NUMERICAL ANALYSIS OF A FRACTIONAL STEP θ-method FOR FLUID FLOW PROBLEMS A Dissertation Presented to te Graduate Scool of Clemson University In Partial Fulfillment of te Requirements for te Degree Doctor of Pilosopy Matematical Sciences by Jon C. Crispell August 008 Accepted by: Dr. Vincent J. Ervin, Committee Cair Dr. Eleanor W. Jenkins, Co-Cair Dr. Cristoper E. Kees Dr. Hyesuk K. Lee Dr. Daniel D. Warner
ABSTRACT Te accurate numerical approximation of viscoelastic fluid flow poses two difficulties: te large number of unknowns in te approximating algebraic system corresponding to velocity, pressure, and stress, and te different matematical types of te modeling equations. Specifically, te viscoelastic modeling equations ave a yperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approac is to use a fractional step θ-metod. Te θ-metod is an operator splitting tecnique tat may be used to decouple matematical equations of different types as well as separate te updates of distinct modeling equation variables wen modeling mixed systems of partial differential equations. In tis work a fractional step θ-metod is described and analyzed for te numerical computation of bot te time dependent convection-diffusion equation and te time dependent equations of viscoelastic fluid flow using te Jonson-Segalman constitutive model. For convection-diffusion te θ-metod presented allows for a decoupling witin time steps of te parabolic diffusion operator from te yperbolic convection operator. Te yperbolic convection update is stabilized using a Streamline Upwinded Petrov-Galerkin SUPG-metod. Te analysis given for te convection-diffusion equation serves as a template for te analysis of te more complicated viscoelastic fluid flow modeling equations. Te θ-metod implementation analyzed for te viscoelastic modeling equations allows te velocity and pressure approximations witin time steps to be decoupled from te stress, reducing te number of unknowns resolved at eac step of te metod. Additionally te θ-metod decoupling results in te approximation of te nonlinear viscoelastic modeling system using only te solution of linear systems of equations. Similar to te sceme implemented for convection-diffusion, te yperbolic constitutive equation is stabilized using a SUPG-metod. For bot te convection-diffusion and te viscoelastic modeling equations a priori error estimates are establised for teir θ-metod approximations. Numerical ii
computations supporting te teoretical results and demonstrating te θ-metods are also included. iii
To My Parents
ACKNOWLEDGMENTS Tis work would not ave been possible witout te support of many oters. I would like to start by acknowledging my advisors: Dr. Vince Ervin, and Dr. Lea Jenkins. I do not ave te words to express te debt of gratitude I feel for te knowledge tey ave sared, guidance tey ave given, and patience and kindness tey ave sown me. Tey bot possess a love of matematics tat comes troug in and outside te class room. It as been a privilege to be teir student, and I ope to work wit bot Dr. Jenkins and Dr. Ervin in te future. I would like to tank Dr. Cris Kees for taking me under is wing during my summer internsips wit te Army Corps of Engineers in Vicksburg, as well as for serving on my committee. It was an onor to work under is guidance, and I look forward to future opportunities to work wit Dr. Kees. I would also like to tank te oter members of my committee Dr. Dan Warner and Dr. Hyesuk Lee. Teir experience, suggestions, and expertise ave been a tremendous benefit. Trougout te completion of tis work I was supported by grants from te National Science Foundation and te U.S. Army Researc Office. Tese grants allowed me to focus my efforts over te last four years on researc. I will be forever grateful for aving ad tat opportunity. I would like to tank my parents, my broter Jared you will always be my Captain, and Grandma for all teir support and love. Teir encouragement and reassurance trougout my life as always been a blessing. Tey believed in me before I believed in myself. I can t tank tem enoug. Over te last few years I ad te privilege to get to know Sandy Berry and er family during my summer stays in Vicksburg Mississippi. Se and er family made a boy from te Nort feel at ome in te Sout. v
During my time at Clemson I was lucky enoug to sare an office wit Jason Howell. Jason is a tremendous colleague and friend. Knowing im as enanced my life bot personally and professionally. My friends ave been a immense benefit wile completing tis work. I couldn t ave done it witout teir support. Specifically, I would like to tank Denise Guinnane for er constant encouragement and unwavering patience. vi
TABLE OF CONTENTS Page TITLE PAGE....................................... i ABSTRACT........................................ ii DEDICATION....................................... ACKNOWLEDGMENTS................................. LIST OF TABLES..................................... LIST OF FIGURES.................................... iv v ix x CHAPTER. INTRODUCTION................................. Motivation................................... Te Fractional Step θ-metod........................3 Te Jonson-Segalman Model for Viscoelastic Fluid Flow........ 5. Matematical Notation........................... 6. CONVECTION-DIFFUSION.......................... 0. Introduction................................. 0. Convection-Diffusion Model and Notation..................3 θ-metod for Convection-Diffusion...................... Variational Formulation........................... 3.5 Unique Solvability of te Sceme......................6 A Priori Error Estimates.......................... 5.7 Optimal θ Value............................... 8.8 Numerical Results.............................. 3 3. VISCOELASTIC FLUID FLOW........................ 7 3. Introduction................................. 7 3. θ-metod for Viscoelastic Fluid Flow................... 7 3.3 Matematical Notation........................... 9 3. Unique Solvability of te Sceme..................... 3 3.5 A Priori Error Estimates for te Conservation Equations and Constitutive Equation............................... 37 3.5. Analysis of te conservation equations.............. 38 3.5. Analysis of te constitutive equation............... 59 3.6 Establising te Full θ-metod Error Estimate.............. 5 vii
Table of Contents Continued Page 3.6. Te first time step 0, t]..................... 8 3.6. Te general time step n t, n t].............. 53 3.7 Induction Hypotesis............................ 63. NUMERICAL RESULTS FOR VISCOELASTIC FLUID FLOW...... 65. Example.................................. 65.. Approximating ũ and p wit known σ............. 66.. Approximating σ wit u and p known.............. 68..3 Full θ-metod approximation for viscoelasticity......... 68. Example.................................. 7 5. SUMMARY AND FUTURE WORK...................... 75 APPENDICES 5. Convection-Diffusion Equation....................... 75 5. Viscoelastic Fluid Flow........................... 75 5.3 Two-Pase Flow in a Porous Media.................... 76 A. Common Bounds................................. 78 B. Initial Bounds for te Conservation Equations................. 8 C. Initial Bounds for te Constitutive Equation.................. 87 BIBLIOGRAPHY..................................... 95 viii
