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1 A Fractional Step θ-metod for Convection-Diffusion Equations Jon Crispell December, 006 Advisors: Dr. Lea Jenkins and Dr. Vincent Ervin

2 Fractional Step θ-metod Outline Crispell,Ervin,Jenkins Motivation Convection-Diffusion Problem Definition Te Fractional Step θ-metod Specialized Solution Tecniques Error Estimate Idea of Proof Numerical Results Summary December, 006 slide

3 Fractional Step θ-metod Motivation Crispell,Ervin,Jenkins Te Time Dependent Jonson-Segalman Model for Viscoelastic fluid flow: ) σ σ + λ t + u σ + g a σ, u) αdu) = 0 in Ω ) u Re t + u u + p α) Du) σ = f in Ω u = 0, in Ω u = 0, on Ω σ = σ Ω on Ω ux, 0) = u 0 in Ω σx, 0) = σ 0 in Ω were g a σ, u) = a σ u + u T σ ) + a Du) = ) u + u T σ u + σ u T ) December, 006 slide 3

4 Problem Definition A Starting Point Crispell,Ervin,Jenkins Te Time Dependent Convection Diffusion Problem u t u + b u + cu = f in Ω 0, T ] subject to ux, t) = 0, x Ω 0, T ) ux, 0) = u 0 x), x Ω December, 006 slide 4

5 θ-metod Te Fractional Step Metod Crispell,Ervin,Jenkins Writing in abstract form: u + F u, x, t) = 0 in Ω 0, T ] t subject to ux, t) = 0 x Ω 0, T ) ux, 0) = u 0 x) x Ω Additively split F : were F u, x, t) = F u, x, t) + F u, x, t) F u, x, t) = u + c u f F u, x, t) = b u + c u December, 006 slide 5

6 θ-metod Te Fractional Step Metod Crispell,Ervin,Jenkins Coose a value of θ 0, /). Step. Compute an approximation to u n+θ). u n+θ) θ t u n) + F n+θ) = F n) t n+) θ t t n+ θ) Step. Compute an approximation to u n+ θ)). θ) t u n+ θ) u n+θ) θ) t + F n+ θ) = F n+θ) t n+θ) θ t t n Step 3. Compute an approximation to u n+). u n+) u n+ θ) θ t + F n+) = F n+ θ) December, 006 slide 6

7 θ-metod Special Solution Tecniques Crispell,Ervin,Jenkins Streamline Upwinding: Variation Formulation Step : u n+ θ) u n+θ) θ) t = Test against v + δb v instead of v) ), v + f n+θ) c un+θ) b u n+ θ) ), v + δb v ), v + δb v + u n+θ) ), v + c ) un+ θ), v + δb v u n+θ) ), δb v were f, g) = Ω fg da December, 006 slide 7

8 Error Estimates Optimal θ Value Crispell,Ervin,Jenkins Second order in t Taylor series during analysis: Te first order terms in te expansions te coefficients of t) all reduce to a multiple of: and tis as roots of θ = ± /. θ 4θ +, Optimal coice: For θ in 0, /) te error is O t) ). θ = December, 006 slide 8

9 Error Estimates Optimal θ Value Crispell,Ervin,Jenkins Consider te numerical solution to te convection diffusion problem sown: Figure : ux, t) = 0xy x) y)e x4.5 ) t 4) at t = 0 wit te conditions tat b = [ ] T, c =, k =, δ = 0, and f set accordingly. December, 006 slide 9

10 Error Estimates θ in te Fractional step metod Crispell,Ervin,Jenkins Convergence Rate at time T = θ =.0 θ =.5 θ =.9 θ = Optimal θ =.30 θ =.35 θ = zoom / Figure : Experimental Convergence Rates. December, 006 slide 0

11 Error Estimates θ in te Fractional step metod Crispell,Ervin,Jenkins 3 x 0 4 u u 0,0 at T = θ Value Figure 3: Error u u 0,0 as a function of θ. December, 006 slide

12 Error Estimates Some Definitions Crispell,Ervin,Jenkins Defining te following norms: v,k := sup 0<t<T T v 0,k := 0 v, t) k v,k := max <n<n vn k v, t) k dt ) / v 0,k := N t v n k n= ) / December, 006 slide

13 Teorem Error Estimate Crispell,Ervin,Jenkins Teorem. For a sufficiently smoot solution u, and t C, te fractional step θ-sceme approximation, u for convection-diffusion, converges to te true solution u on te interval 0, T ] as t, 0, and satisfies te error estimates: were u u,0 G t,, δ) + C k+ u,k+ ) u u 0, G t,, δ) + C k u,k+ ) ) 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 Using Linears: u u 0, = O t, tδ,, δ) Move to Results December, 006 slide 3

14 Teorem Outline of Analysis Crispell Teorem is establised in te following steps:. Use linear combinations of variational formulations to set up telescoping summation.. Bound all resulting terms. 3. Bring bounded terms togeter in a single expression for te error. 4. Obtain a stability result necessary for te application of te Gronwall s lemma. 5. Apply interpolation properties and lemmas from te appendix of tis document to establis Teorem. December, 006 slide 4

15 Teorem Telescoping Sum Crispell From te linear combinations in proof of Teorem : u n+θ) u n) ) θ, v θ t u n+ θ) u n+θ) ) θ), v θ) t u n+ θ) u n+θ) ) θ, v θ t Summing 3),4), and 5): ) + θa u n+θ), u n), v ) + θ) A u n+ θ), u n+θ), v ) + θa 3 u n+), u n+ θ), v 3) 4) 5) u n+) u n) t ) ), v + A u n+), u n+ θ), u n+θ), u n), v December, 006 slide 5

