FUNDAMENTALS OF FINITE DIFFERENCE METHODS

Transcription

1 FUNDAMENTALS OF FINITE DIFFERENCE METHODS By Deep Gupta 3 rd Year undergraduate, Mechanical Engg. Deptt., IIT Bombay Supervised by: Prof. Gautam Biswas, IIT Kanpur

2 Acknowledgements It has been a pleasure preparing this lecture. I take this opportunity to thank my supervisor Prof. Gautam Biswas (Professor,IIT Kanpur) for his enthusiastic support. A major chunk of this lecture is inspired by his lecture notes. Excerpts from Prof. Atul Sharma's (Asstt. Professor, IIT Bombay) lectures have been included with his kind permission and profuse thanks to him for the same. Last but not the least invaluable reference from J.W. Anderson's "Computational Fluid Dynamics" was indispensable in the completion of this lecture.

3 Contents The presentation discusses: Partial Differential eqns: classification and equations involved in fluid mechanics with boundary/initial conditions. Concept of discretization. Finite Difference Quotients, starting from Taylor s series. Finite Difference eqns.: concept, basic aspects like consistency and convergence. Explicit and Implicit methods of discretization: a comparison. Errors and stability analysis, including analysis of parabolic and hyperbolic equations. Fundamentals of Fluid Flow Modeling: Transportive, Conservative properties and Artificial Viscosity. Upwind Scheme of Discretization (First and Second). 3

4 Partial Differential Equations (Classification) Consider a second order differential eqn., φ φ φ φ φ φ =, Linear & non-linear eqns. Quasi-linear eqns. Homogeneous or non-homogeneous. Can be further classified as: B AC = parabolic 4 0, B 4 AC < 0, elliptic B 4 AC > 0, hyperbolic ( ) A B C D E F G x y x x y y x y 4

5 Partial Differential Equations (Examples) Unsteady N-S eqns are elliptic in space and parabolic in time. Laplace and Poisson eqns are elliptical in nature. φ φ 0, forlaplace' s + = x y Constt., forpoisson' s Heat Conduction eqn.(-d) is parabolic, φ t = B φ x 5

6 Partial Differential Equations (examples) Fluid flow problems have non-linear terms advection (for momentum eqn.) or convection (for energy eqn.). φ φ φ φ φ + u + v = B + S + t x y x y Parabolic in time and elliptical in space, turns hyperbolic for high speed flows. 6

7 Boundary/Initial Conditions No. of B.C.s=order of highest derivatives in differential equations. Unsteady problems require initial conditions. Spatial B.C.s in flow/ heat transfer problems (types): Dirchlet:- Φ=Φ (r) A Neumann:- Φ n=φ (r) A Mixed:- a(r)φ+b(r) Φ n=φ 3 (r) A 3 7

8 Discretization A closed form mathematical expression, such as a partial differential equation, is approximated by analogous expressions which prescribe values at only a finite no. of discrete points or volumes in domain. Analytical solution: continuous. Numerical solution: Discrete. Discretization Techniques Finite Difference Finite Volume Finite Element 8

9 Finite Difference Method (FDM) :Grid Generation x and y need not be uniform, only for simplifying the programming. Uniform spacing: Transformed Computational plane Non-Uniform spacing: Physical plane. 9

10 FDM: Finite Difference Quotient Taylor Series Expansion Example: f(x)=sin x x=0.; x=0.0; f(x+ x)=0.983; f(x)=0.95; π f =f =0.95(3.76%error f 3 = (0.775% error) f 4 =0.984(0.0% error) 0

11 FDM:Finite Difference Quotient (Cont.) Taylor Series expansion I Order Forward Difference

12 FDM:Finite Difference Quotient (Cont.)

13 FDM:Finite Difference Quotient (Cont.) 3

14 FDM:Finite Difference Quotient (Cont.) II Order Central Difference for mixed Derivative xy φ + i+, j+ i, j i+, j i, j+ ij, φ φ φ φ ( ) ( ) = +Ο x, y 4 xy Δ Δ ΔΔ Higher-order-accurate difference equations Disadvantages: Require more grid points, thus require more computer time for each time wise or spatial step. Advantages: Require a smaller no. of total grid points in a flow solution to obtain overall accuracy. 4

15 FDM: Polynomial approach One Sided Finite Differences, other side points not known. Not restricted to be used at boundaries only. u y For example, assume, u = a+ by+ cy After subsequent solving, get u 3u + 4u u = +Ο Δ y Δy Similarly, for third order accuracy, ij, ( y) 3 u + 8u 9u + u ij, ij, + ij, + ij, + 3 = +Ο Δ 6Δy ( y) 3 5