LIST OF TABLES Table Page. Approximation errors and convergence rates for Example........... Approximation errors and convergence rates for Example.......... 5.3 Approximation errors and convergence rates for Example 3.......... 6. Approximation errors and convergence rates for u ũ and p p at T =........................................ 67. Approximation errors and convergence rates for σ σ at T =...... 69.3 Approximation errors and convergence rates for u u,0, σ σ,0, and p p,0 at T =............................ 70. Approximation errors and convergence rates for u u 0,, u u 0,0, σ σ 0,0, and p p 0,0 at T =.................... 7 ix
LIST OF FIGURES Figure Page. Experimental convergence rates........................... Error u u 0,0 as a function of θ.........................3 Plots of te true and approximate solution for convection-diffusion Example.. Plots of te true and approximate solution for convection-diffusion Example 3. 6. Optimal θ value................................... 66. Plot of : contraction domain geometry...................... 7.3 Sample contraction mes.............................. 73. Streamlines and magnitude of velocity contours for u at t = 0.5 and t = 0.375. 73.5 Streamlines and magnitude of velocity contours for u at t = 0.5 and t = 0.75. 7.6 Streamlines and magnitude of velocity contours for u at t = and t =.... 7 x
CHAPTER INTRODUCTION. Motivation Viscoelastic fluids are encountered everyday. Examples of tese fluids include: food products like egg wites, mayonnaise, and molten cocolate, biological fluids suc as mucus and blood, as well as industrial materials like paint, adesives, multi-grade oils, and polymers [8]. From tis list of examples it can be seen tat viscoelastic fluids sare te properties of bot viscous fluids and elastic solids. In order to describe te motion of a viscoelastic fluid te constitutive law modeling te relationsip between te fluid s stress and deformation must incorporate a istory of te fluids deformation. Te complicated nature of te equations governing viscoelastic flow make finding analytical solutions difficult, if not impossible. Hence, te study of good numerical metods to obtain approximations to te modeling equations of viscoelastic flow is a topic of interest for bot academic and industrial researcers, as it can give useful insigts into applications suc as biological fluid modeling, as well as fiber and film processing. To tis end, in tis dissertation a fractional step θ-metod is examined for numerically simulating viscoelastic fluid flow. Computational metods for modeling viscoelastic fluid flow are difficult for a variety of reasons. Te modeling equations assuming slow flow 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 fluids velocity and te extra stress. Te numerical approximation requires te determination of te fluid s velocity, pressure and stress a symmetric tensor. For an accurate approximation a direct approximation tecnique requires te solution of a very large nonlinear system of equations at eac time step. Te fractional step θ-metod [55, 56, 58] is an appealing numerical approximation tecnique for several reasons. Te θ-metod decouples te approximation of velocity and
pressure from te approximation of stress, tereby reducing te size of te algebraic systems wic ave to be solved at eac sub-step. Tis decoupling also allows for 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 [58] is tat by decoupling velocity and pressure from te stress, te algebraic systems to be solved at eac sub-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 [6, 6]. 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 [7], and Pan-Tien and Tanner [50], as well as te Jonson and Segalman [36] 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 [] using a discontinuous Galerkin DG formulation of te constitutive equation. In [5] Sandri presented analysis of te steady state problem using a streamline upwind Petrov-Galerkin SUPG metod of stabilization. Te timedependent problem was first analyzed by Baranger and Wardi in [5], 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 []. Analysis of a modified Euler-SUPG approximation to te transient viscoelastic flow problem was presented by Bensaada and Esselaoui in [6]. Te temporal accuracy of te approximation scemes studied in [5, 6, ] are all O t. Te work of Macmoum and Esselaoui in [0] 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 [9] 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 [9] 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 [7]. Te metod is widely used for te accurate approximation of te Navier-Stokes equations NSE [33, 59, 60]. In [39], 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 [] and sown to be second order. Te implementation of te fractional step θ-metod in [58] 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 [] 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, ]. Prior to presenting analysis of te fractional step θ-metod for viscoelastic fluid flow in Capter 3, a preliminary analysis is conducted for a linear convection-diffusion problem in Capter. Te linear convection-diffusion equations are a coupled yperbolic/parabolic system and terefore sare some of te caracteristics observed in te viscoelastic model. Te analysis conducted for te convection-diffusion problem serves as a template for te work done regarding viscoelasticity. Capter 5 contains a brief summary and a description of future work. Te remainder of te current Capter gives an introduction to te θ-metod, 3
establises te matematical model considered for viscoelastic fluid flow, and presents relevant matematical notation need in order to formulate te problem in an appropriate matematical setting.. Te Fractional Step θ-metod In order to introduce te fractional step θ-metod consider te following time dependent partial differential equation written in te abstract form: u t F u, x, t = 0 in Ω 0, T ] subject to ux, t = 0 x Ω 0, T ux, 0 = u 0 x x Ω. In order to implement te θ-metod F is additively split suc tat F u, x, t = F u, x, t F u, x, t. Discretizing te temporal derivative and coosing a value of θ 0, /, te θ-metod advances across a generic n t time step of size t according to te following tree step procedure: Step. Compute an approximation to u nθ : u nθ θ t u n F nθ = F n. t n θ t Step. Compute an approximation to u n θ : u n θ u nθ θ t F n θ = F nθ. t n θ t nθ θ t Step 3. Compute an approximation to u n : θ t u n u n θ θ t F n = F n θ. t n
