic flow was first done by Baranger and Sandri in [3] using a discontinuous Galerkin DG formulation of te constitutive equation. In [3] Sandri presented analysis of te steady state problem using a streamline upwind Petrov-Galerkin SUPG metod of stabilization. Te time-dependent problem was first analyzed by Baranger and Wardi in [4], using an implicit Euler temporal discretization and DG approximation for te yperbolic constitutive equation. Ervin and Miles analyzed te problem using an implicit Euler time discretization and a SUPG discretization for te stress in [3]. Analysis of a modified Euler-SUPG approximation to te transient viscoelastic flow problem was presented by Bensaada and Esselaoui in [5]. Te temporal accuracy of te approximation scemes studied in [4, 5, 3] are all O t. Te work of Macmoum and Esselaoui in [3] examined time dependent viscoelastic flow using a caracteristics metod tat as accuracy O / t + t. Ervin and Heuer proposed a Crank-Nicolson time discretization metod [] wic tey sowed was O t. Teir metod uses a tree level sceme to approximate te nonlinear terms in te equations. Consequently teir approximation algoritm only requires linear systems of equations be solved. In [7] Bonito, Clément, and Picasso use an implicit function teorem to analyze a simplified time dependent viscoelastic flow model were te convective terms were neglected. Te fractional step θ-metod was introduced, and its temporal approximation accuracy studied, for a symmetric, positive definite spatial operator, by Glowinski and Périaux in [6]. Te metod is widely used for te accurate approximation of te Navier-Stokes equations NSE [0, 36, 37]. In [], Klouček and Rys sowed, assuming a unique solution existed, tat te θ-metod approximation converged to te solution of te NSE as te spatial and mes parameters went to zero, t 0 +. Te temporal discretization error for te θ-metod for te NSE was studied by Müller-Urbaniak in [5] and sown to be second order. Te implementation of te fractional step θ-metod in [35] for viscoelasticity differs significantly from tat for te NSE. For te NSE at eac sub-step te discretization contains te stabilizing operator u. For te viscoelasticity problem te middle substep wen resolving te stress is a pure convection transport problem tat requires stabilization in order to control te creation of spurious oscillations in te numerical approximation. Marcal and Crocet [4] were te first to use streamline upwinding to stabilize te yperbolic constitutive equation in viscoelastic flow. A second common approac to stabilizing te convective transport problem is to use a discontinuous Galerkin DG approximation for te stress [, 3]. In [0] te autors sowed a fractional step θ-metod for a linear convection-diffusion problem tat is second order accurate wit respect to te temporal discritization. Similar to te viscoelastic model, te linear convection-diffusion equations are a coupled yperbolic/parabolic system. Using te θ-metod s additve split allowed te distinct modeling equation penomonon yperbolic

3 convective transport and parabolic diffusion to be updated separatly. A SUPG approximation tecnique was used to stabilize te resulting yperbolic solve. Our work on te convection-diffusion equations was extended in our preliminary analysis of a θ-metod for viscoelastic fluid flow in [] were a priori error estimates for a Stokes-like problem assuming known stress, and constitutive model assuming known velocity and pressure were establised. In tis work we couple results similar to tose obtained in [] to obtain an a priori error estimate for te θ-metod applied to viscoelastic fluid flow. Te remainder of tis document is organized as follows: in Section te matematical model and θ- metod for viscoelastic fluid flow are introduced. Section 3 gives te matematical notation need in order to formulate te problem in an appropriate matematical setting. Te unique solvability and a priori error estimates for te θ-metod applied to te viscoelastic modeling equations is presented in Section 4. Numerical computations confirming te teoretical results and demonstrating te θ- metod are given in Section 5. Te Matematical Model and θ-metod Approximation In tis section te modeling equations for viscoelastic fluid flow as well as a fractional step θ-metod approximation sceme are presented. Te Jonson-Segalman Model for Viscoelastic Fluid Flow Te non-dimensional modeling equations for a viscoelastic fluid in a given domain Ω R d d =, 3 using a Jonson-Segalman constitutive equation are written as: σ σ + λ t + u σ + g aσ, u αdu = 0 in Ω,. u Re t + u u + p α du σ = f in Ω,. u = 0 in Ω,.3 u = 0 on Ω,.4 u0, x = u 0 x in Ω,.5 σ0, x = σ 0 x in Ω..6 Here. is te constitutive equation relating te fluids velocity u to te stress σ, and. and.3 are te conservation of momentum and conservation of mass equations. Te fluid pressure is denoted by p. Te Weissenberg number λ is a dimensionless constant defined as te product of a caracteristic strain rate and te relaxation time of te fluid [6]. Note tat if te value of λ is set to zero in. te Navier-Stokes modeling equations are obtained [6, 3]. Re denotes te fluid s Reynolds number were Re = LV ρ µ, and ρ = fluid density, L = caracteristic lengt scale, µ = fluid viscosity, V = caracteristic velocity scale. 3

