Discrete Fourier Transform { } ( ) sin( ) Discrete Sine Transformation. n, n= 0,1,2,, when the function is odd, f (x) = f ( x) L L L N N.

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1 Dscrete Fourer Trasform Refereces:. umercal Aalyss of Spectral Methods: Theory ad Applcatos, Davd Gottleb ad S.A. Orszag, Soc. for Idust. App. Math umercal smulato of compressble flows wth smple boudares. I. Galerk (spectral) represetato, Stud. Appl Math., 9 (97). Dscrete Se Trasformato whe the fucto s odd, f (x) f (x) k,,,,, / x,,,,, f ( x ) a s k x f ( x) a s k x a s k x x / a f( x )s k x <proof> a f( x )s k x f ( x ) a s k x <proof> l cos + l f( x )s s ( k x ) a s( k x ) ( k x ) m m s a s k x k x m a δ a m m s ( ) s ( k x) δ m m l l l l l l l l l s s cos s cos s cos s cos l l l l 4l l s + s + s s + s s 5l l 6l 4l + s s + s s l ( ) ( ) ( ) + + s s l s + l s l + ( ) s l s l s + l + + { } l l l s + + s ( ) l s ( l + )

2 s ( ) s ( k x) δ m m l cos + l s ( ) s ( k x) δ m m l cos + l <proof> f m: s s cos cos ( ) ( kmx) { (( k km) x) (( k + km) x) } ( m) ( + m) cos cos <proof> f m: s ( ) s( ) cos( ) + ( ) + cos ( ) m m ( ) Dscrete Cose Trasformato whe the fucto s eve, f (x) f (x) f ( x) bcos( ) b b b eve odd / x,,,,, b ( b b + b), for, 4,6, b b + b b b, for,,5, f ( x) bcos( odd ) b f( x )cos k x b b + b m m m odd b b + b m m m+ odd m m m odd b b + b <proof> b f( x)cos( ) m m b f( x) bmcos b + bmcos m m m b + bm cos m m m cos cos cos + (for ) m m { ( ) } b b + bm k f ( x ) b cos k x ( m) m m m

3 <proof> b b b + m m m+ odd m ( )cos b f x k x m b bmcos cos ( m+ ) ( m) bm cos + cos m m+ m b + b m ( ) + ( ) m m k f ( x ) b cos k x Dscrete Fourer Trasformato f ( x) aexp k x,,,,, f ( x ) a exp k x a f x k x exp a f x k x <proof> exp f ( x ) a exp k x f mod(, ) exp otherwse <proof> f mod(, ), wrte l f x k x a k x k x ( ) exp( ) mexp( m ) exp( ) m exp cos( l) + s ( l) ( + ) a exp ( m) m m am m m δ a f mod(, ), wrte θ exp cos θ + s θ { } ( ) + l cos + l

4 Dscrete Fourer Trasformato a a f ( x ) a exp k x a f x k x : complex umber exp * a (complex cougate) ( data) f f (x) s real. oly data eeds storg the amout of computatos s o the order of. Fast Fourer Trasform (FFT) the radx- Cooley-Tukey algorthm exp a f x k x f ( x ) exp f ( x) exp + f ( x) exp eve odd M m m+ + M M ( ) f ( xm) exp f ( xm+ ) exp m M m M Fast Fourer Trasform (FFT) the radx- Cooley-Tukey algorthm m m a f x f x M M a M M ( m) exp + exp ( m+ ) exp m m E + exp O M E, O : DFT of sze M /,,,, M M ( M + ) m f ( xm ) exp m M ( M + ) ( M + ) m + exp f ( xm+ ) exp m M M+ M a Fast Fourer Trasform (FFT) the radx- Cooley-Tukey algorthm m E exp O M M f ( xm ) exp m M M+ M m exp f ( xm+ ) exp m M,,,, M K If, for stace, the amout of computatos becomes o the order of log

5 Covoluto For stace : u u uv + + t x y Stadard fte dfferece schemes v.s. spectral methods Example: a perodc fucto of perod uxt (, ) a( t)exp vxt (, ) b( t)exp uxtvxt (, ) (, ) c( t)exp arbs r r + s c ~ the amout of computatos s o the order of fx a exp( x) a expwx w w Spectral method: dw f( x) a exp( x ) a exp wx + w+ dw Fte dfferece methods: f f( x ),,,,, x Frst dervatve Exact : f( x) a expwx f ( x) w a expwx w d order cetral dfferece: f f f f ( x) + f( x) a expwx w aexpwx+ expwx w aexpwx expwdx expwdx w dw spectral: f ( x) w a exp wx w+ dw spectral exact except w ± a wx w w exp s s( w) aexpwx w exact wa dx 5

6 4rd order cetral dfferece: f( x) a expwx w spectral f f ( x ) f 8f + 8f f + + 8s( w) s( w) aexp wx 6 w exact wa kη req' d ~ ωε w k ε tolerace ωε k η smallest legthscale compact fte dfferece schemes: ele S.K. (99) J. Comp. Phys.,6-4 Frst dervatve: β f +α f + f +α f +βf + + f f f f f f a + b + c 6h 4h h w k Questo: What choces of values of parameters a,b,c,α,β possess the best accuracy (resoluto)? () () () (v) (v) a / 4/9 5/6 7/ b /9 /5 /5.4 c -/8 /.4559 α /4 / /8 /.58 β /.959 order