16 Finite Difference Equations Concept: All partial derivatives finite-difference quotients. Consider unsteady,-d heat conduction: T = α t T x Time t is a marching variable. Using FTCS (Forward Time Central Space ) method of discretization: T T α = 0= t x T PartialDifferentialEq. n+ i DifferenceEq. ( n n n T ) i+ Ti + Ti n T α i TE.. + Δt Δx ( ) 6

17 Finite Difference Equations: Consistency and Convergence Consistency: As the mesh is refined, lim PDE FDE = lim ( TE) = 0 ( ) mesh 0 mesh 0 Depends on Differencing Scheme, for eg., DuFort-Frankel scheme is not consistent for D unsteady state heat conduction eq. Convergence: Approximate sol. approaches the exact one for each value of the independent variable as grid spacing tends to zero, i.e., n u = u x, t asδx, Δt 0 ( ) i i n 7

18 Explicit and Implicit Methods Explicit method: Consider -D unsteady state heat conduction equation, T = α t F.D.E. is (FTCS): T n+ i n Ti = Δt α T x ( n n n T ) i+ Ti + Ti Δx ( ) T n i The eq. can be written as: n n n α Δt T n i+ Ti + T + i = Ti + ( Δx) Being Parabolic,this eq. lends itself to a marching solution. ( )( ) 8

19 Explicit and Implicit Methods (Cont.) Implicit Method: Considering the same -D heat conduction eq., but with different discretization, This method is known as Crank-Nicolson implicit scheme T n+ i ( n n n n n n T ) i+ + Ti + Ti Ti + Ti + Ti n T α i = Δt Δx ( ) Obtained by expressing the spatial differences in terms of averages between n & n+ time levels. 9

20 Explicit and Implicit methods (Cont.) Implicit Method (cont.): The eq. can be written as, n n r ( α Δt + n+ n n+ n n+ n T ) where r= i Ti = Ti+ + Ti+ Ti Ti + Ti + Ti Δx or, ( ) ( ) n+ + r n+ n+ n r n n Ti + Ti Ti+ = Ti + Ti + rti+ r r applied at all grid pts.,i.e., from i= to i=k+ to get the tridiagonal matrix,which can be solved. n+ n B() 0 0 K 0 T ( C() + A) n n B() K T3 C() n+ n 0 B(3) K 0 T4 = C(3) M M M M M M M n+ n K Bk ( ) T k ( Ck ( ) + D) where, A & D= Boundary conditions C(k)=R.H.S. of the eq. at i=k+ 0

21 Explicit and Implicit Methods (Cont.) Implicit Method for -D conduction: T T T P.D.E.= = α + t x y Applying Crank-NicolsonScheme, F.D.E.= n+ n Ti, j Ti, j α ( δ )( n n ) x δ + = + y Ti, j + Ti, j Δt where, The system is not n n n T, n i+ j T i, j + T i, j tridiagonal (5 unknowns) δ x T i, j = Δ x δ ( ) T T + T n n n n i, j + i, j i, j y T i, j = ( Δ y ) T, T, T, T & T n+ n+ n+ n+ n+ ij, i+, j i, j ij, + ij,

22 Explicit and Implicit Methods (Cont.)

23 Explicit and Implicit Methods (Cont.) 3

24 Explicit and Implicit Methods: Comparison Implicit Method Advantage: Fewer time steps for calculations over a time interval. Disadvantage: - Complicated to set up the program. -Computer time required each step is much high. -For larger Δt, truncation error is high. Explicit method Advantage: Simple solution algorithm. Disadvantage: Requires many time steps to carry out the calculations over a given time interval, due to restrictions on Δt imposed by stability constraints. 4

25 Explicit and Implicit methods: The Choice Time marching broadly has purposes: To obtain a steady state solution by calculating the flow in steps of time, until a final steady-state flow is approached. No need for timewise accuracy approach the final steady state IMPLICIT Preferred. To obtain an accurate timewise solution of an inherently unsteady flow, Timewise accuracy is absolutely necessary less Truncation Error EXPLICIT Preferred. 5

26 Errors and Stability Analysis Errors: Given, A=analytical sol of P.D.E. D=exact sol. of F.D.E. N=numerical sol. from a real computer with finite accuracy. Then, Discretization Error=A-D =Truncation Error+error introduced due to treatment of boundary condition. Round-off Error=ε=N-D Numerical error introduced for a repetitive no. of calculations. or, N=D+ε 6

27 Error and Stability Analysis (Cont.) Considering the F.D.E. of -D heat conduction, Numerical sol. N must satisfy the above eq., thus, But D is the exact sol., therefore, Subtracting the above two,ε also satisfies the F.D.E. 7

