JOURNAL OF UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA

34 1 Vol. 34,No. 1 2 0 0 4 2 JOURNAL OF UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA Feb. 2 0 0 4 :025322778 (2004) 0120029209 WENO Ξ,, (, 230026) :Weighted Essentially Non2Oscillatroy (WENO). WENO,,,. :WENO ;;Runge2Kutta :V211. 1 :A AMS Subject classif ication( 2000) : 65M06,65M99,35L65 1 0,,,,,. 90, ENO ( Essentially Non2Oscillatory) [4,5 ] WENO (Weighted ENO). WENO,,,,. Liu, Osher Chan [12 ] WENO,Jiang,Shu, Hu Qiu WENO [1,3,6,7,14,16 ]. WENO,, CPU,. Levy Puppo Russo [10 ] TVD2Runge2 Kutta,,WENO, [8,9,13 ],WENO,,. 1 WENO u t + f ( u) x = 0 (1) Ξ :2002204220 : (10071083) :,,1978,. E2mail :xuzl @mail. ustc. edu. cn

30 34 WENO [7 ]. x = x j, f ( u) x x = x j 1 x ( f^ j +1/ 2 - f^j - 1/ 2), f^j +1/ 2 WENO. f ( u) 0,k, S r ( j) = { x j - r, x j - r +1,, x j - r + }, r = 0,1,,, (2) f^j +1/ 2,k, j +1/ 2 = p ( r) ( x j +1/ 2 ) = f^ ( r) i = 0 c ri f ( u j - r + i ), r = 0,1,,, (3) p ( r) ( x) S r. WENO f^ j +1/ 2 = r i = 0 f^( r) j +1/ 2, (4) r,., r 0, r = 1. i = 0 f ( u),d r, (Linear Weights), f^ j +1/ 2 = d r i = 0 f^( r) j +1/ 2 (5),f^ j +1/ 2 2. d r, T = r = 0 S r 2 P( x), d r. k = 3, P( x j +1/ 2 ) = d r p ( r) ( x j +1/ 2 ) (6) i = 0 d 0 = 1 10, d 1 = 3 5, d 2 = 3 10. (7) P( x) 2,.,,. Jiang Shu r r, r = h l = 1 2 l - 1 Ij d l d x l p ( r) ( x) 2 d x (8), p ( r) ( x) I j = ( x j - 1/ 2, x j +1/ 2 ) S r ( j). k = 3,, r 0 = 13 12 ( f j - 2-2 f j - 1 + f j ) 2 + 1 4 ( f j - 2-4 f j - 1 + 3 f j ) 2, 1 = 13 12 ( f j - 1-2 f j + f j +1 ) 2 + 1 4 ( f j - 1 - f j +1 ) 2, 2 = 13 12 ( f j - 2 f j +1 + f j +2 ) 2 + 1 4 (3 f j - 4 f j +1 + f j +2 ) 2. r = r s = 0 s, r = d r (+ r ) 2, r = 0,1,,, (9)

1 WENO 31,,= 10-6. : ( ), r d r,2 ; ( ),WENO ENO,. f ( u) 0,,,f ( u) 0 WENO. f ( u), f ( u) = f + ( u) + f - ( u), ( f + ) ( u) 0, ( f - ) ( u) 0. WENO f + ( u) f - ( u),. Lax2Friedrichs, f ( u) = 1 2 [ f ( u) u ], = max u f ( u). (10),f ( u) Jacobi. [ 7 ] : r = d k r + O ( h ), r = 0,1,,, WENO 2. r = D (1 + O ( h ) ), D r. Taylor. TVD2Runge2Kutta u (1) = u n +t L ( u n ), u (2) = 3 4 u n + 1 4 u (1) + 1 4 t L ( u (1) ), ( l) r u n +1 = 1 3 u n + 2 3 u (2) + 2 3 t L ( u (1) ). ( l) Runge2Kutta. r,, ( r l) = r, ( r l) = r, k = 0,1,2, l = 1,2.,Euler,,. ( r l),. = r = D (1 + O ( h ) ).,,WENO ENO,,,. k = 3, 1, r / 1 0, r = 0,2.,t ct 1, c CFL, TVD2Runge2Kutta. f i (1) G l ( u) f i (1) = f ( u i (1) ), Taylor, = f i n +t G 1 ( u i n ) + (t 2 2) G 2 ( u i n ) + (t 3 3!) G 3 ( u i n ) +, = 5 l t f ( u) L ( u). Taylor (1) r = r (1 + O ( h) ), r = 1,2,3. (1) WENO,Taylor r