It will be seen in te analysis and corresponding computations for bot convectiondiffusion and viscoelastic fluid flow, tat for te optimal coice of θ = / te fractional step θ-metod acieves a second-order temporal discretization error..3 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 u Re t u u t u σ g aσ, u αdu = 0 in Ω,. p α du σ = f in Ω,. u = 0 in Ω,.3 u = 0 on Ω,. 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 is a dimensionless constant defined as te product of a caracteristic strain rate and te relaxation time of te fluid [8]. Note tat if te value of is set to zero in. te well known modeling equations for a Newtonian fluid, introduced by Navier 87 and Stokes 85, are obtained [5, 53]. Re denotes te fluids Reynolds number were Re = LV ρ µ, and ρ = fluid denisty, L = caracteristic lengt scale, µ = fluid viscosity, V = caracteristic velocity scale. Te body forces acting on te fluid are given by f, and α 0, denotes te fraction of te total viscosity tat is viscoelastic. 5
Te g a term and deformation tensor du are defined as: g a σ, u := a σ u u T σ a uσ σ u T and 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 term u u in. to be neglected. Note tat wen te parameter a = in g a σ, u, an Oldroyd B constitutive model is obtained. Proofs of te existence and uniqueness of solutions to.-.6 can be found in [, 8, 5].. Matematical Notation Let L r Ω be defined as te space of all functions f on a domain Ω in R d suc tat for r < f r da <. Ω For te case of L Ω te space of all essentially bounded functions on Ω te norm f = ess sup fx x Ω is used, were te essential supremum of f on Ω is te greatest constant lower bound K for wic fx K almost everywere on Ω [, 8]. Te L Ω inner product and norm are denoted by,, and u respectively. For functions u and v u, v := for vector functions u and v u, v := Ω Ω uv da, and u := u da Ω, u v da, and u := u u da Ω, 6
and for tensors σ and τ σ, τ := Ω σ : τ da, and σ := σ : σ da Ω. Te products are defined as follows for vectors u and v, and tensors σ and τ u v = d i= u i v i, and σ : τ = d d i= j= σ ij τ ij. Denote te set of L functions tat vanis on te boundary by L 0Ω = { f L Ω : f = 0 on Ω }. 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 0Ω = L Ω. Te notation H k Ω is used to represent te Sobolev space W k, and k denotes te norm in Hk. 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. 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 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 nn For 0 θ, te temporal operator d θv n is defined as n= k. d θ v n := vn v n θ, θ t and if a full time step is examined define d t v n := vn v n. t It is necessary to use inverse estimates to establis relationsips between various finite element space norms. Tus, for P, a vector space of functions defined on a bounded domain ˆK in R d, Lemma.. Inverse Estimate [] Let {T }, 0 <, denote a quasi-uniform family of subdivisions of a polyedral domain Ω R d. Let ˆK, P, N be a reference finite element suc tat P W l p ˆK W m q ˆK, were p, q, and 0 m l. For K T, let K, P K, N K be te affine equivalent element, and V = {v : v is measurable and v K P K K T }. Ten tere exists C = Cl, p, q suc tat for all v V. K T v p W l pk /p C m lmin 0, d p d q K T v q W m q K /q.7 Te discrete Gronwall s inequality [3] is used wen establising a priori error estimates for te fractional step θ-metod applied to bot convection-diffusion and te viscoelastic fluid modeling equations. 8
Lemma.. Discrete Gronwall s Inequality Let t, H, and a n, b n, c n, γ n for integers n 0 be nonnegative numbers suc tat l l l a l t b n t γ n a n t c n H for l 0. Suppose tat tγ n < n, and set σ n = tγ n. Ten } l l l a l t b n exp t σ n γ n { t c n H for l 0..8 9
CHAPTER CONVECTION-DIFFUSION. Introduction Te fractional step θ-metod for te linear convection-diffusion equation is analyzed in tis capter. In general, operator splitting metods for convection-diffusion problems can be divided into two approaces: i additive decomposition metods and ii product decomposition metods. Additive decomposition metods rewrite te spatial operator as a sum of several operators. At eac sub-step in te approximation algoritm te spatial operator is replaced by its additive decomposition, wit some of te operators evaluated at te current time i.e. treated implicitly and te oters at past times i.e. treated explicitly. Examples of tis approac are te Alternating Direction Implicit ADI metods [7, 0, 3, 9] and te IMplicit EXplicit IMEX scemes [, 3]. Wit product decomposition metods, to advance te approximation from time t n to t n, a pure convective operator is applied to obtain an initial estimate at t n. Tis estimate is ten taken as te initial data at t n and a pure diffusion operator is used to determine te approximation at t n. Examples of tis approac include te work of Dawson and Weeler [5, 6], Kan and Liu [38], and Evje and Karlsen [3]. A survey of tese metods can be found in []. Te fractional step θ-metod studied ere for convection-diffusion is an additive decomposition metod, wit te features of a product decomposition metod. In te first and tird sub-steps of te tree sub-step algoritm a pure diffusion problem is approximated. In te second sub-step a pure convection problem is approximated. Analysis of te convection-diffusion problem will provide insigt for te viscoelastic fluid flow problem because te approximation sceme studied for convection-diffusion is similar to tat in [57] for viscoelastic fluid flow. Te middle sub-step in bot applications is a pure convection transport problem, and te first and tird sub-steps are parabolic problems. In tis work 0
te θ-metod is applied to te linear convection-diffusion equations is torougly outlined. A detailed analysis can be found in [] and [3]. Te remainder of tis capter is organized as follows. In te next section te convection-diffusion problem is specified, and te matematical notation used tat is specific to its analysis is given. In Section.3 te fractional step θ-metod for te convectiondiffusion equation is described, computability of te algoritm is sown, and a priori error estimates for te metod are establised. A discussion on te optimal coice of te θ parameter is given in Section.7. Numerical examples demonstrating te metod are presented in Section.8.. Convection-Diffusion Model and Notation Te linear convection-diffusion equation may be written as u t u b u c u = f in Ω 0, T ]. ux, t = 0, x Ω 0, T ]. ux, 0 = u 0 x, x Ω,.3 were b = [b x, t, b x, t] T is an incompressible velocity field i.e b = 0, cx, t c min is an absorption coefficient, and fx, t is a given body force. Te function space { } X := H0 Ω := u H Ω : u = 0,. Ω is used in te variational formulations in te analysis. Te corresponding finite element space X X is written as: X := { v X C Ω : v K P k K K T }. Let U be te L projection of u onto X, and use u n := u, n t. Used in te error analysis are Λ n and E n defined by Λ n := u n U n, E n := U n u n.
