SHUILI XUEBAO. KFVS( Kinetic Flux Vector Splitting) Boltzmann, KFVS( Kinetic Flux Vector Splitting) BGK(Bhatnagar2Gross2Krook) (Well2balanced),

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1 SHUILI XUEBAO 7 7 : (2006) KFVS 2, (, 0020 ;2, ) :Boltzmann, KFVS( Kinetic Flux Vector Splitting),, KFVS,, : ; ; ; KFVS ; :TV4 :A 90, [,2 ] Boltzmann, KFVS( Kinetic Flux Vector Splitting) BGK(Bhatnagar2Gross2Krook) Ghidaoui [ ] [4 ], BGK, (Well2balanced),,2002, BGK [5 ] KFVS [6 ] KFVS BGK Boltzmann,, KFVS ( ),,,KFVS, BGK / [6 ], KFVS KFVS, Boltzmann 9f 9t + u 9f + v 9f 9y + 9< 9f 9u + 9< 9f 9y 9v = 0 () : f ; u v x y ; <,, < x = g ( S 0 x - S fx ), < y = g ( S 0 y - S fy ) ; g ; S fx S fy x y ; S 0 x S 0 y x y, S 0 x = - 9bΠ, S 0 y = - 9bΠ9y, b f x y t u v, q [,4 ],Maxwellian q = h ( ) e - [ ( u- U) 2 + ( v- V) 2 ] (2) : : (400600) ; (M40054) : (96 - ),,,, E2mail org 858

2 := Π( gh) ; U V x y ; h,f = q, + + 9q 9t u 9q + v 9q + 9< 9q 9u + 9< 9q 9y 9v u v d ud v = 0 () [,4] (2) (), Boltzmann, 9h + 9hU + 9hV = 0 9t 9y 9hU + 9 9t ( hu2 + 2 gh2 ) + 9hUV = gh ( S 0 x - S fx ) (4) 9y 9hV + 9hUV + 9 9t 9y ( hv2 + 2 gh2 ) = gh ( S 0 y - S fy ), Boltzmann Boltzmann, Boltzmann (4), Boltzmann () + + () (, u, v) T, + + 9E 9t + 9F + 9G 9y = S (5) : E = [ h, hu, hv ] T ; F = uf d ud v, u f d ud v, - uvf d ud v -, v 2 f d ud v T ; S = 0, gh ( S 0 x - S fx ), gh ( S 0 y - S fy ) uvf d ud v T T + + ; G = vf d ud v, - Boltzmann (5),,, i i,, (5), 9 E 9 t + ( Fcos+ Gsin ) d l = κ S i d xd y (6) i : i ; (cos,sin ) ;d l (6), F n = Fcos+ Gsin, i = E n i - t E n+ F nj l j j = + t κ i - S oi d xd y + ts fi (7) : t ; j i j ; l j ; n (7),,Godunov TVD MacCormack BGK, KFVS 2 KFVS (7),KFVS f x - t ( x ), x,u v, Boltzmann () f ( x ij, t, u, v) = f 0 ( x 0,0, u 0, v) (8) : 0 ; f 0 ; x ij i j ; x 859