32 34 t = O ( h)., (1) r / 1 (1) 0, r = 0,2. (2) r / 1 (2) 0, r = 0,2.,,,. WENO. ( ),,,. ( ),,. Roe.,,WENO,. 2 WENO, TVD2Runge2Kutta, WENO. 1, u t + u x = 0, (11) u ( x,0) = sin (2x),,T = 2. 1 WENO. 2 Burgers u t + ( u 2 2) x = 0, (12) u ( x,0) = 0. 5 + sin (x),. t = 0. 5/, Burgers,2. t = 1. 5/,. 1 t = 0. 5/,,.

1 WENO 33 3 Euler 1 Burgers Fig. 1 Dimensional linear scalar quantity Burgers equations v E t + v v 2 + p v ( E + p) x = 0, (13) p : v p, E = - 1 + 1 2 v 2,= 1. 4. Lax Sod (, v, p) = (0. 445,0. 698,3. 528), x 0 (, v, p) = (0. 5,0,3. 571), x > 0, (, v, p) = (1. 0,0,1. 0), x 0 (, v, p) = (0. 125,0,0. 1), x > 0, 1. 6 2. 0, 2.,. 2 Euler Riemann Fig. 2 The Riemann value problem for Eulerian equations 4 Euler, (, v, p) = (3. 857 143,2. 629 369,10. 333 333), x < 4 (, v, p) = (1 +sin (5 x),0,1), x 0

34 34 = 0. 2.,,.,. t = 1. 8 3, 2000 Runge2Kutta WENO., N = 400. 3 Euler Fig. 3 The collision problem between shock and density wave for Eulerian equation 5. Euler ( l, v l, p l ) = (1,0,1000), x 0. 1, ( c, v c, p c ) = (1,0,0. 01), 0. 1 < x < 0. 9, ( r, v r, p r ) = (1,0,100), x > 0. 9. [ 0,1 ],. 0. 038.,,, [7 ].,CFL, CFL,,,., [16 ] CFL (CFL = 0. 6),4,. 6 Burgers u t + u2 2 x u ( x, y,0) = 0. 5 + sin ( x + y) / 2 + u2 2 y = 0, (14),[0,4 ] [0,4 ]. t = 1. 5/5.. 7. Mach 10,60, = 1. 4, p = 1. 0. [0,4 ] [0,1 ],1 6 < x < 4., 0 < x < 1 6,, Mach 10,,

1 WENO 35 4 Euler Fig. 4 Interacting strong blast waves of Eulerian equation,,t = 0. 2. 6 7, 30. : ( ),.... 2 G CPU 512M IBM, Com2 paq Visual Fortran 6. 5, Lax, WENO 20 25 %,,,6 %. 5 Burgers,80 80 Fig. 5 22dimensioral Burgers equations 6 1. 5 22. 7 30, Fig. 6 Double Mach reflection paroblem ; Dansity;,, contour lines from= 1. 5 to= 22. 7

36 34 7 1. 5 22. 7 30 Fig. 7 Double Mach reflection problem. Blon2up reqion around the double Mach system.,. Density; 30 contour lines from= 1. 5 to= 22. 7 ( ).. WENO CFL, Courant,., WENO CFL. 8 Lax Fig. 8 Lax shock wave pipe problem