4 Te body forces acting on te fluid are given by f, and α 0, denotes te fraction of te total viscosity tat is viscoelastic. Te g a term and deformation tensor du are defined as: and g a σ, u := a σ u + u T σ + a uσ + σ u T du = u + u T. Te gradient of u is defined suc tat u i,j = u i / x j. For te remainder of tis document slow or inertialess flow is assumed, allowing te inertial term u u in. to be neglected. Note tat an Oldroyd B constitutive model is obtained wen a = in g a σ, u. Proofs of te existence and uniqueness of solutions to.-.6 can be found in [4, 7, 30]. θ-metod for Viscoelastic Fluid Flow Te fractional step θ-metod additively decomposes equations. and.. Use te splitting parameters ω and γ 0,, and define: Constitutive equation: Conservation of momentum: Gσ := ωσ,.7 Gσ := ωσ + λ u σ + g a σ, u αdu..8 Fu := γ α du σ f,.9 Fu := γ α du..0 Let t denote te temporal increment between times t n and t n+, and for c {θ, ω, γ, a, α} let c := c. Also, let f n := f, n t. Te θ-metod approximation for viscoelasticity may ten be described as follows. See also [, 33, 35]. θ-metod algoritm for viscoelasticity Step a: Update te stress. λ σn+θ σ n + Gσ n+θ = Gσ n. Step b: Solve for velocity and pressure. Re un+θ u n + p n+θ + Fu n+θ = Fu n, u n+θ = 0. 4

5 Step a: Update te velocity and pressure. Re un+ θ u n+θ θ t + p n+ θ + Fu n+ θ = Fu n+θ, u n+ θ = 0. Step b: Solve for te stress. λ σn+ θ σ n+θ θ t + Gσ n+ θ = Gσ n+θ. Step 3a and Step 3b: Te temporal advancement to time t n+ is completed by repeating, Step a, and Step b wit n and n + θ replaced by n + θ and n +, respectively. Tis decomposition of te constitutive and conservation of momentum equations results in te approximation of te non-linear system of equations only requiring te solution of linear systems of equations. 3 Matematical Notation Te L Ω inner product and norm are denoted by,, and u respectively. Te standard Sobolev space [] of order k is denoted Wp k Ω, and its norm is given by W k p. Wen p = and k = 0 ten W 0 Ω = L Ω. Te notation H k Ω is used to represent te Sobolev space W k, and k denotes te norm in H k. Te following function spaces are defined for use in te analysis: X := H0 Ω := { u H Ω : u = 0 on Ω }, { S := σ = σ ij : σ ij = σ ji ; σ ij L Ω; i, j d } { σ = σ ij : u σ L Ω, u X }, { } Q := L 0Ω = q L Ω : q dx = 0, Ω { } Z := v X : q v dx = 0, q Q. Ω Te spaces X and Q satisfy te inf-sup condition inf sup q, v β > q Q v X q v A variational formulation of te inertialless version of modeling equations.-.3, found by multiplication by test functions and integrating over Ω, is: Given u 0 X and σ 0 S find 5

6 u, σ, p : 0, T ] X S Q suc tat σ λ t, τ + σ, τ α du, τ + λ u σ + g a σ, u, τ = 0, τ S, 3. u Re t, v p, v + α du, dv + σ, dv = f, v, v X, 3.3 u, q = 0, q Q, 3.4 u0, x = u 0 x, 3.5 σ0, x = σ 0 x. 3.6 As te velocity and pressure spaces X and Q satisfy te inf-sup condition 3., an equivalent variational formulation to is given by: Given u 0 Z and σ 0 S find u, σ : 0, T ] Z S suc tat λ σ t, τ + σ, τ α du, τ + λ u σ + g a σ, u, τ = 0, τ S, 3.7 u Re t, v + α du, dv + σ, dv = f, v, v Z, 3.8 u0, x = u 0 x, 3.9 σ0, x = σ 0 x. 3.0 To describe te finite element framework used in te analysis, let T be a triangulation of te discretized domain Ω R d. Ten Ω = K, K T. It is assumed tat tere exist constants c and c suc tat c K c ρ K, were K is te diameter of triangle K, ρ K is te diameter of te greatest ball spere included in K, and = max K T K. Let P k A denote te space of polynomials on A of degree no greater tan k and C Ω d te space of vector valued functions wit d components wic are continuous on Ω. Ten te associated finite element spaces are defined by: X := {v X C Ω d : v } K P k K K T, S := {τ S C Ω d d : τ } K P m K K T, Q := { q Q C Ω : q K P q K K T }, Z := {v X : q, v = 0 q Q }. Analogous to te continuous spaces assume tat X and Q satisfy te discrete inf-sup condition: inf q Q q, v sup β > v X q v Several different continuous and discrete norms are used in te analysis. Wen vx, t is defined on te entire time interval 0, T, define v,k := sup v, t k, v 0,k := 0<t<T T / v, t k dt, v t := v, t. 0 6

