Finite difference approximation

Transcription

1 Finite difference approximation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore January 12, / 21

2 Finite difference method (FDM) Consider an ODE with some boundary conditions. au + bu + cu = 0 x (0, 1) Basic idea of finite difference method: Discretize the domain Ω = (0, 1) with N points known as grid or mesh Usually 0 = x 1 <... < x N 1 < x N = 1 x j+1 x j = h = x Approximate derivatives with finite differences Du j u (x j ) D 2 u j u (x j ) Satisfy the ODE at the grid points in an approximate sense: U j u(x j ) a j D 2 U j + b j DU j + c j U j = 0 2 j N 1 Somehow solve for the unknown values U 2, U 3,..., U N 1. 2 / 21

3 Finite difference approximation Derivatives as limits (we take h > 0 always) u u(x j + h) u(x j ) u(x j ) u(x j h) (x j ) = lim = lim h 0 h h 0 h Finite difference approximation: Take finite h; many possibilities Forward difference (FD) : Backward difference (BD) : D + x u j = u j+1 u j h D x u j = u j u j 1 h Central difference (CD) : D 0 xu j = u j+1 u j 1 2h Since they were obtained from the definition of the derivative, they must provide consistent approximation of the derivative. 3 / 21

4 Consistency of forward difference D + x u j Taylor formula around x = x j Error u j+1 = u j + hu j + h2 2 u j + O ( h 3) u j+1 u j h = u j + h 2 u j + O ( h 2) D + x u j u j = h 2 u j + O ( h 2) Or, use Taylor formula with remainder term Typical error estimate D + x u j u j = h 2 u (ξ), ξ [x j, x j+1 ] D x + u j u j 1 2 h sup u (x) = O (h) x If u C 2 (0, 1): D + x is consistent and first order accurate 4 / 21

5 Consistency of backward difference D x Taylor formula around x = x j Error u j 1 = u j hu j + h2 2 u j + O ( h 3) u j u j 1 h = u j + h 2 u j + O ( h 2) D x u j u j = h 2 u j + O ( h 2) Or, use Taylor formula with remainder term Typical error estimate D x u j u j = h 2 u (ξ), ξ [x j, x j+1 ] Dx u j u j 1 2 h sup u (x) = O (h) x If u C 2 (0, 1): D x is consistent and first order accurate 5 / 21

6 Consistency of central difference D 0 x Taylor formula around x = x j Error u j+1 = u j + hu j + h2 2 u j + h3 6 u j + O ( h 4) u j 1 = u j hu j + h2 2 u j h3 6 u j + O ( h 4) u j+1 u j 1 2h = u j + h2 6 u j + O ( h 4) D 0 xu j u j = h2 6 u j + O ( h 3) Or, using Taylor formula with remainder term Dxu 0 j u j 1 3 h2 sup u (x) = O ( h 2) x If u C 3 (0, 1): D 0 x is consistent and second order accurate 6 / 21

7 Consistency of central difference when u / C 3 (0, 1) What if u C 2 (0, 1) but u / C 3 (0, 1) u j+1 = u j + hu j + h2 2 u (ξ), ξ [x j, x j+1 ] u j 1 = u j hu j + h2 2 u (η), η [x j 1, x j ] Error u j+1 u j 1 2h = u j + h 4 [u (ξ) u (η)] D 0 xu j u j = h 4 [u (ξ) u (η)] Or Dxu 0 j u j 1 2 h sup u (x) = O (h) x If u C 2 (0, 1): D 0 x is consistent and first order accurate 7 / 21

8 Generation of FD formulae Suppose we want to approximate u (x j ) using u j 2, u j 1, u j Assume a formula with unknown coefficients a, b, c Du j = au j 2 + bu j 1 + cu j u j, (non-centered) Use Taylor formula around x = x j [ Du j = a u j 2hu j + 2h 2 u j 4 3 h3 u j + O ( h 4)] [ +b u j hu j h2 u j 1 6 h3 u j + O ( h 4)] +cu j = (a + b + c)u j (2a + b)hu j + (2a b)h2 u j ( 4 3 a b)h3 u j + O ( h 3) 8 / 21

9 Generation of FD formulae Choose a, b, c so that a + b + c = 0 (2a + b)h = 1 2a b = 0 Unique solution is a = 1 2h, b = 2 h, c = 3 2h Hence the FD formula is Error estimate Du j = u j 2 4u j 1 + 3u j 2h Du j u j = ( 4 3 a b)h3 u j + O ( h 3) = 1 3 h2 u j + O ( h 3) If u C 3 (0, 1): Consistent and second order accurate 9 / 21

