3-2. FVM (finite volume methods) with unstructured meshes Consider 2D shallow water equations(in fact, depth-averaging 3D inviscid incompressible flow considering Coriolis force, and the change of river-bed U1+(F(U)2+(G(U),=S h nul (17a) U=hu, F()=hu/2+igh?G(U) h S=-gh as+ fih uv hv+28 hb-fuh or their compact form U,+V F(U=S, F(U=F(U),G(U) (17b) Integrating above equations over a control volume of unstructured mesh(Fig. 1) and using green Theorem, we can obtain dU dt ∑ lk∈ov
3-2.FVM (finite volume methods) with unstructured meshes Consider 2D shallow water equations (in fact, depth-averaging 3D inviscid incompressible flow considering Coriolis force, and the change of river-bed ) (17a) or their compact form (17b) Integrating above equations over a control volume of unstructured mesh (Fig.1) and using Green Theorem, we can obtain (18) 2 2 1 2 2 2 1 2 ( ( )) ( ( )) 0 , ( ) , ( ) , b b t x y z x z y U F U G U S h hu hv U hu F U hu gh G U hvu S gh fvh hv huv hv gh gh fuh ∂ ∂ ∂ ∂ + + = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = = + = = ⎜ ⎟ − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ + ⎜ ⎟ − − ⎝ ⎠ ( ) , ( ) ( ( ), ( )) U F t U S F U F U G U ∗ ∗ + ∇ ⋅ = = 1 ( ) | | | | k i k i i k l k V V i l V dU F n l Sdv dt V ∗ ∈∂ = − ∑ ⋅ + ∫
(0,200)(100,200)(200.200) (100,170) (100,95) (0,0)(100,0)(200,0) 3° Fig.2 The numerical simulation results by FVM(JwWang&RXLiu
(0, 200) (100, 200) (200, 200) (100, 170) (100, 95) (0, 0) (100, 0) (200, 0) Flow 0 50 100 150 200 0 25 50 75 100 125 150 175 200 6 8 10 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 25 50 75 100 125 150 175 200 9 . 2 1 8. 85 8.49 8 . 1 4 8 . 1 4 7.78 7 .42 7 .06 6.7 6.34 5 .27 6.34 5.99 5 .63 5 . 2 7 5.63 5.27 1 .2 5 1 .5 .7 5 2 0 1 5 30 45 6 0 75 9 0 0 1 0 2 0 30 40 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 1 .06 1 .12 1 .36 1 .79 1.92 1.67 1.55 1 .55 1.55 1.61 1.67 1.67 1.61 1 .67 1.67 1.36 1.42 1.36 1.42 1.42 1.61 Fig.2 The numerical simulation results by FVM (J.W.Wang&R.X.Liu)
3-3. Rational approximation methods, high order compact and Pad schemes The runge(gibbs)phenomeno of polynomial interpolation iPo(r) f(x)= x∈[-5,5] Ps(R) 3-2
3-3.Rational approximation methods, high order compact and Pad schemes The Runge (gibbs) phenomeno of polynomial interpolation 2 1 ( ) , [ 5,5] 1 f x x x = ∈ − +
S KLele(1992, J. Comput. Phys., Vol 103, pp16-42) High order compact- or Pade finite difference scheme Bm2+2)+a(+)+f厂=a+b43+c4 B(m2+2)+a(m+m")+"=a3+b22+cx3 By Taylor expansion one can obtain the relations between the coefficient relations a+b+C=1+2a+2B a+2 b+3"c=2 m (a+2 B),(m+2)-order approximation error AF'=BF,F=(1,…,f1 (21) AF"=BF,F=(f1…,)
S.K.Lele (1992, J. Comput.Phys.,Vol.103,pp16-42) High order compact- or Pade finite difference scheme. Let (19) By Taylor expansion one can obtain the relations between the coefficient relations (20) or (21) 1 1 2 2 3 3 1 1 2 2 3 3 2 2 2 2 2 1 1 2 4 6 2 2 2 2 2 1 1 4 9 ( ) ( ) ( ) ( ) i i i i i i i i i i i i i i i f f f f f f i i i i i x x x f f f f f f f f f i i i i i x x x f f f f f a b c f f f f f a b c β α β α + − + − + − + − + − + − − − − + − + − ∆ ∆ ∆ − + − + − + + − + − ∆ ∆ ∆ ′ ′ + + ′ + ′ + ′ = + + ′ + + ′ ′′ + ′′ + ′′ = + + ( 1)! ! 1 2 2 2 3 2 ( 2 ), ( 2) m m m m m abc a b c m order approximation error α β α β + + + = + + + + = + + − ( ) 1 1 1 , , , T AF x BF F N f f ∆ − ′ = = " ( ) 1 1 1 , , , T AF x BF F N f f ∆ − ′′ = =
Fig 3 Steady-state stream and vorticity distributions for lid-driver cavity problem at Re=1oo(left)and 1000 (right )simulated by Compact method
Fig.3 Steady-state stream and vorticity distributions for lid-driver cavity problem at Re=100(left) a nd 1000 (right) simulated by Compact method