7 For N ZZ + let T/N = t, and define te discrete norms v,k := max v n k N, v 0,k := t v n n N For 0 θ, te temporal operator d θv n is defined as n= k. and for a full time step define d θ v n d t v n := vn v n θ, := vn v n. t 4 Analysis Te analysis for te θ-metod for viscoelastic fluid flow is accomplised in tree parts: te analysis of te Stokes-like problem using a known true stress; te analysis of te constitutive model assuming a known true velocity and pressure; and finally a coupling of tese estimates to establis te full a priori error estimates for te θ-metod. Te a priori error estimates establised for te Stokes-like problem, constitutive model, and te full θ-metod for viscoelastic fluid flow are given in Teorems 4.5, 4.6, and 4.7 respectively. Te proofs for Teorems 4.5, and 4.6 follow work done by te autors in []. A discussion of te proof for Teorem 4.7 is given in Section 4.. Detailed proofs of tese teorems are given in [9]. For te analysis it is elpful to define ũ := discrete approximation using true σ, σ := discrete approximation using true u, û := ũ u, ˆσ := σ σ. Note tat u, p, and σ denote approximations obtained by implementing te full θ-metod for viscoelastic flow described by Steps a - 3b above. Letting U and S denote te L projections of u and σ onto Z and S respectively, define: Λ n = u n U n, E n = U n ũ n, Γ n = σ n S n, F n = S n σ n, e n u = un ũ n, en σ = σn σ n. 4. Unique Solvability of te Sceme Computability of te algoritm is establised before te error estimates are presented. Computability implies tat te coefficient matrix associated wit te variational formulation of eac step of te θ-metod algoritm is invertible. To stabilize te yperbolic constitutive equation a streamline upwind Petrov-Galerkin SUPG discretization is used to avoid spurious oscillations in te 7

8 approximation. Tis is implemented by testing all terms in te constitutive equation except te discretized temporal derivative against modified test elements of te form τ µ δ were τ δ µ := τ + δu µ τ, 4. and δ is a small positive constant. Note tat δ = 0 gives te standard Galerkin metod. Te variational formulations for te steps in te θ-metod approximation are as follows. Step a: Find σ n+θ λ σ n+θ, τ + ω Step b: Find u n+θ Re u n+θ, v Step a: Find u n+ θ Re S suc tat = λ σ n+θ, τ n δ, τ ω λ g a σ n σ n Z suc tat + γ α γ α u n+ θ, v Step b: Find σ n+ θ λ du n+θ σ n λ u n, τ δ n + α, τ δ n, un, dv = Re f n+θ, v du n, dv + Z suc tat + γ α γ α σ n+ θ + λ Step 3a: Find σ n+ λ σ n+, τ λ du n+θ S suc tat, τ + ω + ω g a σ n+ θ = S suc tat u n+ θ du n+ θ, dv + σ n+ θ, τ n+ θ δ, u n+ θ σ n+, τ n+ θ δ λ, τ n+ θ δ, dv f n+θ, v + λ σ n+θ = λ λ σ n+ θ, τ n+ θ δ u n σ n, τ δ n, τ δ n, v du n σ n+θ, τ S. 4., dv, v Z. 4.3 Re = u n+θ, v, dv, v Z. 4.4 u n+ θ α, τ σ n+ θ ω, τ σ n+θ σ n+ θ du n+ θ g a σ n+ θ + α, τ n+ θ δ, τ δ n+ θ σ n+θ, τ n+ θ δ ω σ n+ θ, τ S. 4.5, τ n+ θ δ, u n+ θ, τ n+ θ δ, τ S. 4.6 du n+ θ, τ n+ θ δ 8

9 Step 3b: Find u n+ Z suc tat Re u n+, v + γ α γ α du n+ du n+ θ, dv, dv = Re + f n+, v u n+ θ, v σ n+, dv, v Z. 4.7 Te following induction ypotesis is used wen proving te lemmas tat establis te existence and uniqueness of te solutions to Induction Hypotesis Under te assumptions of Teorem 4.7 tere exists a constant K suc tat for n =,..., N u n, u n+θ, and K. 4.8 un+ θ Induction Hypotesis is justified in Section 4.3. Lemma 4. Step a Assume Induction Hypotesis is true. For δ C and t sufficiently small tere exists a unique solution σ n+θ S satisfying 4.. Proof: Equation 4. can be written as A were σ n+θ, τ = λ λ σ n, τ ωσ n u n σ n + g a σ n, un A σ n+θ, τ := λ σ n+θ, τ δ n, τ δ n, τ + α + ω du n σ n+θ, τ δ n, τ δ n., τ S, 4.9 Here 4.9 represents a square linear system of equations Ax = b. Wit te coice τ = σ n+θ, te individual terms in A are λ σ n+θ, σ n+θ λ = σ n+θ, ω, σ n+θ = ω σ n+θ, σ n+θ and ωδ σ n+θ, u n σ n+θ ωδ d C ωδk d σ n+θ. u n C σ n+θ Provided δ C and t λ/θωk d C, ten A σ n+θ, σ n+θ > 0. Tus, ker A = {0}. It follows tat 4. as a unique solution. Lemma 4. Step b Tere exists a unique solution u n+θ Z satisfying