10 Some remarks Approximate u (x j ) using u j 2, u j 1, u j, u j+1, u j+2 Du j = 1 2h (u j 2 8u j 1 + 8u j+1 u j+2 ) = u (x j ) + O ( h 4) central difference approximation Note the anti-symmetric structure of the formula To obtain higher order accuracy, we need to increase the stencil size Higher order accuracy requires higher regularity/smoothness of the function Central difference formulae give more accuracy with compact stencil as compared to one-sided or non-centered stencil 10 / 21

11 Differentiation via interpolation Construct approximation to u (x j ) using {u j k,..., u j 1, u j, u j+1,..., u j+l } k + l + 1 points Write an interpolating polynomial p j (x) = j+l r=j k L r (x)u r, L r (x s ) = δ rs L r (x) = (x x j k)... (x x r 1 )(x x r+1 )... (x x j+l ) (x r x j k )... (x r x r 1 )(x r x r+1 )... (x r x j+l ) Approximation of derivative u (x j ) p j(x j ) = Can be used on non-uniform grids also. j+l r=j k L r(x j )u r 11 / 21

12 Approximation of second derivative Finite difference u (x j ) = lim h 0 u (x j + h/2) u (x j h/2) h u (x j ) u (x j + h/2) u (x j h/2) h u j+1 u j h uj uj 1 h h D 2 u j = u j 1 2u j + u j+1 h 2 = u (x j ) + O ( h 2) Note the symmetric structure of the formula Other methods Generate formulae by matching Taylor series Or, use the method of interpolating polynomial 12 / 21

13 Non-uniform grids Non-uniform grids essential to capture rapid variations, e.g., boundary layers x 1 < x 2 <... < x N, Finite difference u j u j 1 x j x j 1 = u (x j ) + O (x j x j 1 ), x j+1 x j = h j constant u j+1 u j x j+1 x j = u (x j ) + O (x j+1 x j ) u j+1 u j 1 x j+1 x j 1 = u (x j ) + O (x j+1 x j 1 ) In170 general, Basic Discretization all of thesetechniques are first order accurate. Central difference does not give high accuracy as in case of uniform grids. j 1 j j 1 y j d x i x i 1 Figure Representative grid in a boundary layer region above a solid wall. 13 / 21

14 Smooth non-uniform grids Example: Suppose x [0, 1]. Do a change of variable, e.g., x = f(ξ) = 1 [1 cos(πξ)], ξ [0, 1] 2 Make a uniform grid in ξ-space, also called computational space ξ 1 < ξ 2 <... < ξ N, ξ j ξ j 1 = ξ = 1 N 1 Associated grid in physical space ξ x Finite difference approximation x j = f(ξ j ), du dx = du dξ dξ dx 1 j N du dx (x j) u j+1 u j 1 dξ 2 ξ dx (x j) 14 / 21

15 Spectral analysis f : [0, 1] R a periodic function. Discrete Fourier series approximation f = N/2 k= N/2 ˆf k e i2πkx, ˆf k = ˆf k How does the finite difference approximate derivatives of Fourier modes f(x) = e i2πkx, 0 k N/2 Exact derivative Central difference df dx (x j) = i2πke i2πkxj D 0 xf j = f j+1 f j 1 2h = ei2πk(xj+h) e i2πk(xj h) 2h = i sin(2πkh) e i2πkxj h Dxf 0 j = sin(2πkh) 2πkh i2πkei2πkxj = sin(2πkh) df 2πkh dx (x j) 15 / 21

16 Spectral analysis Ratio of numerical to exact amplitude (Note: h = 1/N) w = 2πkh, ε d = sin(2πkh) 2πkh = sin(w) w = w (w) w, 0 w π ε d 1 ε d = sin w w 0 π w Central finite difference is able to resolve only small wavenumbers k, i.e., w π, i.e., very smooth functions For large k, i.e., w π, the functions are rapidly varying. Finite difference damps these high frequency modes excessively. Desirable to have ε d 1 over a larger range of w Very important for convection problems 16 / 21