Note tat Λ is te error between te true solution and te best solution in te discrete space, and E is te difference between te acieved approximation and te best approximation available in te discrete space. Having defined Λ te following interpolation properties of finite element spaces will be utilized and can be found in []: N N t Λ n tci k u, t n k Ck u 0,k.5 N N t Λ n tci k u, t n k Ck u 0,k.6 N Λ n Λ n t t n= C k u t 0,k.7 were k denotes te order of te elements used in te interpolation space..3 θ-metod for Convection-Diffusion Using te outline of te θ-metod given in Section., an abstract representation of.-.3 is: u t F u, x, t = 0 in Ω 0, T ].8 subject to ux, t = 0, x Ω 0, T.9 ux, 0 = u 0 x, x Ω,.0 were F = u b u cu f. Split F additively as: F u, x, t = F u, x, t F u, x, t. were F u, x, t = u c u f. F u, x, t = b u c u..3
For 0 θ, te fractional step metod for advancing te approximation of un to u n is described by tese steps: Step. Compute te approximation u nθ : u nθ θ t u n F nθ = F n d θ u nθ u nθ c unθ Step. Compute te approximation u n θ : u n θ u nθ θ t d θ u n θ F n θ = F nθ b u n θ c un θ Step 3. Compute te approximation to u n : u n u n θ F n = F n θ θ t d θ u n u n c un = f nθ b u n = f nθ u nθ = f n b u n θ Analysis requires variational formulations of eac substep. c un.. c unθ..5 c un θ..6 u nθ. Variational Formulation Te corresponding discrete variational formulation of..6 is: Determine X, u n θ d θ u nθ c unθ, v X, and u n u nθ X satisfying, v = f nθ b u n c un, v, v X,.7 d θ u n θ =, v f nθ c unθ, v b b u n θ c un θ, v b, v u nθ u nθ, δb v, v X,.8 3
d θ u n Note te following: c un, v =, v f n b u n θ u n c un θ, v, v X..9 To stabilize te convection transport equation.5, a Streamline Upwind Petrov Galerkin SUPG metod is used [35]. Te term v b is defined as v b := v δb v. Te term u, δb v is defined elementwise as see [3]: u, δb v := u δb v da. K T Te solution ux, t of.,. satisfies te continuous variational formulation u t, v u, v b u, v cu, v = f, v, v X..0 K.5 Unique Solvability of te Sceme Te first step in te analysis is to sow tat te sceme.7.9 is computable. Tat is, te associated coefficient matrices on te left and sides of.7 -.9 are invertible. Lemma.5. Tere exists a unique solution u nθ X satisfying.7. Proof: Equation.7 can be equivalently written as Au nθ, v = f nθ θ t un b u n c un, v, v X,. were Aw, z = θ t w, z w, z c w, z. Note tat. represents a square linear system of equations Ac = f. Te fact tat Aw, w = θ t w, w w, w c w, w > 0 guarantees tat kera = {0}. It follows tat.7 as a unique solution. Te unique solvability of.9 follows exactly as for.7. For.8 te same approac,
togeter wit te divergence free assumption for b i.e. b = 0, establises te unique solvability. Te unique solvability of te θ-metod algoritm and te subsequent a priori error estimates for te convection-diffusion equation may be sown witout te divergence free assumption for b provided te time step t is assumed sufficiently small..6 A Priori Error Estimates Te accuracy of te approximation from.7-.9 is addressed using an a priori error estimate. Tis estimates given in Teorem.6., and a discussion of te proof is presented below. Te complete proof is given in []. Teorem.6. For a sufficiently smoot solution u, wit t C, te fractional step θ-metod approximation u given by.7-.9 converges to u on te interval 0, T ] as t, 0, and satisfies te error estimates: u u,0 G t,, δ,. u u 0, G t,, δ,.3 were G t,, δ = C t u ttt 0,0 u tt 0, u tt 0,0 f tt 0,0 C tδ u t 0, u t 0, u t 0,0 f t 0,0 C k u t 0,k C k u 0,k C k u 0,k Cδ u t 0,0 C k u,k. A key step in te analysis of te fractional step θ-metod is te construction of unit stride expressions for te error at successive time steps. Tese are expressions were a bound is known on te difference of error terms at times t apart i.e. E n E n. Assuming te initial error E 0 = 0, te summation of a sequence of tis form telescopes to E l. Coosing an optimal value for θ, it will be seen tat te first order terms in 5
te interpolation error are zero, yielding a second order temporal discretization. Furter discussion of te optimal value of θ can be found in Section.7. Outline of te proof: Te θ-metod provides approximations u n, u n θ, and u nθ. Linear combinations of equations.7,.8, and.9 are formed to generate te appropriate uniform stride differences between successive error terms. Step θ. Form te following linear combinations of equations.7,.8, and.9 to obtain equations involving u n u n θ, respectively. u n, un θ u n θ, and u nθ θ t.7 θ t.8 θ t.9. wit n n wit n n wit n n θ t.7 θ t.8 θ t.9.5 wit n n wit n n wit n n θ t.7 θ t.8 θ t.9.6 wit n n wit n n wit n n Step θ. Subtract equations.,.5, and.6 from.0 evaluated at te midpoint of eac unit stride. Add, subtract and rearrange terms to give u n u n u n θ u n θ u nθ u nθ u n u n, v u n θ u n θ, v u n θ u n θ, v t G pos u n u n, v = t G rem t, f, u, u n θ, u nθ, u n, v,.7 t H pos u n θ u n θ, v = th rem t, f, u, u nθ = t K rem t, f, u, u n, u n, un θ, v,.8 tk pos u nθ u nθ, v, un θ were G pos, H pos, and K pos denote te positive part of te operators., u n θ, v,.9 6