10 Proof: Equation 4.3 can be written as were A A u n+θ u n+θ, v, v Note tat coosing v = u n+θ A u n+θ, u n+θ = Re = Re u n + f n+θ, v := Re u n+θ u n+θ, v γ α du n, dv, dv, v Z,, v, u n+θ σ n+θ + γ α + γ α du n+θ du n+θ, dv., du n+θ > 0. Tus, kera = {0}, and existence and uniqueness of a solution to 4.3 as been sown. Lemma 4.3 Step a Tere exists a unique solution u n+ θ Z satisfying 4.4. Proof: Write equation 4.4 as A 3 u n+ θ, v = Re + u n+θ f n+θ, v, v γ α σ n+θ, dv, du n+θ, dv were A 3 u n+ θ For v = u n+ θ, A3 unique solution., v u n+ θ Re := u n+ θ, v + γ α du n+ θ, dv., u n+ θ > 0. Tus, kera 3 = {0}, and it follows tat 4.4 as a Lemma 4.4 Step b Assume Induction Hypotesis is true. For δ C and t sufficiently small tere exists a unique solution σ n+ θ S satisfying 4.5. Proof: Write 4.5 as A 4 wit A 4 σ n+ θ, τ σ n+ θ, τ = λ := + λ λ ω σ n+ θ, τ u n+ θ σ n+θ, τ σ n+θ, τ n+ θ δ + ω σ n+ θ, τ n+ θ δ + α σ n+ θ + λ du n+ θ, τ n+ θ δ, τ n+ θ δ g a σ n+ θ, u n+ θ, 4.0, τ δ n+ θ. 0

11 Bounding te terms in A 4 Tus, σ n+ θ, σ n+ θ yields λ σ n+ θ, σ n+ θ ω σ n+ θ, σ n+ θ ω σ n+ θ, δu n+ θ σ n+ θ λ u n+ θ λ u n+ θ σ n+ θ, σ n+ θ σ n+ θ, δu n+ θ σ n+ θ λ g a σ n+ θ, u n+ θ, σ n+ θ λg a σ n+ θ σ n+ θ λ = = ω σn+ θ σn+ θ, ω d δc K σ n+ θ d λc K σ n+ θ, = λδ σ n+ θ, un+ θ 4λ σn+ θ u n+ θ σn+ θ 4 d λ u n+ θ σ n+ θ 4 d λc un+ θ σ n+ θ 4 d λc K σ n+ θ,, u n+ θ, δu n+ θ 4λ σn+ θ u n+ θ δun+ θ σ n+ θ 4 d λ u n+ θ σ n+ θ δun+ θ σ n+ θ 4 dλc K δ ɛ σn+ θ + λɛ δ un+ θ σ n+ θ A 4 σ n+ θ, σ n+ θ λ + ω δ ω + λ d C K 4 dλc δ K ɛ + δλ ɛ,. un+ θ, σn+ θ σ n+ θ Coosing ɛ =, δ C, and t C establises A 4σ n+ θ, σ n+ θ > 0. Hence kera 4 = {0}, implying tat a unique solution exists for 4.5. Te unique solvability of 4.6 and 4.7, representing te tird step in te algoritm, follows exactly as 4. and 4.3..

12 4. A Priori Error Estimates Teorems 4.5 and 4.6 give a priori error estimates for a θ-metod for te Stokes-like problem and te constitutive equation respectively. Tese estimates are used to establis te a priori error estimate for te full θ-metod implementation for viscoelastic fluid flow stated in Teorem 4.7. Teorem 4.5 Assuming σ is known For sufficiently smoot solutions u, σ, p suc tat σ, u t, u tt, u ttt, and u t K, t 0, T ], wit θ = and t C, te fractional step θ-metod approximation, ũ given by Step b, Step a, and Step 3b converges to u on te interval 0, T ] as t, 0, and satisfies te error estimates: were u ũ,0 Fu t,, 4. u ũ 0, Fu t,, 4. Fu t, := C k+ u t 0,k+ + C k u 0,k+ + C q+ p 0,q+ + C t u ttt 0,0 + C t u tt 0, + C t f tt 0,0 + C t C T + C k+ u,k Teorem 4.6 Assuming u and p are known For sufficiently smoot solutions σ, u, p suc tat σ, σ t, σ t, u, u t, u tt, u, u t, and u tt K, t [0, T ], wit θ = and t C, te fractional step θ-metod approximation, σ given by Step a, Step b, and Step 3a converges to σ on te interval 0, T ] as t, 0 and satisfies te error estimates: σ σ,0 Fσ t,, δ, 4.4 were σ σ 0,0 Fσ t,, δ, 4.5 Fσ t,, δ := C t σ ttt 0,0 + σ tt 0, + σ t 0, + σ 0, + σ tt 0,0 + σ t 0,0 + σ 0,0 + C T + C tδ σ 0, + σ t 0, + σ 0,0 + σ t 0,0 + C T + C m+ + δ m σ 0,m+ + C m+ σ,m+ + C m+ σ t 0,m+ + Cδ σ t 0,0. 4.6