1 WENO 37. Gibbs,.,,,,,,.. Lax. TVD2Runge2Kutta, WENO. (8 ),,,.,,,WENO., Runge2Kutta,. 8,,,.. [ 1 ] Balsara D and Shu C W. Monotonicity preserv2 ing weighted essentially non2oscillatory schemes with increasingly high order accuracy [ J ]. J Comput. Phys.,2000,160 : 4052452. [2 ] Deng X and Zhang H. Developing high2order weighted compact nonlinear schemes. J. Com2 put. Phys.,2000,165 : 22244. [3 ] Friedrichs O. Weighted essentially non2oscilla2 tory schemes for the interpolation of mean val2 ues on unstructred grids [ J ]. J Comput. Phys.,1998,144 : 1942212. [4 ] Harten A, Engquist B,Osher S and Chakara2 vathy S. Uniformly high order accurate essen2 tially non2oscillatory schemes [J ]. J Com2 put. Phys.,1987,71 : 2312303. [5 ] Harten A and Osher S. Uniformly high order accurate essentially non2oscillatory schemes, I, SIAM J. Numer. Ana.,1987,24 : 2792309. [ 6 ] Hu C And Shu C W. Weighted essentially non2 oscillatory schemes on triangular meshes[j ]. J Comput. Phys.,1999,150 : 972127. [7 ] Jiang G S and Shu C W. Efficient implementa2 tion of weighted ENO schemes[j ]. J Comput. Phys.,1996,126 : 2022228. [8 ] Levy D,Puppo G and Russo G. On the behav2 ior of the total variation in CWENO methods for conservation laws [ J ]. Appl. Numer. Math.,2000,33 : 4072414. [9 ] Levy D, Puppo G and Russo G. A third order central WENO scheme for 2D conservation laws [ J ]. Appl. Numer. Math., 2000, 33 : 4152421. [10 ] Levy D,Puppo G and Russo G. Central WENO schemes for hyperbolic systems of conservation laws[j ]. Math. Model. Numer. Anal.,1999, 33 : 5472571. [11 ],,., [J ].,2001,31 : 2452262. [ 12 ] Liu X D,Osher S and Chan T. Weighted essen2 tially non2oscillatory schemes [ J ]. J Comput. Phys.,1994,115 : 2002212. [13 ] Qiu J X and Shu C W. On the construction, comparison and local characteristic decomposi2 tion for high order central WENO schemes[j ]. Submit to J Comput. Phys. (54 )

54 34 FB G and t ransverse loading is obtained experimentally. A multi2point dist ribute t ransverse load sensing by means of wavelengt h addressing is also investigated wit h an experiment of two2grat2 ing simultaneous sensing. Key words : fiber Bragg grating ; transverse load ; stress induced birefringence ; resonance wave2 lengt h separation ; multi2point dist ribute sensing (37 ) [ 14 ] Qiu J X and Shu C W. Finite difference WENO schemes with Lax2Wendroff type time dis2 cretizations[j ]. Submit to SIAM J Sci. Com2 put. [15 ] Shi J, Hu C and Shu C W. A technique of treating negative weights in WENO scheme [ R]. ICASE Report No. 2000249. [ 16 ] Shu C W. Essentially non2oscillatory and weighted essentially non2oscillatory schemes for hyperbolic conservation laws [ R ]. ICASE Re2 port No. 97265. [17 ] Wang Z J and Chen R F. Optimized weighted essentially non2oscillatory schemes for linear waves with discontinuity [ J ]. J Comput. Phys.,2001,174 : 3812404. Some Optimal Methods f or WENO Scheme in Hypabolic Conservation La ws XU Zhen2li, Q IU Jian2xian, L IU Ru2xun ( Depart ment of M aths, US TC, Hef ei, A nhui, 230026) Abstract : WENO (weighted Essentially Non2Oscillat roy) is a high2resolution numerical scheme used for solving equations of hyperbolic conservation laws. In t his paper, some optimal st rate2 gies of the WENO scheme of hyperbolic conservation laws are discussed and the time of nonlin2 ear weighted computation and characteristic decomposing is reduced. By some numerical exam2 ples, the feasibility of these strategies is proved and the advantages and disadvantages com2 pared. Key words :WENO scheme ; hyperbolic conservationa law equation ; Runge2Kutta met hod