Step 3θ. Use u u = Λ E, coose v = E n, v = E n θ, and v = E nθ in.7,.8, and.9, respectively to obtain E n E n tg pos E n, E n tr t, f, u, Λ n, Λ nθ, Λ n θ, Λ n, E n, E nθ, E n θ..30 E n θ E n θ th pos E n θ, E n θ tr t, f, u, Λ n θ, Λ n, Λ nθ, Λ n θ, E n θ, E n, E nθ..3 E nθ E n θ tk pos E nθ, E nθ tr 3 t, f, u, Λ n θ, Λ n, Λ nθ, Λ nθ, E n θ, E n, E nθ..3 Use E 0 = 0 and sum.30,.3, and.3 from n = 0 to N, n = to N, and n = to N, respectively. Adding te resulting expressions togeter gives: E N E N θ E θ t G pos E n, E n t H pos E n θ, E n θ n= t K pos E nθ, E nθ n= t R t, f, u, Λ n, Λ nθ, Λ n θ, Λ n, E n, E nθ, E n θ t R t, f, u, Λ n θ, Λ n, Λ nθ, Λ n θ, E n θ, E n, E nθ n= t R 3 t, f, u, Λ n θ, Λ n, Λ nθ, Λ nθ, E n θ, E n, E nθ n= E θ E θ..33 Step θ. Apply suitable inequalities/estimates to te terms in.33. 7
Step 5θ. Apply discrete Gronwall s inequality see Lemma.. wit a l = E N E N θ E θ. Te lemma requires tat t C C δ C3 δ C,.3 were C, C, C 3, and C denote constants independent of te discretization parameters t and and te upwinding parameter δ. If δ is cosen suc tat δ C,.3 becomes C t..35 Tis is computationally very restrictive. Tis constraint is not enforced for te numerical results in Section.8. It is an open question if tis condition is necessary for. and.3. Step 6θ. Use te triangle inequality to obtain te error estimate for u N θ u N θ u θ u θ. u N u N Numerical results are presented in Section.8 for a continuous, piecewise linear approximation to u i.e. k =. For tis case,. and.3 give te following estimate. Corollary For X te space of continuous, piecewise linear functions, t C, δ C, and u sufficiently smoot, te approximation u satisfies te error estimate: u u 0, C t tδ δ, and.36 u u,0 C t tδ δ..37.7 Optimal θ Value In [7] Glowinski and Périaux studied te convergence and stability of te θ-metod for du dt Au = 0, 8
were A was assumed to be a constant p p symmetric, positive definite matrix and u IR p. Te decomposition tey considered see. was, for α 0,, Au = αau αau, i.e. F u, t = αau and F u, t = αau. Using an eigenvalue analysis, te autors were able to establis tat for θ = / te fractional step θ-metod was second order accurate in time, independent of te coice of α. An eigenvalue analysis approac is not possible for te approximation metod described in.7.9. In Step θ above, te linear combinations given in..6 give rise to expressions of te following form: In G rem : θu n θ u nθ u n,.38 θ u n θ θu n u n..39 In H rem : θu n θ u nθ u n θ,.0 θ u n θ θu n θu n θ u n θ.. In K rem : θu n θ u n θ u nθ,. θ u nθ θu n θu nθ u nθ..3 Suitable estimates for tese expressions are obtained using Taylor series expansions about n/ t, n/ θ t, and nθ / t for te G rem, H rem, and K rem expressions, respectively. Te first order terms in tese expansions, i.e. te coefficients of t, all reduce to a constant multiple of θ θ.. Te roots of. are θ = ± /. Tus, in order to ave a second order temporal discretization error te only possible coice for θ satisfying 0 < θ < / is θ = /. Verification tat.38 yields a second order accurate temporal discretization is sown in 9
te proof of Lemma.7.. Second order accuracy for te oter five terms.39 -.3 is sown in a manner similar to te proof of Lemma.7. and is given in []. Lemma.7. For te optimal value θ = / t θu n θ u nθ u n C t u tt 0,0. Proof: Expanding u n and u nθ about u n gives and tn u n = u n u t, t dt t n = u n t un t t u nθ = u n n u t, t dt Using.5 and.6 gives t nθ = u n θ t u n t θu n θ u nθ u n = = = Ω Ω { tn u tt, t t n t dt,.5 t n t n t nθ u tt, t t t nθ dt..6 θu n θ u nθ u n } da { θ u n tn t un t u tt, t t n t dt t n θ u n θ t u n t t n u tt, t t t nθ dt u } n da t nθ Ω { θ θ t u n t θ tn u tt, t t n t dt t n t n θ u tt, t } t t nθ dt da t nθ 0
= { θ Ω tn u tt, t t n t dt t n { Ω { Ω θ t n θ tn t nθ u tt, t t t nθ dt } da u tt, t t n t dt t n t θ n u tt, t } t t nθ dt da θ t nθ tn tn u tt, t dt tn t dt t n t n t θ n t u tt, t n dt t tnθ dt t nθ t nθ { = θ t 3 tn u tt, t dt Ω 3 t n θ 3 } t 3 θ n t u tt, t dt da t nθ = t 3 { tn θ u tt, t dt 3 Ω t n } t θ θ 3 n u tt, t dt da 3 = t3 t nθ t 3 { max θ, θ, θ 3} Ω θ tn t nθ u tt, t dt. } { } tn u tt, t dt da t nθ da Tus, t θu n θ u nθ u n n 0 t θ C t u tt 0,0. tn t nθ u tt, t dt
Te optimal θ value was investigated numerically by calculating experimental convergence rates for te convection-diffusion problem given in..3 for b = [, ] T, c =.0, Ω = 0, 0,, X te space of continuous piecewise linear functions, and f and u 0 determined by te true solution ux, y, t = 0xy x ye x.5 t..7 Te meses used for tese calculations were obtained by dividing te spatial and temporal t discretization parameters on eac successive mes by. As te spatial discretization sceme is second order, it is expected tat te experimental convergence rate is determined by te temporal discretization. Figure. indicates tat wen θ = / te metod as second order convergence wit respect to bot and t. Figure. displays te error u u 0,0 at T = on a mes wit t = /8 and = /30 for different values of θ. Te minimum error corresponds to θ = /...0 3 x 0.0 Convergence Rate at time T =.9.8.7.6 θ =.0 θ =.5 θ =.9 θ = Optimal θ =.30 θ =.35 θ =.0.98.96.9 80 00 0 zoom u u 0,0 at T =.5 50 00 50 00 50 /.5 0. 0.5 0. 0.5 0.3 0.35 0. 0.5 0.5 θ Value Figure. Experimental convergence rates. Figure. Error u u 0,0 as a function of θ.