13 Proofs of Teorems 4.5 and 4.6 are similar to te work done in [] and are presented in detail in [9]. Te error estimates in Teorems 4.5 and 4.6 are used in te a priori error estimate for te full θ-metod stated in Teorem 4.7. Teorem 4.7 For sufficiently smoot solutions σ, u, p suc tat σ, σ t, σ t, u, u t, u tt, u ttt, u, u t, u tt K, t [0, T ], wit θ = and t C, δ C, te fractional step θ-metod approximations u, σ given by Step a - 3b converge to u and σ, respectively, on te interval 0, T ] as t, 0 and satisfy te error estimates: σ σ,0 + u u,0 F t,, δ, 4.7 were σ σ 0,0 + u u 0, F t,, δ, 4.8 F t,, δ := C + δ + δ F u t, + C + δ F σ t,, δ, 4.9 and Fu t, and Fσ t,, δ are defined by 4.3 and 4.6 in Teorems 4.5 and 4.6, respectively. Outline of te proof of Teorem 4.7: Te proof of Teorem 4.7 requires Induction Hypotesis stated in 4.8 and te following additional Induction Hypotesis justified in Section 4.3. Induction Hypotesis Under te assumptions of Teorem 4.7 tere exists a constant K 3 suc tat n σ, σ n+θ, and σ n+ θ < K Make note tat u n u n u n ũ n + û n and σ n σ n σ n σ n + ˆσ n. Teorems 4.5 and 4.6 establis te bounds for u n ũ n and σ n σ n terms. In order to obtain bounds on E θ E and θ in te proof of Teorem 4.5 a Crank-Nicolson sceme was used to analyize Steps b and a of te initial time step. Te same approac is used in te proof of Teorem 4.7. Tus, to obtain bounds bounds on û n and ˆσ n bot te initial time step, and a general or typical time step in te θ-metod are analyized. We proceed wit te following steps: Step θ. Obtain bounds for te terms at terms: ˆσ θ, û θ, ˆσ θ, û θ, ˆσ, and û in te initial time step of te θ-metod. For example a bound on û θ û 0 is found by subtracting te full θ-metod approximation for Step b using approximated stress from te 3

14 variational formulation for Step b using a known stress. Note tat û 0 = 0, coose te test function v = û θ X, and bound eac term in te resulting expression to obtain a bound for û θ. A similar procedure is followed to find bounds for te oter at terms in te initial time step. Implementing a Crank-Nicolson metod for Step a requires σ θ, wic is not obtained until Step b. For te analysis a second order extrapolated value, σ θ ex see [9]. is obtained using σ 0 and σ θ Step θ. Following te work done in Step θ a bound on te difference of succesive θ-metod terms in te general time step are found by subtracting appropriate variational formulations. Coose v = û n+θ, û n+ θ, and û n+ in te difference of te variational fromulations obtained for Step b, Step a, and Step3b respectivly, and coose τ = ˆσ n+θ, ˆσ n+ θ, and ˆσ n+ in te difference of variational formulations obtained for Stepa, Step b and Step 3a respectivly. Appling suitable inequalities/estimates to te six resulting expressions yields te following inequalities: Step a: Step b: ˆσ n+θ Step a: û n+θ Step b: û n+ θ Step 3a: ˆσ n+ θ G t,, δ ˆσ n ˆσ n + G t,, δ û n + C t dû n+θ C t dû n + C3 t û n ˆσ n+θ ˆσ n+θ + H t, δfu t,, 4. + C 4 tfσ t,, δ, 4. û n+θ + C 5 t dû n+ θ C 6 t dû n+θ + C 7 t + C 8 tfσ t,, δ, 4.3 G 3 t,, δ G 5 t,, δ ˆσ n+ ˆσ n+θ ˆσ n+θ ˆσ n+ θ ˆσ n+ θ + G 4 t,, δ + G 6 t,, δ û n+ θ û n+ θ + H t, δfu t,, H 3 t, δfu t,, 4.5 4

15 Step 3b: û n+ C 0 t û n+ θ + C 9 t + C t dû n+ θ dû n+ + C tfσ t,, δ, 4.6 ˆσ n+ were G i t,, δ and H i t, δ terms denote functions, and te C i s denotes constants independent of te descritization and upwinding parameters t,, and δ. Step 3θ. Tree unit strides are constructed by: Summing Summing wit n n + in expressions 4. and 4.. Summing wit n n + in expressions 4., 4., 4.3 and 4.4. Step 4θ. Sum te unit stride expressions obtained in Step 3θ, applying inverse estimates to te rigt and side deformation terms and obtain an expression of te form ˆσ n + ˆσ n++θ ˆσ n+θ + ˆσ n++ θ ˆσ n+ θ + û n+ û n + û n++θ û n+θ + û n++ θ +C 3 t dû n+θ + C 4 t dû n+ θ + C 5 t dû n+ +C 6 t dû n++θ + C 7 t dû n++ θ K t,, δ ˆσ n + K t,, δ ˆσ n+θ + K 3 t,, δ ˆσ n+ +K 4 t,, δ ˆσ n+ θ + K 5 t,, δ ˆσ n++θ + K 6 t,, δ +K 7 t,, δ û n+θ + K 8 t,, δ û n+ θ + K 9 t,, δ +K 0 t,, δ û n++θ + K t,, δ û n++ θ ˆσ n+ û n+ θ û n û n+ +K t,, δf u t, + K 3 t,, δf σ t,, δ, 4.7 were te K i terms denote funcions of te descritization and upwinding parameters t,, and δ. Step 5θ. Sum expression 4.7 from n = 0 to N, and noting û 0 yields a bound at time N t = T., ˆσ0, and ˆσθ are zero, Step 6θ. Apply discrete Gronwall s inequality see [9] wit te restriction tat t C 8 + C 9 + δ C0 4 + δ C, 4.8 If δ is cosen suc tat δ C, 4.8 becomes C t

