Jou rnal of M athem atical Study

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1 Jou rnal of M athem atical Study V ol 38 o 4 D ec 2005 α (, ),, (SVV ), SVV, SVV, SVV - H elm holtz, (SEM ); (SVV ); O A 1,, 1g, ( ),, 80,, ( [ 4 ]),,,, (e g [ 1 ]),, Gibbs,,,,,, (SVV ), SVV, S M Kaber[ 3 ] L SVV [ 2 ] SV, α A ; China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

2 ,,, Xu and Pasquetti[ 6 ] SVV - H elm holtz,, ;, ;, SVV ;, 2 SVV 2 1 u (x, t), 5tu (x, t) + 5x f (u (x, t) ) = 0 Π x +, t > 0 ug# in = 0 u (x, 0) = u0 (x ) Π x + + = [- 1, 1 ], f u, # in = {x 5+; f u (u (x, t) ) > 0}, X u [7 ] (1) 5x f (u (x, t) ) 5x f (u (x, t) ) = v (x, t) 5x u (x, t), v (x, t) = f u, (1) L agrangian D u D t = 0 (2) D gd t L agrangian (2) Q Euler u n+ 1 + Α1u υn υn+ 1- Q + + ΑQ u = 0 (3), Αq, q = 0,, Q,, u n+ 1 u = t n+ 1 u υn+ 1- q u (X (x, t n+ 1, T ) ; T ) T = t n+ 1- q X n (x ) X n (x ) = X (x, t n+ 1 ; t n ) x, X (x, t; T ) d X d T = v (X, T ), 0 < T < t X (x, t; t) = x (4), T t x u υn+ 1- q [7 ], u υn+ 1-1, q = 1,, Q, (3) u n+ 1, (2) + + θ = K k= 1 +θk, + k + Ι = <, Π k, Ι, k Ι China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet (1) (4)

3 4 405 X X = P, K (+) = {Υ C 0 (+) ; Υg+ k P n (+ k ), 1 Φ k Φ K } X 0 = {Ω X, Ω g# in = 0} P (+ k ) + k (2) u X 0, u (g, 0) = I u0, (D tu, Υ ) = 0, Π Υ X 0 (5) (g, g) L 2 -,, (5),, (5) 2 2 (SVV ),, SVV, (5) Ε 5x (Q (5x u ) ) u X H 1 0 (+), (D tu Υ ) + Ε (Q (5x u ), 5x Υ ) = 0, Π Υ X H 1 0 (+) (6) Ε > 0, Q g K = 1, Ε = O (1g ), Q Q Υ Q δ iυ δ il i, Π Υ, Υ= Υ δ il i, L i L egendre, Q δ i Q, i= 1 Q δ i = 0, 0 Φ i Φ m 0 Φ Q δ i Φ 1,m Φ i Φ, Q (i Ε m ) m Q, m = m = g2 m Φ i Φ,, Q δ i, Q δ i = e - ( (i- ) g(i- m ) ) 2, m Φ i Φ m Tadmor [5 ], Ε, (6) (2) [5 ], L 2, T > 0, u (g, T ) u (g, T ), V (u, Υ ) = Ε (Q (5X u ), 5X Υ )L 2 (+), (7) X x, + V V = Ε (Q 1g2 5X u, Q 1g2 5X 5 )L 2 (+) Q 1g2 Q 1g2 Υ Q δ iυ δ il i, Π Υ, Υ= Υ δ il i, = Ε 1 = Ε 1 V Q (5X - 1 u ) 5X Υ dx [ - 1 Q δ i (5X u ) il i (X ) = Ε Q δ i (5X u ) i (5X Υ ) i L i 2 0 (5X Υ ) il i (X ) ]dx China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