.8 Numerical Results Tree examples are used to verify te teoretical convergence rates and demonstrate te effectiveness of te fractional step θ-metod.7.9 for convection diffusion problems. Example is a simple convection-diffusion problem wit a constant velocity field and a constant absorption coefficient. In Example, te diffusion coefficient is several orders of magnitude less tan te magnitude of te velocity field. Te solution in Example 3 represents a steep moving front propagating troug te domain. Te optimal value of θ = / was used in all computations. In eac example a sequence of continuous, piecewise linear approximations u were computed, by dividing te time step t and te spatial mes parameter by two. Te experimental convergence rates are computed for various coices of δ. From Teorem.6., te predicted convergence rates are u u 0, O δ and u u,0 O δ. Example. Tis test problem was described in Section.7 wen te optimal value of θ was examined, and is a modified version of a steady state problem given in [37].Te true solution is given in.7. Te solution is a sligtly skewed bubble function wic decays to zero as t. Te numerical results for tis example are presented in Table.. Example. In tis example, taken from [6], te approximation of ux, y, t satisfies u t k u b u = f in Ω 0, T ],.8 for k = 0.000, b = [ y, x] T, and Ω = 0.5, 0.5 0.5, 0.5. Te true solution is given by σ ux, y, t = σ kt exp x 0.5 ȳ σ,.9 kt were x = x cost y sint, ȳ = x sint y cost and σ = 0.077. Te initial and boundary conditions are given by u 0 x, y = ux, y, 0, and ux, y, t Ω = ux, y, t 0. Te solution represents a Gaussian pulse being convected by a rotating velocity field. Table. lists te errors in te numerical approximation and te experimental convergence rates. Te approximation is illustrated in Figures.3a.3b, for = /6 and δ =. 3
Table. Approximation errors and convergence rates for Example. θ = / Time T =.0 δ t, 0, 8 0, 6 0, 3 80, 6 60, 8 0 u u 0,.09e-.8e-.7e- 5.68e-.83e- Cvge. Rate - 0.9.0.0.0 u u,0.039e- 5.358e-3.359e-3 3.e- 8.537e-5 Cvge. Rate -.9.0.0.0 u u 0, 5.07e- 3.06e-.576e- 7.757e- 3.77e- Cvge. Rate - 0.8.0.0.0 u u,0 7.055e- 3.75e-.5e- 7.78e-3 3.3e-3 Cvge. Rate -.... 3/ u u 0,.337e-.59e-.35e- 5.668e-.83e- Cvge. Rate - 0.9.0.0.0 u u,0 3.78e-.76e- 3.63e-3.38e-3 3.69e- Cvge. Rate -.7.7.7.6 u u 0,.e-.88e-.7e- 5.69e-.83e- Cvge. Rate - 0.9.0.0.0 u u,0.63e- 6.793e-3.709e-3.56e-.058e- Cvge. Rate -.9.0.0.0 0.8 0.6 0. 0. 0 0.8 0.6 0. 0. 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. -0. 0 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. 0 0.6 0.6 0. 0. -0.6-0. -0. 0 0. 0. 0.6-0.6-0. -0. 0 0. -0.6-0. -0. 0 0. 0. 0.6-0.6-0. -0. 0 0. a u at t = 0 b u at t = 0.3 Figure.3 Plots of te true and approximate solution for convection-diffusion Example.
Table. Approximation errors and convergence rates for Example. θ = / Time T = 0.3 δ t, 0, 8 0, 6 0, 3 80, 6 60, 8 0 u u 0,.e-0 7.999e- 3.39e-.53e- 7.60e- Cvge. Rate - 0.6...0 u u,0 8.e-.07e-.7e-.57e-3 6.338e- Cvge. Rate -.0.9..0 u u 0, 9.5e- 7.978e- 5.93e-.079e-.637e- Cvge. Rate - 0. 0. 0.5 0.6 u u,0 6.609e- 5.75e-.353e- 3.0e-.057e- Cvge. Rate - 0. 0. 0.5 0.6 3/ u u 0, 9.899e- 7.375e- 3.9e-.797e- 8.356e- Cvge. Rate - 0. 0.9.. u u,0 6.69e-.58e-.35e- 8.68e-3 3.003e-3 Cvge. Rate - 0.5...5 u u 0,.076e-0 7.59e- 3.8e-.553e- 7.66e- Cvge. Rate - 0.5...0 u u,0 7.0e-.00e-.8e- 3.e-3 7.70e- Cvge. Rate - 0.8.7.0.0 Example 3. Tis example problem of approximates te solution to a steep front moving troug a domain [7]. Set k = 0.0, f = 0, ten ux, y, t = wx, t wy, t, were wη, t = 0.A 0.5B C A B C, Aη, t = exp 0.05η 0.5.95t 0.3/k, Bη, t = exp 0.5η 0.5 0.75t 0.3/k, Cη = exp 0.5η 0.375/k, and b = [wx, t, wy, t] T. Te boundary and initial conditions are determined by te true solution. 5
Tis example does not satisfy te assumptions for Teorem.6., as b 0. Noneteless, te numerical results presented in Table.3 are consistent wit tose predicted in. and.3. Te numerical approximation using = /3 and δ = 3/ is displayed in Figures.a.b. Table.3 Approximation errors and convergence rates for Example 3. θ = / Time T = 0.3 δ t, 0, 8 0, 6 0, 3 80, 6 60, 8 0 u u 0, 5.93e-.7e-.30e- 6.389e- 3.63e- Cvge. Rate - 0.9..0.0 u u,0.03e-.663e- 6.805e-3 3.e-3.576e-3 Cvge. Rate -.3.3..0 u u 0,.775e-.999e-.68e- 9.099e-.76e- Cvge. Rate - 0.7 0.8 0.9 0.9 u u,0.693e-.89e-.580e- 8.6e-3.56e-3 Cvge. Rate - 0.7 0.8 0.9 0.9 3/ u u 0,.659e-.586e-.67e- 6.63e- 3.3e- Cvge. Rate - 0.8.0.0.0 u u,0.05e-.730e- 6.900e-3 3.e-3.503e-3 Cvge. Rate -..3.. u u 0,.867e-.656e-.89e- 6.36e- 3.56e- Cvge. Rate - 0.9.0.0.0 u u,0 3.975e-.6e- 6.70e-3 3.75e-3.565e-3 Cvge. Rate -.3.3..0 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. 0 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. 0 0.9 0.8 0.7 0.6 0.5 0. 0.3 0. 0. 0.8 0.8 0 0. 0. 0.6 0.8 0 0. 0. 0.6 0 0. 0. 0.6 0.8 0 0. 0. 0.6 a u at t = 0 b u at t = 0.3 Figure. Plots of te true and approximate solution for convection-diffusion Example 3. 6