28 Errors and Stability Analysis (Cont.) As solution progresses from step n to n+,it is: Stable: If the ε i s shrink, or remain the same. Unstable: If the ε i s grow larger. Condition for stability: 8

29 Error and Stability Analysis (Contd.) εi s are of the form: This random variation of εi s can be expressed as a Fourier series: Sequential Addition of sine and cosine functions with sequentially increasing wave no. where k m is wave no 9

30 Error and Stability Analysis (Cont.) Wave Number, k m : Consider the sine function, m is related with the no. of wave fitted in a given interval. Assuming the time-wise distribution is exponential in t, Since the F.D.E. is linear, can deal with one term only. 30

31 Errors and Stability Analysis (Cont.) Substituting the value of ε m (x,t) in the F.D.E., 3

32 Errors and Stability Analysis (Cont.) Substituting value of ε m (x,t) in the condition for stability, G is called Amplification Factor. can be possibilities: Called Von Neumann stability analysis. 3

33 Stability of Hyperbolic equations Considering the first order wave eq. u u + c = 0 t x Applying Von Neumann stability analysis to its FTCS discretization shows unconditional unstability. So Lax method of discretization is used, where u(t) is represented by an average value between grid pts. i+ and i-,in time derivative, i.e., ut () = u + u ( n n ) i+ i 33

34 Stability of Hyperbolic Equations (Contd.) Then, u = t u u + u Δt ( + ) n+ n n i i i n n n n u i u n+ ui+ + ui Δt + i ui = c Δx Assuming error of the same form, ( ) at ε m x t = e e, m Amplification factor becomes, ( ) ik x ( ) sin ( ) e at Δ = cos kδ x ic kδx m From stability requirement, finally get, C Δ t = c Δ x C is Courant Number. The condition is CFL (Courant-Freidrichs- Lewy) condition. m 34

35 Stability of hyperbolic Equations (Contd.) Consider second order eq., u u = c t x If v=δu/δt and w=c(δu/δx),then the system of eqns can be written as, u t which is a first order eqn. v 0 c where & A u= w u + = x [ A] 0 = c 0 Eigenvalues of matrix A : det[a-λi]=0, or λ = c 0 Roots are λ =+c & λ =-c, representing traveling waves with speeds c & -c. The characteristic lines are ct (right-running) x = ct (left-running) 35

36 Stability of Hyperbolic Equations (Contd.) Courant No. (C)< Numerical Domain includes all the Analytical Domain, hence STABLE Courant no. (C)> Numerical Domain does not include all the Analytical Domain, hence UNSTABLE 36

37 Stability Analysis-Real Perspective From the definition of Round Off error ε, an incorrect impression is formed that no instabilities would be there, if perfect computer is used with no round-off error. General concept of numerical stability is, in reality,based on the timewise behavior of the solution itself. Stability doesn t inherently depend on behavior of roundoff error per se. General Von Neumann analysis is, where solution itself is written as a Fourier series, i. e., Ux = ik x Ve m where Vm m ( ) m is the amplitude of the mth harmonic of the solution. 37

38 Fluid Flow Modeling Fluid Flow problems complex. G.E. s form a non-linear system. Model eqn. must have a convective, diffusive and time-dependent term. Burger s eqn satisfies all requirements, + u = υ t x x ζ ζ ζ For inviscid flow, it is replaced by Euler s eqn., ζ ζ + u = 0 t x 38

39 Fluid Flow Modeling :Conservative Property Consider the vorticity transport eqn., ω t = V. ω + υ ω ( ) Integrating over fixed region R, finally get, ω d R= V ω. nda + υ ω. nda ( ) ( ) t R A0 A0 R This implies that the time rate of accumulation of ω in equals net advective flux rate of ω across A 0 into Rand net diffusive flux rate of ω across A 0 into R 39

40 Conservative Property (Contd.) Concept of conservative property is to maintain this integral relation in finite difference representation. Considering the conservative form (which follows the said property) of Burger s eqn., i. e., ω = t x ( uω ) Using FTCS method, the F.D.E. is, n+ n n n n n ωi ωi ui+ ωi+ ui ωi = Δt Δx 40

41 Conservative Property (Contd.) Consider a region R from i=i to i=i.evaluating Δ t i = I the integral ω Δ x as, i = I I I I n+ n ( ) n ( ) n ωi Δ x ωiδ x = uω uω i i+ Δt i= I i= I i= I R 4

42 Conservative Property (Contd.) Summation of R.H.S. finally gives, ( uω ) ( uω ) = I I + Rate of accumulation of ω i in R is equal to net advective flux rate across the boundary of running from i=i to I. F.D.E. has preserved integral conservation relations, hence possesses conservative property. 4