7 Pseudo-spectral method alasg error trucated wave doma grd shft techque mxed method da Burger's equato: t x x u u u + ν uxt (, ) a( t)exp u ( x, t) b( t)exp exp exp exp dt Galerk: ( k x) + k b t ( k x) ν k a t ( k x) exp k x dx m da k b νk a dt + orthogoalty Covoluto Pseudo-spectral method uxt (, ) a( t)exp vxt (, ) b( t)exp uxtvxt (, ) (, ) c( t)exp c a b r s r r+ s ux aexp vx bexp wx ux vx c w( x )exp k x

8 Pseudo-spectral method Pseudo-spectral method c a exp k x b exp k x exp k x r r s s r s abexp ( k + k k ) x r s r s r s ( r+ s) Γ( rs,, ) exp f mod( r+ s, ) Γ ( rs,, ) otherwse r+ s ab r sexp r s for rs,, <, < r+ s < ab Γ r s r s ( r, s, ) Γ r, s, δ ( r+ s ) +δ ( r+ s ) +δ ( r+ s + ) Pseudo-spectral method r s r s (,, ) c a b Γ r s c a b + a b + a b r s r s r s rs, rs, rs, r+ s r+ s r+ s + c c + a b + a b r s r s rs, rs, r+ s r+ s + Pseudo-spectral method possble soluto: trucated wave doma < K < rs,, < K K < r+ s < K < Γ rs,, δ ( r+ s) choose K (alasg error)

9 Grd Shft Techque Cosder x x + / ( + ) x +Δx u u( x ) a ( t)exp k x ( + / ) exp exp u u x a k Δx k x smlarly w w v v( x ) b ( t)exp k x ( + / ) exp exp v v x b k Δx k x u v c uv exp( ) c + a b + a b r s r s rs, rs, r+ s r+ s + u v cˆ exp u v + / uv exp( + /) exp ˆ a exp k x b exp k x exp k x ( + / ) ( + / ) ( + / ) r r s s r s abexp k + k k x ( + / ) r s r s rs, ( k Δx ) exp cˆ u v k x c c ab ab r s r s rs, rs, r+ s r+ s + c c + a b + a b r s r s rs, rs, r+ s r+ s + - ab exp( ( k + k k ) Δx ) exp ( k + k - k ) x r s r s rs, ( r+ s) ( r+ s) kr + ks k Δ x ( r s ) f r+ s or + or c c + c ( ˆ ) 6 FFTs are requred to obta alasg-error-free coeffcets.

10 Exteded to D Smulatos: ux ( ) aexp k x ( ) vx bexp k x ( ) wx ux vx c exp k x ( ) c rs, r+ s a( r) b( s) (D covoluto) Pseudo-spectral Method: c u x v x k x exp ( ) (,, ),,,. x,, x, x, x Δ Δ Δ Pseudo-spectral Method: alasg errors c r rs, s a( r) exp( k r x ) b s k x k x exp( s ) exp ( ) a( r) b( s) exp ( k r + k s k ) x ( ) k k k x r s r + s + ( r + s ) exp ( k + k k) x exp ( r + s ) r s { } δ r + s +δ r + s + +δ r + s c a r b s r s Γ(,, ) rs, Γ δ + +δ + + +δ + ( r, s, ) { ( r s ) ( r s ) ( r s ) } 6 terms cotrbutg to alasg errors 4

11 Grd shft techque: x Δ x ± Δ x ± Δ x ± ( ( ), ( ), ( ) ) ( m ) 8 combatos (8 shfted grds) m,,,,8 ( m) m exp( ) u a k x ( m) m exp( ) v b k x c u v k x exp ( ) ( m) ( m) ( m) ( m) (4 FFTs) 8 c c 8 ( m) m Mxed techque: Two grds used oly: x ( Δx, Δx, Δx ) x + Δ x + Δ x + Δ x + ( ( ), ( ), ( ) ) c u x v x k x exp ( ) ( ) ( ) exp( ) cˆ u x v x k x Δx k exp Δx k exp ( ) ( ) exp( ) cˆ u x v x k x cˆ u x v x k x exp ( ) ( ) exp + + ( ) ( r s ) ( ) ( r s ) δ + + c a( r) b( s) δ r + s + + rs, δ + ( r s ) ( ) ( r s ) δ + cˆ a( r) b( s) δ r + s + rs, δ + c c c a r b s r s + ˆ + δ ( + σ) rs, 5

12 Trucated wave doma: + ˆ + δ ( + σ) σ c c c a r b s r s rs, : exactly oe zero compoet ad the other two are ether + or a total of alasg-error terms Doma D: < ± <, f, a r b s r s D The all alasg-error terms become zero. Proof : suppose σ, σ r + s or r + s or The, oe of the followg must be true: r s r s + + or r s r s ( r + s ) ( r + s ) ± r r s s ± + ± + ± < Mxed techque: completely alasg-error free 6 FFTs ( grds) + trucated wave doma Doma D: largest scrbed spercal wave doma (also true for D flows) < ± <, + + < 9 6

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