16 Tis is computationally very restrictive. Tis constraint is not enforced for te numerical results in Section 5. It is an open question if tis condition is necessary for te estimates given in Terorms 4.5 and 4.6. Step 7θ. Use te triangle inequality and te results of Teorems 4.5 and 4.6 to establis te Teorem Induction Hypotesis Verification of Induction Hypotesis for u. Assume tat Induction Hypotesis olds true for n =,,..., N. Interpolation properties and inverse estimates see [8] give u N u N u N + u N E N + Λ N + û N + K C d E N + C d Λ N + C d û N + K Applying te discrete Gronwall s inequality in te proof of Teorem 4.5 establises C d E N d C k + q+ d + t d, 4.3 and applying te discrete Gronwall s inequality in te proof of Teorem 4.7 gives C d û N C d k + m+ d + q+ d Using interpolation properties C d Tus, expression 4.30 yields u N C Λ N C k+ d + t d + tδ d + δ m d + δ d. 4.3 d k + m+ d + q+ d u N k+ d k+ C t d + tδ d + δ m d + δ d + K, 4.34 an expression independent of N. Hence, provided k, m d δ are cosen suc tat Ten Similarly it follows tat d k, m+ d, q+ d, and t, δ d, u N+θ u N < K + 7C., and u N+ θ < K + 7C. Te verification of Induction Hypotesis follows in a similar manner. d, q, and values of, t and 6

17 5 Numerical Results To verify numerical convergence rates for te θ-metod applied to viscoelastic fluid flow numerical results are presented for an example wit a known analytical solution. Finite element computations were done using te FreeFem++ integrated development environment [8]. Continuous piecewise quadratic elements were used for modeling te velocity, and continuous piecewise linear elements were used for te pressure and stress. Te constitutive equation was stabilized using a SUPG discretization wit parameter δ. For te optimal value of θ = / te local temporal discretization errors are O t, and tis optimal θ value was used in all computations. A numerical study verifying te optimal value of θ for viscoelastic fluid flow is given in []. Te constitutive and conservation equation splitting parameters ω and γ were set to / for te computations presented in Tables 5., 5., 5.3, and 5.4. Te full analysis presented by te autors in [9] and discussed above used a Crank-Nicolson metod in Steps b and a of te initial time step. Te computations presented in tis section were all implemented using te algoritm as stated in Section. Example Te teoretical convergence rates were verified by considering fluid flow across a unit square wit a known solution. Let Ω = 0, 0,, Re =, α = /, λ =, and a =. Te true solution is e u = x+y t x xy y e x+y t x xy, 5. y p = cosπxy y, 5. σ = αdu. 5.3 Remark: A rigt-and-side function is added to. and f in. is calculated using Tree sequences of computations were performed to verify te results of Teorems 4.5, 4.6, and 4.7. i Teorem 4.5: approximation of ũ and p, assuming σ, ii Teorem 4.6: approximation of σ, assuming u and p, iii Teorem 4.7: approximation of u, p and σ. 5. Approximating ũ and p wit known σ Numerical results for te approximation of velocity, ũ, and pressure, p for a known stress, σ, are presented in Table 5.. Tese results correspond to te analysis of Step b, Step a, and Step 3b as stated in Teorem 4.5. Note te following corollary to Teorem 4.5. Corollary 5. For X te space of continuous, piecewise quadratic functions, and Q te space of continuous, piecewise linear functions, t C, and σ, u, p sufficiently smoot, tere exists a 7

18 constant C suc tat te approximation ũ satisfies te error estimate: u ũ,0 + u ũ 0, C t Te numerical convergence rates observed in Table 5. are consistent wit tose predicted in Corollary 5.. Table 5.: Approximation errors and convergence rates for u ũ and p p at T =. t,, 4 4, 8 8, 6 6, 3 3, 64 64, 8 u ũ 0,.455e e-.837e e e-4.39e-4 Cvge. Rate u ũ 0,0.5039e-.8838e e e-4.97e e-6 Cvge. Rate u ũ,0.4446e e e-4.303e-4.740e e-6 Cvge. Rate p p 0, e-0.3e-.4973e e e e-3 Cvge. Rate p p, e e-.667e e e-3.95e-3 Cvge. Rate Approximating σ wit u and p known Te following corollary is obtained from Teorem 4.6. Corollary 5. For S te space of continuous, piecewise linear functions, t C, and σ, u, p sufficiently smoot, tere exists a constant C suc tat te approximation σ satisfies te error estimate σ σ,0 + σ σ 0,0 C t + tδ + δ + + δ. 5.5 Te numerical convergence rates presented in Table 5. are consistent wit tose predicted in Corollary 5.. In Table 5. te effect of te upwinding parameter δ on σ σ 0,0 and σ σ,0 is clearly evident. 5.3 Full θ-metod approximation for viscoelasticity Tables 5.3 and 5.4 contain te results for te approximation of u, p and σ using te θ-metod described by Step a - Step 3b. Corollary 5.3 is obtained from Teorem