43 Conservative Property (Contd.) Conservative Property depends on the form of continuum eqn. used. For e.g., consider non-conservative form : ω ω = u t x After using FTCS technique and then applying integration as earlier, finally yields, I n n n n The F.D.E. fails to preserve the integral Gauss-divergence property, i.e., the conservative property. i= I u ω u ω i i i i+ 43

44 The Upwind Scheme and Transportive Property FTCS to Burger s eqn, found to be unconditionally unstable. Thus, Upwind Scheme of discretization is used, n+ n n n ζi ζi uζi uζi = + viscous terms u>0 Δt Δx n+ n n n ζi ζi uζi + uζi = + viscous terms u<0 Δt Δx Transportive Property: If the effect of perturbation is convected only in the direction of velocity. F.D.E. formed by FTCS method violates the said property. 44

45 Transportive Property (Contd.) Consider a perturbation ε m =δ in ζ. For u>0, ε m =δ (perturbation at m th space location), all other ε=0. Now, checking the upwind scheme, at the down stream location (m+) n+ n ζm + ζm+ 0 uδ uδ = =+ Δt Δx Δx which is acceptable. 45

46 Transportive Property (Contd.) At point m of disturbance, i.e., perturbation is being transported out. At upstream (m-), ζ ζ n + n m ζ m u δ 0 u = = δ Δt Δx Δx n+ m n ζ m 0 0 = = Δt Δx no perturbation effect is carried upstream. Upwind Method maintains unidirectionality. 0 46

47 Upwind Scheme and Artificial Viscosity From Taylor Series, Substituting in F.D.E. obtained by upwind scheme, or, ζ ζ n + i n i ± ( Δt) ( Δ t ) n ζ ζ = ζ i + Δ t + + t t ( Δ x ) n n ζ ζ ζ i x x i x n i = ± Δ + + ( Δx) ζ ζ 3 u ζ ζ 3 Δ t + +Ο ( Δ t) = Δx +Ο ( Δx) Δt t t Δx x x +[diffusive terms] ζ ζ uδt ζ ζ = u + uδx + ν +Ο Δx t x x Δ x x n i n i ( ) L L 3 47

48 Artificial Viscosity (Contd.) The eqn. may be written as, ζ = u ζ + ν ζ e + ν ζ + higher-order terms t x x x where, ν e = ( ) and C (Courant number)=u t/ x uδx C Artificial Viscosity depends on discretization procedure. For steady state, i.e., ζ/ t=0, ν e =u x/. For a -D convective-diffusive eqn., following the similar steps, the eqn. obtained, ζ u ζ v ζ ( ) ζ = + νex + ν + ( ν ) ey + ν ζ Where, t x y x y ν ex = ( ), uδx C x with C x =u t/ x, C y =v t/ x ν ey = uδy( C y ) ; 48

49 Second Upwind Differencing (Hybrid Scheme) Usage of higher order upwind method of differencing, or method which improves the accuracy, is favored. According to the method of second upwind differencing, u ζ u ζ u ζ = x Δx ( ) i, j R R L L where, u R =(u i,j +u i+,j )/ ; u L =(u i,j +u i-,j )/ ; and, ζ R =ζ i,j for u R >0; ζ R =ζ i+,j for u R <0; ζ L =ζ i-,j for u L >0; ζ L =ζ i,j for u L <0; 49

50 Hybrid Scheme (Contd.) For u R >0 and u L >0,get, For unsteady x-direction x momentum eqn., i.e., ζ=u, and introducing a factor η, which can express eqn. as a weighted average of central and upwind differencing, u x i, j where 0<η<. ( uζ ) u, u+, u, u, ij+ i j ij+ i j = ζij, ζi, j x Δx ( ui, j ui+, j) η ( ui, j ui+, j) = + + 4Δx ( ) ( u ) i, j ui, j η ui, j ui, j + The accuracy of the scheme can always be increased by a suitable adjustment of η value. 50

51 Summary The Presentation showed: Finite Difference methods is one of the important discretization technique in CFD, and is based on replacing partial derivatives with difference quotients. Consistency and convergence of a F.D.E. depend on the Truncation Error & differencing scheme used. Explicit and Implicit methods are different CFD techniques, and their use depend on the type of problem being solved. The exact form of stability criterion depends on the form of differential eqn, e.g.,von Neumann stability criteria for parabolic and CFL for hyperbolic eqns Fluid flow can be modeled using a simpler eqn, called Burger s eqn. Conservative property depends on form of continuum eqn. used. Space centred differences are more accurate than upwind differences, as indicated by Taylor s series, the whole system is not more accurate if Transportive Property is looked upon. So, a combination is maintained using second upwind differencing. 5

52 Thank You for your kind Attention And Welcome for Any Questions, Comments or Suggestions. 5