17 tion 3.1. These generalizations are derived by writing approxima- =c~+~-~-~+bf,+2-.f;~2+afi+i-~-i regarded way to as c 6h 4h 2h schemes. If th in terms tridiagonal s (2.1) tional st Pf:-*+cI +fl+olfi+1 +Bfi+2 schemes are g The The relations between the coefficients a, b, c and a, B are order formal =c~+~-~-~+bf,+2-.f;~2+afi+i-~-i regarded derived by matching the Taylor series coefficients of various of sixth-order 6h 4h 2h schemes. orders. The first unmatched coefficient determines the may be furthe formal truncation error of the approximation (2.1). These eighth-order tridiagona (2.1) schemes Match terms constraints in Taylor are: scheme. series about x i on both sides The relations between the coefficients a, b, c and a, B are First order the for t derived a+b+c=1+2a+2p by matching the Taylor series (second coefficients order) of (2.1.1) generated various of sixth-o by one-parameter orders. The first unmatched coefficient determines the may be formal a + 22b truncation + 32c = 2 error G (a + of 2 p) the approximation (fourth order) (2.1). (2.1.2) schemes These eighth-ord is ob constraints are: scheme. B=O, a First a + 24b + 34c = 2 z (a + 24/?) (sixth order) (2.1.3) a+b+c=1+2a+2p (second order) (2.1.1) The generated truncation otherwise one-param the a+26b+3 c=2~(a+2 B) (eighthorder) (2.1.4) a + 22b + 32c = 2 G (a + 2 p) (fourth order) (2.1.2) from schemes here on) described be a + 28b + 38c = 2 E (a + 28/?) (tenth order). (2.1.5) indicated B=O in t a + 24b 34c = 2 z (a + 24/?) (sixth order) (2.1.3) within a class As The a -+ trun 0 th Compact tions schemes of the form: (Lele) 17 / 21

18 Compact schemes (Lele) Example: Three point stencil, β = b = c = 0 αf i 1 + f i + αf i+1 = a f i+1 f i 1 2h Two unknowns a, α, we need two equations and a = 1 + 2α, a = 2 3! 2! α = 6α = a = 3 2, α = 1 4 Need some boundary condition: periodic or one-sided formulae Stencil is {i 1, i, i + 1} Implicit; solve for {f i } by solving tri-diagonal matrix problem Can be done very efficiently using Thomas Tridiagonal Algorithm The approximation is fourth order accurate. Classical Pade scheme 18 / 21

19 es the may be further specialized into a one-parameter family of and (2.1.13) ). Compact These eighth-order 5 pentadiagonal 7 schemes g or (2a a single - 1) h*f tenth-order 9) scheme. analyzed Th By choosing the leading scheme. schemes order truncation (Lele) error coefficient vanishes and family of eight the scheme First is the formally tridiagonal sixth-order schemes are described. These are (2.1.14) 5 7 accurate. ;hlofilk Its coefficients a (2.1.1) Tridiagonal are This eighth-ord generated schemes: by p = β 0. = If 0 a further choice of c = 0 is made, a one-parameter (a) family of fourth-order tridiagonal This particu /I? (2.1.2) Fourth order, schemes a=$, one is parameter obtained. j?=o, For a+, α these schemes b+, c=q (2.1.7) and analyze By choos b=& the The leading specific B=O, order tridiagonal a=f(a+2), truncation schemes error b=f(4a-1), obtained coefficient for a = c=o. vanishes $ and a (2.1.6) = and i family of (2.1.3) the were scheme given is by formally Collatz [22, sixth-order p accurate. Its coefficients The specific e are This eighth- With The /3 truncation = 0 and c error #O the on the family r.h.s. of of schemes (2.1) (unless (2.1.6) stated is scheme (2.1.8) Sixth extended order otherwise to the a two-parameter term truncation error family will be of used fourth-order in this sense this one-param (2.1.4) tridiagonal from a=$, here schemes. on) j?=o, for Contained this a+, scheme and within b+, for other these c=q schemes is a (2.1.7) one- to be By choosing parameter described family below of sixth-order are listed in tridiagonal Table I. The schemes. stencil For sizes generated. Th (2.1.5) indicated in the table are the maximum stencil sizes needed The this specific (sixth-order) within tridiagonal family a class of schemes. schemes obtained for a = $ and a = i accuracy amo Sixth order, one parameter α were given As by a -+ Collatz 0 this family [22, merges p coefficients into the well-known fourth- The specific of hen the With order /3 = central 0 and P=O, difference c #O the a=$(a+9), scheme. family Similarly of schemes for (2.1.6) a = $ the is scheme (2.1 e solved extended classical to Padt a two-parameter scheme is recovered. family Furthermore, of fourth-order (2.1.8) a=;, b=h(32a-9), c=&(-3a+l). for a = f this one-par tridiagonal schemes. Contained within these is a one- By choos Among the parameter The sixth-order family tridiagonal of sixth-order scheme tridiagonal (2.1.7) is a schemes. member For of sented generated. by (2. this this (sixth-order) family (with c family = 0, a = f). This sixth-order family can be formal accuracy accura coefficients 19 / 21