CHAPTER 3 VISCOELASTIC FLUID FLOW 3. Introduction In tis capter a fractional step θ-metod approximation for an inertialess or slowflow model of te viscoelastic governing equations given by.-.6 in Capter is formulated. At eac time increment te θ-metod decouples te velocity-pressure and stress updates. Additionally te systems for te velocity-pressure update and te stress update are linear. Motivated by tis decoupling, te analysis of te metod is approaced similarly. Analysis of a Stokes system for te velocity and pressure is examined first, assuming te stress is known. Te yperbolic constitutive model is analyzed next, assuming te velocity and pressure are known. Te estimates obtained wen analyzing eac of te decoupled parts are ten used to establis a priori error estimates for te full implementation of te θ-metod. Te remainder of tis capter is organized as follows. In Section 3. te θ-metod for te viscoelastic modeling equations is formulated. Section 3.3 contains te matematical notation and variational formulations used in te analysis. Computability of te algoritm is sown in Section 3.. A priori error estimates for te θ-metod are stated and proved in Section 3.5. Numerical examples demonstrating te θ-metod applied to viscoelastic fluid flow are presented in Capter. 3. θ-metod for Viscoelastic Fluid Flow Te non-dimensional modeling equations for inertialess i.e. u u 0 in., and is ignored viscoelastic fluid flow using a Jonson-Segalman constitutive equation are: 7
σ σ t u σ g aσ, u αdu = 0 in Ω, 3. Re u p α du σ t = f in Ω, 3. u = 0 in Ω, 3.3 u = 0 on Ω, 3. u0, x = u 0 x in Ω, 3.5 σ0, x = σ 0 x in Ω. 3.6 Te fractional step θ-metod requires an additive decomposition of equations 3. and 3.. Use te splitting parameters ω and γ 0,, and define: Constitutive equation: Gσ := ωσ, 3.7 Gσ := ωσ u σ g a σ, u αdu. 3.8 Conservation of momentum: Fu := γ α du σ f, 3.9 Fu := γ α du. 3.0 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 [, 55, 58]. θ-metod algoritm for viscoelasticity Step a: Update te stress. σnθ σ n θ t Gσ nθ = Gσ n. 8
Step b: Solve for velocity and pressure. Re unθ u n θ t p nθ Fu nθ = Fu n, u nθ = 0. Step a: Solve for velocity and pressure. Re un θ u nθ θ t p n θ Fu n θ = Fu nθ, u n θ = 0. Step b: Solve for te stress. σn θ σ nθ θ t Gσ n θ = Gσ nθ. 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. 3.3 Matematical Notation 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 > 0. 3. q Q v X q v 9
A variational formulation of 3.-3.3, found by multiplication of te modeling equations by test functions and integrating over Ω, is: Given u 0 X and σ 0 S find u, σ, p : 0, T ] X S Q suc tat σ t, τ σ, τ α du, τ 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. 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 3.-3. is given by: Given u 0 Z and σ 0 S find u, σ : 0, T ] Z S suc tat σ t, τ σ, τ α du, τ 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 Recall te finite element framework introduced in Capter were T is a triangulation of te discretized domain Ω R d suc tat Ω = K, K T. As P k A denotes 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 30
associated finite element spaces are defined by: X := S := {v X C Ω d : v } K P k K K T, {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 > 0. 3. v X q v Te analysis for te θ-metod is accomplised in tree parts: te analysis of Stokeslike 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 establises te full a priori error estimates for te θ-metod. 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, Γ n = σ n S n, E n = U n ũ n, F n = S n σ n, e n u = un ũ n, en σ = σn σ n. Te following properties of Sobolev and finite element spaces are needed in te analysis. From [6] for u, p H k Ω d H q Ω tere exists U, P Z Q suc 3
tat u U C k u W k, 3. u U W C k u W k, 3.3 p P C q p W q. 3. A skew symmetric formulation is used wen bounding some of te convective terms [9]. Define cu, σ, τ := u σ, τ, 3.5 cu, σ, τ := cu, σ, τ cu, τ, σ, 3.6 and note tat: cu, σ, τ = cu, σ, τ wen u = 0 in Ω, and u = 0 on Ω, cu, σ, σ = 0. 3. Unique Solvability of te Sceme Before deriving te error estimates computability of te algoritm is establised. 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 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 δ µ were τ δ µ := τ δu µ τ, 3.7 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. 3