3 , t - t x x, x = x + u ( t - t ) + 2 < x ( t - t ) 2 ; ( ) u, u = u + < x ( t - t ),,t Taylor, t q ( x, t, u, v) q ( x, t, u, v) [ + 2< x ( u - U) ( t - t ) ] (9) :2< x ( u - U) ( t - t ) q Maxwellian,< x,(9) q ( x, t, u, v) = q ( x, t, u, v) [ + 2 < x ( u - U) ( t - t ) ] (0), ( u, u 2, uv) T < x < x = g ( S 0 x - S fx ) l ( - H[ x ]) + g ( S 0 x - S fx ) r H[ x ] () : H[ x ]Heaviside ; l r (8) f 0, f 0 = q l ( + a l ( x - x ij ) ) ( - H[ x - x ij ]) + q r ( + a r ( x - x ij ) ) H[ x - x ij ] (2) : q l q r Maxwellian ; a a : ( m, m 2, m, m 4 ), m - 4 U 9h ( hu) = m + m 2 u + m v + m 4 ( u 2 + v 2 ) () = ( U 2 + V 2 ), m = - 4V 9h ( hv), x ij = 0 9h - 2 U 9 ( hu), m 4 = 9h - 2V 9 ( hv), m 2 =, (2) t = 0 f 0, (8) t x = 0 u f 0 ( - ut - 2 < xt 2,0, u - < x t, v) = q l 0 + a l ( - ut - + q r 0 + a r ( - ut - 2 < xt 2 ) - H - ut - 2 < xt 2 ) H - ut - 2 < xt 2 2 < xt 2 q l 0 ( - uta l ) H[ u ] + q r 0 ( - uta r ) ( - H[ u ]) (4) t 2 t = 0 t, q l 0 q r 0, : q l 0 = q l l ( + 2 < x ( u - U l ) t) ; q r 0 = q r r ( + 2 < x ( u - U r ) t),(4) : f 0 ( - ut - 2 < xt 2,0, u - < x t, v) q l l ( + 2 < x ( u - U l ) t) ( - uta l ) H[ u ] + q r r ( + 2 < x ( u - U r ) t) ( - uta r ) ( - H[ u ]), f 0 ( - ut - 2 < xt 2,0, u - < x t, v) q l ( + 2 l < x ( u - U l ) t - uta l ) H[ u ] + q r ( + 2 r < x ( u - U r ) t - uta r ) ( - H[ u ]) (5) (5), a l a r Heaviside, < x < x = - g ( S 0 x - S fx ) l H[ u ] + ( S 0 x - S fx ) r ( - H[ u ]) (6) (5) (8), x = 0 f (0, t, u, v) (8) 860 F h ( t) F hu ( t) F hv ( t) ij + + = u u - - v f ij (0, t, u, v) d ud v (7)

4 [5 ], u =,2 u = 5Π4, uv =,,,,, i j, n : z n + h n i + b i = h n j + b j = z ; ( U,V) n i = ( U,V) n j = 0 (8) h n+ i = z ; ( U,V) n+ i = 0 (9) [77 ] 20 90, [,5,7 ] BGK [5 ] KFVS [6 ], E. Audusse [8 ] KFVS,, ( 2 gh2 ) + gh b = gh z (20), b ij = b ji = 2 ( b i + b j ) (2) [8 ], Zhou [ ] (Surface Gradient Method, [5 SGM) ] (Water Level2bottom topography Formulation, WLF), 2, h ij = ( h i + b i - b ij ) (22), (20) ( 2 gh2 ) = - gh,,(7) : S 0 ij = 0 2 gh2 ij E n+ i = E n i - t F nj l j j = + t b j = S 0 ij l j +ts fi (2) (2) n (8), (2), 0 = - t j = 2 gh2 ijl j + t j = 2 gh2 ijl j (24) [9 ], (9), 2, MUSCL, [ 20 ] z ij, h ij = z ij - b ij, 86

5 b ij (2),, S 0 ij = 0-2 g ( h ij + h i ) ( b ij - b i ), (25), n (8), (2) 0 = - t 2 gh2 ijl j - t h i ) ( b ij - b i ) l j, j = j = j = (25) 2 g ( h ij + 2 g ( h ij + h i ) ( z ij - z i ) l j = 0 (26) (20),, z ij = z i,, (9) 4 4,,, [ 224 ] Osher [2 ] HLL [22 ] Roe [2,24 ] Riemann Godunov ; [7 ] Riemann 40m, 0m 25275m = 895, 0, z = m, u = 857mΠs, v = 0, Froude Fr = 274 ; 9zΠ = 9uΠ = 9vΠ = 0, 9 600, 2 0, z = [25 ] 499m, U = 795mΠs, Fr = 2074 Hager,= 0, z = 5m, U = 7956mΠs Fr = , 72km, 080, 200m, ,, [9 ] [26 ; ; Riemann ] ,, s, 8m/ s, 5 6, ( ) 862