4 = Ε 1-1 [ g K > 1 Q δ i (5X u ) il i (X ) Q δ i = Ε (Q 1g2 5X u, Q 1g2 5X Υ )L 2 (+) (5X Υ ) il i (X ) ]dx SVV K > 1, V, Ε m, x = f (X ) + + k Q f, u of P (+), 5x u P (+) g = f - 1, X = g (x ), 5x g 5X f = 1, 5x g > 0 Υ υ = Υof, 5x u = 5x g (x ) 5X u υ (X ) -, g, Q (5x u ) (x ) = 5x g Q δ i (5x u υ ) il i (X ), + k V V k = Ε k (5x g Q (5X u υ ), 5x f 5 π x g 5 X Υ υ ) L 2 (+) = Ε k 5x g (Q (5X u υ ), 5X Υ υ ) L 2 (+) V k = Ε k 5x g (Q 1g2 (5X u υ ), Q 1g2 5X Υ υ ) ) L 2 (+) = Ε k (5x gq 1g2 (5X u υ ), Q 1g2 (5X Υ υ ) ) L 2 (+) (8) Ε k, Ε k = O ( L k 2 ), L k + k - g, (7) (8), V ( ), (8) V k,, V V k K = k= 1 = K k= 1 Ε k (5x gq 1g2 (5X u υ ), Q 1g2 (5X Υ υ ) ) L 2 (+) (9) 3 3 1, [8 ] Gibbs,, X u [6 ] H elm ho ltz [6 ], u n+ 1 (3), u n+ 1 u λn+ 1, u λn+ 1 - u n+ 1 X H 1 0 (+) (u λn+ 1, Υ ) + V (u λn+ 1, Υ ) = (u n+ 1, Υ ), Π Υ X H 1 0 (+) (10) V (9), u λn China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

5 4 407 g 5+ = u n+ 1 g 5+, Π n = 0, 1, u λn+ 1 F H u λn+ 1 = F H u n+ 1, (11) F H H elm holtz V, u n SVV SVV (6) (10) {u εn }n Ε 1 SVV (6) Q B D, u εn ( uεn+ 1 + Α1u n + + n+ 1- Q ΑQ u {u n }nε 1 (5), u n, Υ ) + V (u εn+ 1, Υ ) = 0, Π Υ X H 1 0 (+) (12) ( un+ 1 + Α1u υn υn+ 1- Q + + ΑQ u, Υ ) = 0, Π Υ X 0 (13) (12) u n, u λn = F H u n, u n+ 1 ( un+ 1 + Α1u n + + n+ 1- Q ΑQ u Υ ) = 0 (14) (11), [ (uλn+ 1, Υ ) + V (u λn+ 1, Υ ) ] + ( Α1u n + + n+ 1- Q ΑQ u, Υ ) = 0 ( uλn+ 1 + Α1u n + + n+ 1- Q ΑQ u, Υ ) + V (u λn+ 1, Υ ) = 0, (15) u εn {u λn }n Ε 1 {u εn }nε 1, SVV (6) 3 3 (10) u n+ 1, u λn+ 1, u λn+ 1 - u n+ 1 X H 1 0 (+), (u λn+ 1, Υ ) + V (u λn+ 1, Υ ) = (u n+ 1, v ), Π Υ X H 1 0 (+) (16) (g, g) V k (u λn+ 1 V (u λn+ 1, Υ ) (8) V k (u λn+ 1, Υ ) = K V k (u λn+ 1, Υ ), k= 0, Υ ) Gauss - L obatto, V k (u λn+ 1, Υ ) = Ε k (5x gq 1g2 (5X u λn+ 1 ), Q 1g2 (5X Υ υ ) ) (17) L agrangian X H 1 0 (+), u λn+ 1, Υ, + k V k (u λn+ 1, Υ ) Ε k Θig k i (QD ) im ( p = 0 (QD ) ipu λ p), Π m = 1,, China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

