3. Recent development of the numerical simulation method for cfd The numerical test and study on the shallow water wave problems is one of the most active topics in computational hydraulics. USing numerical simulation and numerical analysis, taking the suitable simplified model, scientists have got a lot of significant information for various complicated shallow water wave phenomena n recent years, especially, many impressive and wonderful numerical simulated results for 2d or 3D discontinuous problems are continually reported In the section. several efficient simulation methods will be introduced
3.Recent development of the numerical simulation method for CFD The numerical test and study on the shallow water wave problems is one of the most active topics in computational hydraulics. Using numerical simulation and numerical analysis, taking the suitable simplified model, scientists have got a lot of significant information for various complicated shallow water wave phenomena. In recent years, especially, many impressive and wonderful numerical simulated results for 2D or 3D discontinuous problems are continually reported. In the section, several efficient simulation methods will be introduced
3-1.ENo (essentially non-oscillatory schemes )and weighted ENO The polynomial interpolation is the foundation of most numerical methods although it can introduces spurious oscillations Preserve the monotonic character for designing finite difference schemes is the key to avoid introducing spurious oscillatory mechanism. Harten et al (1998 developed the reconstruction methods and found the procedure to overcome spurious oscillation ENO, especially weighted eNo basically realized the object by controlling the scale of every Newton difference-quotients and selecting suitable node-stencils during the process heightening the order of the polynomial
3-1.ENO (essentially non-oscillatory schemes) and weighted ENO The polynomial interpolation is the foundation of most numerical methods although it can introduces spurious oscillations. Preserve the monotonic character for designing finite difference schemes is the key to avoid introducing spurious oscillatory mechanism. Harten et al (1998) developed the reconstruction methods and found the procedure to overcome spurious oscillation. ENO, especially weighted ENO basically realized the object by controlling the scale of every Newton difference-quotients and selecting suitable node-stencils during the process heightening the order of the polynomial
1. ENO method Consider the initial value problem of the scalar conservation law u, +(f(u)=0, xER, t>0;a=a(u)=df(u/au (11) (x.0)=a(x) Numerical-flux scheme u -rlf(u-f(unI ENO method adopts a Newton polynomial to reconstruct the numerical flux f((u 1). Choose a initial two-cell stencil 2). Extend the stencil to left and right side to construct new stencils S2={U{ 3) Compute the two quotients corresponding two cell-stencils and compare them x1)-N(x-12x) NR(SR= And choose the one having min-absolute-value and expand Newton polynomial p(x)=l1+N(x,x1+1)(x-x,)+N21(x-x1)(x-x1) 4). Substitutex=x r and calculate the value l2=P(x12),f(l+n2)=f(P(x12)
1. ENO method Consider the initial value problem of the scalar conservation law (11) Numerical-flux scheme: . (12) ENO method adopts a Newton polynomial to reconstruct the numerical flux : 1). Choose a initial two-cell stencil 2). Extend the stencil to left and right side to construct new stencils 3). Compute the two quotients corresponding two cell-stencils and compare them And choose the one having min-absolute-value and expand Newton polynomial 4). Substitute and calculate the value 0 ( ( )) 0, , 0; ( ) ( ) / ( .0) ( ) t x u f u x R t a a u f u u u x u x + = ∈ > = = ∂ ∂ = 1 1 2 2 1 [ ( ) ( )] n n i i i i u u r f u f u + + − = − − 1 1 { , } i i S I I = + 1 2 ( ) i f u + 2 2 1 1 1 2 { } { , }; { , } { } L i i i R i i i S I I I S I I I = ∪ − + = + ∪ + 1 1 1 1 2 2 1 1 2 2 1 2 1 1 1 2 1 ( , ) ( , ) ( , ) ( , ) ( ) ; ( ) i i i i i i i i L L R R i i i i N x x N x x N x x N x x N S N S x x x x + − + + + + − + + − − = = − − 1 2 1 1 1 ( ) ( , )( ) ( )( ) i i i i i i i i p x u N x x x x N x x x x = + + − + − + − − i 1/2 x x = + 1/ 2 1/ 2 1/ 2 1/ 2 ( ), ( ) ( ( )) n n i i i i u P x f u f P x + + = = + +
2. Weighted eno methods Weighted eno saves all of possible cell-stencil and combines their corresponding effects to construct a weighted approximation polynomial For example: for 3 three-cell stencil case 1.1 (13) 1)Construct the three Newton polynomials P(x)by means of consistency conditions (r) (14) P1(5)=a0+a15+a22,5=x-x 2)Calculate the values l1-1+-l 1,1,l},an u (15) ={/2,l+2l2 1+=+--l 3)Weighted plus to get u, +/2 the numerical flux (r) and f(u n) (16)
2. Weighted ENO methods Weighted ENO saves all of possible cell-stencil and combines their corresponding effects to construct a weighted approximation polynomial. For example: for 3 three-cell stencil case (13) 1) Construct the three Newton polynomials by means of consistency conditions (14) 2) Calculate the values (15) 3) Weighted plus to get the numerical flux and (16) 2 2 2 (1) 2 1 ( 2 ) 1 1 (3) 1 2 { , , }, { , , }, { , , } i i i i i i i i i S I I I S I I I S I I I = = − − − + = + + ( ) ( ) 2 ( ) 0 1 2 1 ( ) , ( ) , j r j r I i r i P x dx u j S x x x P a a a x ξ ξ ξ ξ = ∈ ∆ − = + + = ∆ ∫ ( ) P x r 1 2 1 2 1 2 2 (1) (1) 2 1 2 1 2 (2) (2) 1 1 1 1 2 (3) (3) 1 2 1 2 1 7 11 { , , }, 3 6 6 1 5 1 { , , }, 6 6 3 1 5 1 { , , }, 3 6 6 i i i i i i i i i i i i i i i i i i i i i S I I I u u u u S I I I u u u u S I I I u u u u − − + − − − + − + + + + + + + = = − + = = − + + = = + − 1 1 2 2 3 ( ) 1 n r i i r r u w u + + = = ∑ 1/ 2 n i u + 1 2 ( ) i f u +
Fig. I The numerical results of circle dam-breaking by WENO (Z.L. Xu&R X Liu
Fig.1 The numerical results of circle dam-breaking by WENO (Z.L.Xu&R.X.Liu)