19 Table 5.: Approximation errors and convergence rates for σ σ at T =. δ t,,, 4 4, 8 8, 6 6, 3 0 σ σ 0,0.35e e-.99e e-3.830e-3 Cvge. Rate σ σ,0.74e e-.646e e-3.387e-3 Cvge. Rate σ σ 0,0.0070e e e-.6980e e-3 Cvge. Rate σ σ,0.6409e e- 3.00e-.553e e-3 Cvge. Rate σ σ 0,0.074e e-.3575e e e-3 Cvge. Rate σ σ,0.6474e e-.70e e-3.769e-3 Cvge. Rate σ σ 0,0.0346e e-.045e- 5.38e e-3 Cvge. Rate σ σ,0.6595e e-.6546e e-3.399e-3 Cvge. Rate Corollary 5.3 For X te space of continuous, piecewise quadratic functions, S and Q te space of continuous, piecewise linear functions, t C, and σ, u, p sufficiently smoot, tere exists a constant C suc tat te approximations u and σ satisfy te error estimates u u,0 + u u 0, C t + tδ + δ + + δ, σ σ,0 + σ σ 0,0 C t + tδ + δ + + δ. Te numerical convergence rates in Tables 5.3 and 5.4 are consistent wit Corollary 5.3. Pressure is treated implicitly in all steps of te algoritm, and a first order convergence rate is observed for bot p p 0,0 and p p,0. As was te case in Section 5. te effect of te upwinding parameter δ can be seen in Tables 5.3 and Conclusions A fractional step θ-metod was analyzed for inertialess viscoelastic fluid flow modeled using a Jonson-Segalman constitutive equation. Te θ-metod presented was proved to be second order accurate wit respect to te temporal discretization, and allowed for te decoupling of te velocity and pressure update from te stress update. By decoupling te modeling system s variables, te θ-metod resulted in smaller approximating algebraic systems. An additional benefit of te metod is tat only linear systems of equations are solved at eac step in te approximation. 9

20 Table 5.3: Approximation errors and convergence rates for u u,0, σ σ,0, and p p,0 at T =. δ t,, 4 4, 8 8, 6 6, 3 3, 64 0 u u,0.608e e-4.559e e e-6 Cvge. Rate σ σ, e-.6455e e-3.373e-3.894e-4 Cvge. Rate p p, e0.87e e e-3.8e-3 Cvge. Rate u u,0.8998e-3.7e e e-4.85e-4 3 Cvge. Rate σ σ,0 6.38e-.954e-.3939e e e-3 Cvge. Rate p p, e0.345e-.560e e e-3 Cvge. Rate u u,0.788e e-4.485e e-5.8e-5 Cvge. Rate σ σ, e-.7633e- 5.50e e e-4 Cvge. Rate p p, e0.8850e e e-3.8e-3 Cvge. Rate u u,0.6484e e e e e-6 Cvge. Rate σ σ, e-.6543e e-3.38e e-4 Cvge. Rate p p, e0.834e e e-3.8e-3 Cvge. Rate

21 Table 5.4: Approximation errors and convergence rates for u u 0,, u u 0,0, σ σ 0,0, and p p 0,0 at T =. δ t,, 4 4, 8 8, 6 6, 3 3, 64 0 u u 0, 5.36e-.3580e e e-4.46e-4 Cvge. Rate u u 0,0.847e e-4.669e e e-5 Cvge. Rate σ σ 0, e-.9868e e e e-4 Cvge. Rate p p 0,0.0359e.5906e e e e-3 Cvge. Rate u u 0, 5.655e-.835e- 7.98e-3 3.0e e-3 3 Cvge. Rate u u 0, e-3.435e e-4.809e e-4 Cvge. Rate σ σ 0, e e-.6088e e e-3 Cvge. Rate p p 0,0.0360e.630e-.9587e e e-3 Cvge. Rate u u 0, 5.440e-.48e e-3.7e e-4 Cvge. Rate u u 0,0 3.54e e e e-5.485e-5 Cvge. Rate σ σ 0, e-.466e e-3.00e e-4 Cvge. Rate p p 0,0.0360e.5990e e e e-3 Cvge. Rate u u 0, e-.395e- 3.60e e-4.544e-4 Cvge. Rate u u 0,0.996e e e e-5.08e-5 Cvge. Rate σ σ 0, e-.004e e e e-4 Cvge. Rate p p 0,0.0359e.598e- 8.43e e-3.789e-3 Cvge. Rate