Step a: Find σ nθ S suc tat θ t σ nθ, τ ω = θ t Step b: Find u nθ Re θ t u nθ, v γ α Step a: Find u n θ Re u n θ θ t, v γ α Step b: Find σ n θ θ t σ n θ Step 3a: Find σ n θ t σ n, τ σ nθ, τ n δ σ n, τ ω g a σ n σ n, un Z suc tat γ α du nθ du n, dv Z suc tat, τ δ n, τ δ n α, dv = Re θ t f nθ, v u n γ α du n θ, dv du nθ, dv S suc tat, τ g a σ n θ = ω u n θ ω f nθ, v σ n θ, τ n θ δ, u n θ θ t S suc tat σ n, τ n θ δ, τ δ n θ σ nθ σ n θ, τ n θ δ, τ = θ t α ω α du n σ n, τ δ n u n, v σ nθ, τ δ n, τ S. 3.8, dv, v Z. 3.9 Re = u nθ θ t, v, dv, v Z. 3.30 σ nθ u n θ du n θ σ n θ σ nθ, τ n θ δ σ n θ, τ g a σ n θ ω, τ n θ δ, τ n θ δ, τ S. 3.3 σ n θ, τ n θ δ, u n θ, τ n θ δ, τ S. 3.3 du n θ, τ n θ δ 33
Step 3b: Find u n Z suc tat Re θ t u n, v γ α γ α, dv = Re θ t f n, v du n du n θ, dv u n θ, v σ n, dv, v Z. 3.33 Te following induction ypotesis is used wen proving te lemmas tat establis te existence and uniqueness of te solutions to 3.8-3.33. Induction Hypotesis Under te assumptions of Teorem 3.6. tere exists a constant K suc tat for n =,..., N u n, u nθ, and un θ K. 3.3 Te justification of Induction Hypotesis is establised in Section 3.7. Lemma 3.. Step a Assume Induction Hypotesis is true. For δ C and t sufficiently small tere exists a unique solution σ nθ S satisfying 3.8. Proof: Equation 3.8 can be written as A σ nθ, τ = σ n θ t, τ u n σ n g a σ n, un ωσ n, τ δ n, τ δ n α du n, τ δ n, τ S, 3.35 were A σ nθ, τ := θ t σ nθ, τ ω σ nθ, τ δ n. Here 3.35 represents a square linear system of equations Ax = b. Wit te coice τ = σ nθ, te individual terms in A are θ t ω σ nθ σ nθ, σ nθ, σ nθ = θ t = ω σ nθ σ nθ,, 3
and ωδ σ nθ, u n σ nθ ωδ d C ωδk d σ nθ. u n C σ nθ Provided δ C and t /θωk d C, ten A σ nθ, σ nθ > 0. Tus, ker A = {0}. It follows tat 3.8 as a unique solution. Lemma 3.. Step b Tere exists a unique solution u nθ Z satisfying 3.9. Proof: Equation 3.9 can be written as A u nθ, v = Re u n θ t f nθ, v, v γ α du n, dv, dv, v Z, σ nθ were A u nθ, v Note tat coosing v = u nθ A u nθ, u nθ = Re θ t := Re θ t u nθ u nθ, v, u nθ γ α γ α du nθ du nθ, dv., du nθ > 0. Tus, kera = {0}, and existence and uniqueness of a solution to 3.9 as been sown. Lemma 3..3 Step a Tere exists a unique solution u n θ Z satisfying 3.30. Proof: Write equation 3.30 as A 3 u n θ, v = Re θ t u nθ f nθ, v, v γ α σ nθ, dv, du nθ, dv were A 3 u n θ, v Re := u n θ θ t, v γ α du n θ, dv. 35
For v = u n θ, A3 u n θ 3.30 as a unique solution., u n θ > 0. Tus, kera 3 = {0}, and it ten follows tat Lemma 3.. Step b Assume Induction Hypotesis is true. For δ C and t sufficiently small tere exists a unique solution σ n θ S satisfying 3.3. Proof: Write 3.3 as wit A σ n θ, τ = θ t ω σ nθ, τ σ nθ, τ n θ δ α du n θ, τ n θ δ, 3.36 A σ n θ, τ := θ t u n θ Bounding te terms in A σ n θ σ n θ, τ σ n θ, τ n θ δ, σ n θ σ n θ θ t, σ n θ ω σ n θ, σ n θ ω σ n θ, δu n θ σ n θ u n θ u n θ σ n θ, σ n θ ω yields = θ t = ω σ n θ, δu n θ σ n θ σ n θ, τ n θ δ g a σ n θ σn θ σn θ, u n θ, ω d δc K σ n θ d C K σ n θ = δ un θ,, σ n θ, τ δ n θ,,. 36
g a σ n θ, u n θ, σ n θ σn θ u n θ σn θ d u n θ σ n θ d C un θ σ n θ d C K σ n θ, Tus, g a σ n θ σ n θ, u n θ, δu n θ σn θ u n θ δun θ σ n θ d u n θ σ n θ δun θ σ n θ dc K δ ɛ σn θ ɛ δ un θ σ n θ A σ n θ, σ n θ θ t ω δ ω d C K dc δ K Coosing ɛ ɛ δ ɛ un θ. σn θ σ n θ =, δ C, and t C establises A σ n θ, σ n θ > 0. Hence kera = {0}, implying tat a unique solution exists for 3.3. Te unique solvability of 3.3 and 3.33, representing te tird step in te algoritm, follows exactly as 3.8 and 3.9.. 3.5 A Priori Error Estimates for te Conservation Equations and Constitutive Equation A priori error estimates are formulated first for a θ-metod for te Stokes-like problem given by 3.9, 3.30, and 3.33, assuming te stress is known. Ten a priori error estimates for a θ-metod for te constitutive equation given by 3.8, 3.3, and 3.3 37
are found assuming te velocity and pressure are known. Tese estimates are given in Teorems 3.5. and 3.5.. 3.5. Analysis of te conservation equations Teorem 3.5. 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: u ũ,0 Fu t,, 3.37 were u ũ 0, Fu t,, 3.38 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. 3.39 Proof: For notational simplicity in tis proof te tilde is dropped from ũ and p, te discrete approximations of velocity and pressure using te true stress. Following te analysis of te θ-metod for te convection-diffusion equation, linear combinations are formed to obtain expressions for u n u n, u n θ u n θ, and u nθ u nθ. Linear combination : Consider te linear combination θ3.9 θ 3.30 θ3.33. 3.0 wit n n wit n n wit n n 38