20 Compact schemes (Lele) TABLE Truncation Error for the First Derivative Schemes I SANJIVA Max. 1.h.s. Max. r.h.s. Scheme stencil size stencil size Truncation error in (2.1) (2.1.6) (2.1.7) $ (3a - 1) h4f 5 $ h6fc7 K. LELE Pentadiagonal schemes general this fourth-order family is given by a=f(4+2ab=$(-1+4a+2 Schemes of sixth-order parameters a and p. They a (2.1.8) (2.1.8)&a =; (2.1.9) F(-8a+3)hlf 7 $haf P 7 ~(-1+3a-l2/?+10c)h~ a=i(9+a-208), b= c=h(1-3a The tridiagonal sixth-order within (2.1.10). Another su c = 0. This sixth-order pent (2.1.10) 5 7 B=h(-1+3a), (2.1.11) 5 5 i (9a - 4) hqf b=&( (2.1.12) (2.1.13) ;,(,I 7 g (2a - 1) h*f 9) This family limits to the (2.1.7) as fi -+ 0 or a = $. Th ficient for (2.1.11) vanishes scheme. This eighth-order (2.1.14) 5 7 ;hlofilk a=$, p=$, This particular scheme is a 20 / 21

21 f t ) The difference schemes (2.1) correspond to Compact schemes: Spectral analysis tion with Fourier coefficients -& = iwfk. The differencing error of the first derivative paring the Fourier coefficients of the derivative obtained from the differencing scheme (fb)rd with the exact Fourier DIFFERENCE SCHEMES 21 coefficients &. For central difference schemes it may be shown that (f;),, = iw fk, where the modified wavenumber w is 0 real-valued. Each finite difference scheme corresponds to r; a particular function w (w). Exact differentiation corresponds to the straight line w = w. Spectral methods provide w = w for w # x (and w = 0 for w = x). The range e of wavenumbers [27r/N, wr] over which the modified wavenumber w (w) approximates the exact differentiation w (w) = w within a specified error tolerance defines the set of well-resolved waves. While, the value wr, i.e., the shortest well resolved wave, certainly depends on the specific error tolerance it is quite reasonable to keep this error tolerance fixed when different finite difference schemes are compared. It should also be noted that wf depends only on the scheme w,(w) = a sin(w) + (b/2) sin(2w) + 0 (c/3) sin(3w) scheme may be assessed by comr; 1 + 2cr cos(w) + 28 cos(2w). (3.1.4) Plots of the modified wavenumber w against wavenumber w are presented in Fig. 1 for a variety of schemes. In this manner the resolution characteristics of different schemes can be compared. From this plot the fraction r,, representing the fraction of poorly resolved waves and the resolving efficiency e, = 1 - ri is determined. This is done and not on the number of points N used in the descretization. In the following the error tolerance is defined as: Iw (w)-wi <E 1, W Resolving COMPACT FINITE DIFFERENCE SCHEMES 21 Scheme (3.1.3) FIG. 1. Plot of modified wavenumher vs wavenumher for first derivative approximations: (a) second-order central differences; (b) fourthorder central differences; (c) sixth-order central differences; (d) standard The fraction I-, = 1 - wf/7c represents the fraction of Padt scheme (p=o = c, x= f); (e) sixth-order &diagonal scheme poorly resolved waves for the first derivative scheme. This 0 (/? = 0 = c, a = f); (f) eighth-order tridiagonal scheme (8 = 0); (g) eighthfraction d is also independent of the number of points N. The order pentadiagonal scheme (c = 0); (h) tenth-order pentadiagonal fraction 0.0 e, = w,/rc 0.5 = 1 - I.0 rl may 1.5 be regarded 2.0 as a 2.5 measure 3.0 of scheme; (i) spectral-like pentadiagonal scheme (3.1.6); (j) exact differentiathe resolving efficiency of a Wovenumber scheme. We note that the com- tion. putational FIG. 1. Plot efficiency of modified of a wavenumher scheme is proportional vs wavenumher to for the first derivative resolving approximations: efficiency but (a) also second-order depends on central the operation differences; (b) count fourth- for three different values of the error tolerance E, 21 viz., / 21 iti 1:; (e) (f) (g) (h) 6) 0 d I Wovenumber E