22 References [] R.A. Adams. Sobolev spaces. Academic Press, New York-London, 975. Pure and Applied Matematics, Vol. 65. [] J. Baranger and A. Macmoum. Existence of approximate solutions and error bounds for viscoelastic fluid flow: caracteristics metod. Comput. Metods Appl. Mec. Engrg., 48- :39 5, 997. [3] J. Baranger and D. Sandri. Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. I. Discontinuous constraints. Numer. Mat., 63:3 7, 99. [4] J. Baranger and S. Wardi. Numerical analysis of a FEM for a transient viscoelastic flow. Comput. Metods Appl. Mec. Engrg., 5-4:7 85, 995. [5] M. Bensaada and D. Esselaoui. Error estimates for a stabilized finite element metod for te Oldroyd B model. J. Mat. Anal. Appl., 35:04 059, 007. [6] R.B. Bird, R.C. Armstrong, and O. Hassager. Dynamics of Polymeric Liquids. Wiley- Interscience, 987. [7] A. Bonito, P. Clément, and M. Picasso. Matematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Mat., 07:3 55, 007. [8] S.C. Brenner and L.R. Scott. Te Matematical Teory of Finite Element Metods. Springer- Verlag, New York, second edition, 00. [9] J.C. Crispell. Numerical analysis of a fractional step θ-metod for fluid flow problem. PD tesis, Clemson University, 008. [0] J.C. Crispell, V.J. Ervin, and E.W. Jenkins. A fractional step θ-metod for convectiondiffusion problems. J. Mat. Anal. Appl., 333:04 8, 007. [] J.C. Crispell, V.J. Ervin, and E.W. Jenkins. A fractional step θ-metod for viscoelastic fluid flow using a SUPG approximation. International Journal of Computational Science, 3:336 35, 008. [] V.J. Ervin and N. Heuer. Approximation of time-dependent, viscoelastic fluid flow: Crank- Nicolson, finite element approximation. Numer. Metods Partial Differential Equations, 0:48 83, 004. [3] V.J. Ervin and W.W. Miles. Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal., 4: electronic, 003. [4] E. Fernández-Cara, F. Guillén, and R.R. Ortega. Matematical modeling and analysis of viscoelastic fluids of te Oldroyd kind. In Handbook of Numerical Analysis, Vol. VIII, pages Nort-Holland, Amsterdam, 00. [5] H. Giesekus. A unified approac to a variety of constitutive models for polymer fluids based on te concept of configuration dependent molecular mobility. Reol. Acta,, 98.

23 [6] R. Glowinski and J.F. Périaux. Numerical metods for nonlinear problem in fluid dynamics. In Supercomputing, pages Nort-Holland, Amsterdam, 987. [7] C. Guillopé and J.-C. Saut. Existence results for te flow of viscoelastic fluids wit a differential constitutive law. Nonlinear Anal., 59: , 990. [8] F. Hect, O. Pironneau, A. Le Hyaric, and K. Otsuka. FreeFem++. ttp:// [9] J.G. Heywood and R. Rannacer. Finite-element approximation of te nonstationary Navier- Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal., 7: , 990. [0] V. Jon. Large Eddy Simulation of Turbulent Incompressible Flows, volume 34. Springer- Verlag, Berlin, 004. Analytical and numerical results for a class of LES models. [] M.W. Jonson and D. Segalman. A model for viscoelastic fluid beavior wic allows nonaffine deformation. J. Non-Newtonian Fluid Mec., :55 70, 977. [] P. Klouček and F.S. Rys. Stability of te fractional step θ-sceme for te nonstationary Navier-Stokes equations. SIAM J. Numer. Anal., 35:3 335, 994. [3] A. Macmoum and D. Esselaoui. Finite element approximation of viscoelastic fluid flow using caracteristics metod. Comput. Metods Appl. Mec. Engrg., 904: , 00. [4] J.M. Marcal and M.J. Crocet. A new mixed finite element for calculation of viscoelastic flow. J. Non-Newtonian Fluid Mec., 6:77 4, 987. [5] S. Müller-Urbaniak. Eine Analyse dex Zweiscritt-θ-Verfarens zur Lösung der instationären Navier-Stokes-Gleicungen, Tesis. University of Heidelberg, 994. [6] J. Neustupa and P. Penel, editors. Matematical Fluid Mecanics: Recent Results and Open Questions. Birkäuser, 00. [7] J.G. Oldroyd. On te formulation of reological equations of state. Proc. Roy. Soc. London, A00:53 54, 950. [8] J.G. Oldroyd. Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. London, A45:78 97, 958. [9] N. Pan-Tien and R.I. Tanner. A new constitutive equation derived from network teory. J. Non-Newtonian Fluid Mec., : , 977. [30] M. Renardy. Matematical Analysis of Viscoelastic Flows, volume 73. Society for Industrial and Applied Matematics SIAM, Piladelpia, PA, 000. [3] R.H. Sabersky and A.J. Acosta. Fluid Flow: A First Course in Fluid Mecanics. Te Macmillan Company, 964. [3] D. Sandri. Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. Continuous approximation of te stress. SIAM J. Numer. Anal., 3:36 377,

24 [33] P. Saramito. A new θ-sceme algoritm and incompressible FEM for viscoelastic fluid flows. RAIRO Modél. Mat. Anal. Numér., 8: 35, 994. [34] P. Saramito. Efficient simulation of nonlinear viscoelastic fluid flows. J. Non-Newt. Fluid Mec., 60:99 3, 995. [35] R. Sureskumar, M.D. Smit, R.C. Armstrong, and R.A. Brown. Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations. J. Non-Newt. Fluid Mec., 8:57 04, 999. [36] S. Turek. A comparative study of time-stepping tecniques for te incompressible Navier- Stokes equations: From fully implicit non-linear scemes to semi-implicit projection metods. Internat. J. Numer. Metods Fluids, 0:987 0, 996. [37] S. Turek. Efficient Solvers for Incompressible Flow Problems, volume 6. Springer-Verlag, Berlin, 999. [38] J.L. Wite and A.B. Metzner. Development of constitutive equations for polymeric melts and solutions. Journal of Applied Polymer Science, 7: ,