Harlow&Welch's MAC(Marker and cell) PIC (particle in cell, Evan and Harlow, 1957), MAC(Marker and cell Harlow and Welch, 1965), FLIC (Fluid in cell, Gentry, Martin and Daly, 1966), ALE (Arbitrary Lagrange and euler) MAC method: Marker technique By tracking these markers based on the velocity-field of flow, we can finely numerically simulate the free surface of moving interface Tracking markers----------Lagrange -computation ax u(x, y, t) n+1 n+1 +1 +.(x(,y(O),)t dy,(t) or m=v(x,y, t) ymt=ymt+l. v(x(t),y(t), t)dt
2-2.Harlow&Welch’s MAC (Marker and cell) PIC (particle in cell,Evan and Harlow,1957),MAC (Marker and cell, Harlow and Welch,1965), FLIC (Fluid in cell, Gentry,Martin and Daly,1966), ALE (Arbitrary Lagrange and Euler). MAC method : Mmarker technique .By tracking these markers based on the velocity-field of flow, we can finely numerically simulate the freesurface of moving interface. Tracking markers----------Lagrange-computation. or (2) ( ) ( , , ) ( ) ( , , ) m m dx t u x y t dt dy t v x y t dt = = 1 1 1 1 1 1 ( ( ), ( ), ) ( ( ), ( ), ) n n n n t n n m m t t n n m m t x x u x t y t t dt y y v x t y t t dt + + + + + + = + = + ∫ ∫
B Leonard's QUICK(quadratic upstream interpolation for convective kinetics) Leonard (1979)used a three-point upstream weighted quadratic interpolation to construct the numerical flux F(Uu) at the discontinuous point (the cell interface )x=x+1/2 gU7+3U1一U1,L1>0 U,+3U-1U,L.,<0 (3) F(U1+12)=[A(U)Ul+1a2 Patanka and Spalding s SIMPLE (Semi-Implicit Method for Pressure-Linked Equation, 1972)
2-3.Leonard’s QUICK (quadratic upstream interpolation for convective kinetics) Leonard (1979) used a three-point upstreamweighted quadratic interpolation to construct the numerical flux at the discontinuous point (the cell interface) . (3) and Patanka and Spalding’s SIMPLE (Semi-Implicit Method for Pressure-Linked Equation,1972) 1/ 2 ( ) F Ui+ 1 1/ 2 x x = + 1/ 2 1/ 2 ( ) [ ( ) ] F Ui i + + = A U ⋅U 1 2 1 2 1 2 6 3 1 8 8 1 1 8 6 3 1 8 8 1 2 8 , 0 , 0 i i i i i i i i i U U U u U U U U u + − + + + + + ⎧ + − > ⎪ = ⎨ + − < ⎪⎩
4. van Leer's MUSCL(monotonic upstream scheme for conservation law) van Leer ( 1979)uses the approximations Uiy at the time level t=t,to directly reconstruct U(, tn+ndx the 2nd polynomial approximation of the integrand of above integral based on characteristics property The third order scheme has the lateral values of the cell interface x=xu12(K=3 is a third order muscl scheme) 2=U1+[(1-x)△U1+(1+x)△U (5a) U/1a2=Ul1-4(1+x)△U1+(1-x)△U In order to restrain the oscillations by inserting a flux limiter is effective strategy i.e. the scheme(5a should be replaced as (5b)
2-4.van Leer’s MUSCL (monotonic upstream scheme for conservation law) van Leer (1979) uses the approximations at the time level to directly reconstruct (4) the 2nd polynomial approximation of the integrand of above integral based on characteristics property. The third order scheme has the lateral values of the cell interface ( is a third order MUSCL scheme) (5a) In order to restrain the oscillations by inserting a flux limiter is effective strategy, i.e. the scheme (5a) should be replaced as (5b) { }n Ui i ∀ n t t = 1 1 1 ( , ) i n i n I U U x t dx x + = + ∆ ∫ 1 1/ 2 4 1 1 1/ 2 1 4 1 [(1 ) (1 ) ] [(1 ) (1 ) ] L i i i i R i i i i U U U U U U U U κ κ κ κ + − + + + = + − ∆ + + ∆ = − + ∆ + − ∆ 1 1/ 2 4 1 1 1/ 2 1 4 1 [(1 ) (1 ) ] [(1 ) (1 ) ] L i i i i R i i i i U U U U U U U U κ κ κ κ + − + + + = + − ∆ + + ∆ = − + ∆ + − ∆ 1 1/ 2 x x = + 13 κ =
2-5. Collela's PPM(piecewise parabolic method) Collela and Woodward (1984) proposed PPM by piecewise parabolic polynomial interpolation to the definition(24) For example the ppm scheme of a scalar equation will be v(x)=l11+5(41+l6(1-5) x △l1=lR;-1L lim (x) lin Im (l+l21) MUSCL van eer 1979) piecewise linear Collet Woodwar piecewise parabolic (1984) Fig. 6 The reconstruction character of left and right limit values forMUSCL and PF
2-5.Collela’s PPM (piecewise parabolic method) Collela and Woodward (1984) proposed PPM by piecewise parabolic polynomial interpolation to the definition (24). For example, the PPM scheme of a scalar equation will be (6) Fig.6 The reconstruction character of left and right limit values forMUSCL and PPM 1 2 1 1 2 2 1 1 2 2 , 6 , , , , , 1 6 , 2 , , ( ) ( (1 )), lim ( ); lim ( ) 6( ( )) i i i L i i i i i i R i L i R i x x L i x x n i i R i L i x x v x u u u x x x x u u u u u x u u x u u u u ξ ξ ξ + − − − + ↑ ↓ ⎧ − ⎪ = + ∆ + − = ≤ ≤ ⎪ ∆ ⎪ ∆ = − ⎨⎪ = = ⎪⎪ = − + ⎩ piecewise linear PPM Collela &Woodwar d (1984) piecewise parabolic MUSCL van eer (1979)
2-6. Harten's TVD(total variation diminishing schemes Harten(1983) first introduced TVD scheme including Limiter Contribution TVD character: Reconstruction: Limiter TVDT(Um)sTW(",T(")=△∑An TTheory For scalar conservation law u,+(f(u)=0, a(u)=af(u)/au Jacobian matrix (8) a scheme consistent with it can be written as -1/2 (l2-121)+D+12(l+1-l1) 9 if the following conditions are satisfied 1+1/20. D12≥0 0≤Cm2+D12s1 it is also a TVd one Monotone scheme iS TVD A TVD Scheme is monotonicity preserving
2-6.Harten’s TVD (total variation diminishing schemes) Harten (1983) first introduced TVD scheme including Limiter Contribution: TVD character; Reconstruction: Limiter TVD: (7) [Theory] For scalar conservation law , (8) a scheme consistent with it can be written as (9) if the following conditions are satisfied (0) it is also a TVD one. Monotone scheme is TVD. A TVD scheme is monotonicity preserving. 1 ( ) ( ), ( ) n n n ni i TV U TV U TV U x U + ≤ = ∆ ∑ ∆ ( ( )) 0, ( ) ( )/ t x u + f u = = a u ∂f u ∂u Jacobian matrix 1 1/ 2 1 1/ 2 1 ( ) ( ) n n n n n n i i i i i i i i u u C u u D u u + = − − − − + + + − 1/ 2 1/ 2 1/ 2 1/ 2 0, 0 0 1 i i i i C D C D + + + + ⎧ ≥ ≥ ⎨⎩ ≤ + ≤