6 , D L egendre, Θi Gauss - Lobatto, u λ p u λn+ 1 p Gauss - L obatto Q Q = M - 1 D iag (Q δ i) 1g2 M M, H elm ho ltz D QD (16) H elm ho ltz SVV Q, Ε k, (16), 4 SVV, Burgers 1 L 2, Ε k = 1g, m = t O SVV SVV F IL TER E E E E E E E E E L 2 -, Ε k = 1gK, m = - 2 t O SVV SVV F IL TER E E E E E E E E E , (1) f (u) = u, v (x, t) 1 u (x, 0) = sin 4 (Πx ) u (x, t) = u (x - t, 0) = sin 4 (Π(x - t) ),,, + = [- 1, 1 ] 10 (K = 10), 9 ( = 9) t = 10-3, Ε k = 1g, m = 1 (O SVV ) (SVV ) (F IL T ER ) t = 1, t = 1 5, t = 2 0 L 2 -,,, China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

7 4 409,,,, Ε k m, 2 Ε k = 1gK, m = - 2,,, 122, SVV F IL TER,,,,, 1 L 2,, L 2, 1 = 1 L 2 - t) 4 2 Burgers Burgers, (1) f (u) = 1 2 u2 f u (x, t) = u (x, u (x, 0) = - sin (Πx ), u ( 1, t) = 0 Burgers,,, ( = 20, K = 20),,Burgers td = 1gΠ x d = 0,,, td 2td = 8, K = 10, = 10-3, Ε = 1gK, m = 2 t = td SEM, SVV 2SEM F IL T ER 2SEM, 3,, China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

8 SEM, SVV F IL T ER, 4 t = 2td,,,, SVV F IL T ER,, 2d O SVV, SVV, F IL T ER SVV,, 3d O SVV, SVV, F IL T ER 4 2td O SVV, SVV, F IL T ER [1 ] Gottlieb D, H esthaven J S Spectral m ethods for hyperbolic p roblem ṡ J Compuṫ A pp l M ath, 2001, [ 2 ] Guo B Y M a H P, T admor E Spectral vanish ing viscosity m ethod for nonlinear conservation law ṡ S IAM J um eṙ A nal, 2001, 39 (4) China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

9 4 411 [ 3 ] O uld Kaber SM A L egender p seudospectral viscosity m ethod J Compuṫ Phyṡ, 1996, [4 ] M aday Y, Patera A T Spectral E lem ent M ethods for the Incomp ressible avier2stokes Equations, 3, A SM E, ed A K oor and J T O den, State2of2A rt Surveys on ComputationalM echanics, 1989 [ 5 ] T admor E Convergence of spectral m ethods for nonlinear conservation law ṡ S IAM J um eṙ A nal, 1989, [ 6 ] Xu C J A filter2based stabiliaxtion m ethod of spectral elem ent computations of h igh R aynolds num ber incom p ressible flow ṡ to appear in J Sci Com puṫ, 2005 [7 ] Xu C J, Pasquetti R O n the efficiency of sem i2imp licit and sem i2l agrangian spectral m ethods for the calculation of incom p ressible flow s, In teṙ J um eṙ M eth F luids, 2001, [ 8 ] V andeven H Fam ily of spectral filters for discontinuous p roblem ṡ S IAM J Sci Compuṫ 6, 1991, Two Stabil ized Spectral Elemen tm ethods for on l inear Con servation Laws Y ang W eicheng X u Chuanju (D epartm ent of M athem atics, X iam en U niversity, X iam en Fujian ) Abstract T h is paper is aim ed to develop efficient stabilization m ethods for the calculation of conservation law s by the spectral elem en t m ethod W e in troduce tw o types of stabilization techn ique spectral visco sity vanish ing (SVV ) and filtering For the SVV, w e generalize the classical deifinition of the SVV operator to the case of m ulti2dom ain Fo r the filtering techn ique, w e in troduce a SVV 2H elm ho ltz operato r, and discuss its relationsh ip w ith the classical spectral filters and SVV 2stabilization Imp lem entation technique of both m ethods are also given F inally w e p resent som e num erical tests to confirm the efficiency of the p roposed m ethodṡ Key W ords SEM ; SVV ; F ilter China Academic Journal Electronic Publishing House All rights reserved http//wwwcnkinet

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