16 Teorem Stability Result Crispell To apply te discrete Gronwall s Lemma: t 7 + c max + θ) c max + θc max + θc ɛ 4 + θ) δ d b C 4 ɛ 4 + 5θc max + θ) ɛ 3 + θ) δ d b C ɛ 5 ) = t C + C + Cδ 4 ) + Cδ Tis gives te condition: t C December, 006 slide 6

17 Fractional Step θ-metod Outline Crispell,Ervin,Jenkins Motivation Convection-Diffusion Problem Definition Te Fractional Step θ-metod Specialized Solution Tecniques Error Estimate Idea of Proof Numerical Results Summary December, 006 slide 7

18 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins Example: Here True solution: u t k u + b u + cu = f spatial domain: Ω = /, /) /, /) temporal domain: convective field: t 0, π/] b = 4y, 4x) T c = 0, and f = 0. u 0 x, y) = exp x x c) + y y c ) ) σ k = 0.000, x c = /4, y c = 0, σ = ux, y, t) = σ σ + 4kt exp x x c) + ȳ y c ) ) σ + 4kt x = x cos4t) + y sin4t), and ȳ = x sin4t) + y cos4t) December, 006 slide 8

19 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins "Mes64Time sol" Animation of Rotating Gaussian pulse. December, 006 slide 9

20 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins Convergence Rate Study Table : Using k = at T = 0.3 δ t, ) u u 0,0 Rate u u 0, Rate u u,0 Rate 0 0, ) 8 4.3e- -.4e-0-8.4e- - 0, ) 6.785e e e-.0 40, ) e e-..7e-.9 80, ) 64.08e e-..57e-3. ).503e e e , 8 Teorem using linear elements: u u 0, = O t, tδ,, δ) Back to Teorem December, 006 slide 0

21 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins Convergence Rate Study Table : Using k = at T = 0.3 δ t, ) u u 0,0 Rate u u 0, Rate u u,0 Rate 0, ) e e e- - 0, ) 6.675e e e , ) 3.8e e e , ) 64.93e e e- 0.5 ) 7.394e e e , 8 Teorem using linear elements: u u 0, = O t, tδ,, δ) Back to Teorem December, 006 slide

22 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins Convergence Rate Study Table 3: Using k = at T = 0.3 δ t, ) u u 0,0 Rate u u 0, Rate u u,0 Rate 0, ) e e e- - 0, ) 6.057e e e , ) e e e-. 80, ) e e e-3.4 ).08e e e , 8 Teorem using linear elements: u u 0, = O t, tδ,, δ) Back to Teorem December, 006 slide

23 Example Problems Rotating Gaussian Pulse Crispell,Ervin,Jenkins Convergence Rate Study Table 4: Using k = at T = 0.3 δ t, ) u u 0,0 Rate u u 0, Rate u u,0 Rate 0, ) e e-0-7.0e- - 0, ) 6.83e e e , ) e e-..8e-.7 80, ) 64.e e-. 3.e-3.0 ).978e e e , 8 Teorem using linear elements: u u 0, = O t, tδ,, δ) Back to Teorem December, 006 slide 3

24 θ-metod Summary Crispell,Ervin,Jenkins Fractional Step θ-metod Less oscillations in solutions tan Crank-Nicolson. For appropriate coices of θ, and δ second order temporal convergence is acieved. Allows for decoupling of operators: Convection from Diffusion Stress from Pressure/Velocity. Linear from Nonlinear Results in: smaller systems to solve application of specialized solution tecniques December, 006 slide 4

25 [],[3],[0],[7],[8],[5],[9],[],[4], [6] Crispell,Ervin,Jenkins *References [] Susanne C. Brenner and L. Ridgway Scott. Te Matematical Teory of Finite Element Metods. Springer-Verlag, New York, 999. [] J.C. Crispell, V.J. Ervin, and E.W. Jenkins. A fractional step θ-metod for convection-diffusion using a SUPG approximation. Tecnical Report TR006 CEJ, Clemson University, 006. [3] Vincent J. Ervin and Norbert Heuer. Approximation of time-dependent, viscoelastic fluid flow: Crank- Nicolson, finite element approximation. Numer. Met. Part. Diff. Eq., 0:48 83, 003. [4] Vincent J. Ervin and William W. Miles. Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal., 4: , 003. [5] R. Glowinski and J. Periaux. Numerical metods for nonlinear problems in fluid dynamics. Proc. Intern. Seminar on Scientific Supercomputers, Paris, Feb. -6:Nort Holland, 987. [6] C. Jonson. Numerical Solutions of Partial Differential Equations by te Finite Element Metod. Cambridge University Press, New York, NY, 987. [7] P. Saramito. A new θ-sceme algoritm and incompressible FEM for viscoelastic fluid flows. Mat. Model. Num. Anal., 8: 35, 994. [8] 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., 8:5704, 999. December, 006 slide 5

26 [],[3],[0],[7],[8],[5],[9],[],[4], [6] Crispell,Ervin,Jenkins [9] S. Turek. Efficient Solvers for Incompressible Flow Problems: An Algoritmic and Computational Approac, volume 34 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 004. [0] Hong Wang, Helge K. Dale, Ricard E. Ewing, Magne S. Espedal, Robert C. Sarpley, and Susuang Man. An ELLAM sceme for advection-diffusion equations in two dimensions. SIAM J. Sci. Comput., 0:60 94, 999. December, 006 slide 6