6 Boltzmann, KFVS, ;, KFVS,,,,, : [ ] Xu Kun. Gas2kinetic scheme for unsteady compressible flow simulations[ R ]. 29 th Computational Fluid Dynamics, Von Karman Institute for Fluid Dynamics Lecture Series 998-0, 998. [ 2 ] Xu Kun. Unsplitting BGK2type schemes for the shallow water equations[j ]. International Journal of Modern Physics C, 999, 0(4) : [ ] Ghidaoui M S, Deng J Q, Gray W G, Xu K. A Boltzmann based model for open channel flows[j ], International Journal for Numerical Methods in Fluids, 200, 5(4) : [ 4 ]. BGK [J ].,2002, (4) : - 7. [ 5 ] Xu Kun. A well2balanced gas2kinetic scheme for the shallow2water equations with source terms [ J ]. Journal of Computational Physics, 2002, 78 : [ 6 ],. KFVS [J ].,2002,7(2) : [ 7 ],. Osher [J ].,994,5(4) : [ 8 ],. [J ].,995,6() : - 9. [ 9 ] Bermudez A, Elena Vazquez M. Upwind methods for hyperbolic conservation laws with source terms[j ]. Computers & Fluids, 994, 2(8) : [0 ] LeVeque R J. Balancing source terms and flux gradient in high2resolution Godunov methods : the quasi2steady wave propagation algorithm[j ]. Journal of Computational Physics, 998, 48 : [ ] Bermudez A, Dervieux A, Desideri J, Vazquez M E. Upwind schemes for two2dimensional shallow2water equations with 86

7 variable using unstructured meshes[j ]. Comput. Methods Appl. Mech. Eng., 998, 55 : [2 ] Vazquez2Cendon M E. Improved treatment of source terms in upwind schemes for shallow2water equation in channels with irregular geometry[j ], Journal of Computational Physics, 999, 48 : [ ] Zhou J G, Causon D M, Mingham C G, Ingraln D M. The surface gradient method for the treatment of source terms in the shallow2water equations[j ], Journal of Computational Physics, 200, 68 : [4 ] Gascon L,Corberan J M. Construction of second2order TVD schemes for nonhomogeneous hyperbolic conservation laws[j ]. Journal of Computational Physics, 200, 72 : [5 ] Hui W H, Pan Cunhong. Water level2bottom topography formulation for the shallow2water flow with application to the tidal bores on the Qiantang river[j ]. Computational Fluid Dynamics Journal, 200, 2() : [6 ],,. Godunov [J ].,200,4(4) : [7 ],,. Godunov [J ].,200,A,8 () :6-2. [8 ] Audusse E, Bristeau M O. A well2balanced positivity preservingsecond2orderscheme for shallow water flows on unstructured meshes[j ]. Journal of Computational Physics, 2005, 206 : -. [9 ] Pan Cunhong, Lin Bingyao, Mao Xianzhong. New development in the numerical simulation of the tidal bore [ A ], Proceedings of the International Conference on Estuaries & Coasts[ C]. Hangzhou :Zhejiang University Press, 200, : [20 ],. [J ],,997, (2) :06 -. [2 ] Zhao D H, Shen H W, Lai J S, Tabios GQ. Approximate riemann solvers in FVM for 2D hydraulic shock wave modeling [J ]. Journal of Hydraulic Engineering, 996, 22(2) : [22 ] Alcrudo F, Garcia2Navarro P. A high2resolution Godunov2type scheme in finite volumes for 2D shallow2water equations [J ]. Int. J. Numer. Meth. Fluids, 99, 6 : [2 ] Hu H, Mingham C G, Causon D M. A bore2capturing finite volume method for open2channel flows[j ],Int. J. Numer. Meth. Fluids, 998, 28 : [24 ] Rogers B, Fasayuki M, Borthwick G L. Adaptive Q2tree godunov2type scheme for shallow water equations[j ], Int. J. Numer. Meth. Fluids, 200, 5 : [25 ] Hager W H, Schwalt S, Jimenez O, Chaudhry M H. Supercritical flow near an abrupt wall deflection[j ], Journal of Hydraulic Research, 994, 2() :0-8. [26 ],,. [J ].,2004, (4) : - 7. Kinetic flux vector splitting scheme for solving 22D shallow water equations with triangular mesh PAN Cun2hong, XU Kun 2 ( Zhejiang Institute of Hydraulics and Estuary, Hangzhou 0020, China ; 2 Hong Kong University of Science and Technology, Kowloon, Hong Kong, China) Abstract :Based on the Boltzman equation, the kinetic flux vector splitting ( KFVS) scheme for solving 22 D shallow water equations with triangular mesh is developed. In order to establish a well2balanced scheme, the source term effect is taken into account explicitly in the flux evaluation. On this basis a special technique for dealing with source term due to bottom topography is adopted and the well2ballanced KFVS scheme with triangular mesh possessing second order accuracy is established. The validity of the proposed method is verified by the comparison of calculation result of a traditional typical example with the field observation data of a tidal bore in Qiantang River. Key words :22D shallow water equations ; kinetic flux vector splitting ( KFVS) scheme ; finite volume method (FVM) ; triangular mesh ; source terms ( : ) 864