W τ R W j

N H = 2

F obj

b q N F aug

F obj b q

Ψ F aug Ψ

( ) ϱ t + + p = 0 = 0 Ω f = Γ Γ b ϱ = (, t) = (, t) Ω f Γ b ( ) ϱ t + + p =

V max 4 3 2 1 0-1 -2-3 -4-4 -3-2 -1 0 1 2 3 4

x 4 x 1 V mn V max (V mn, V max ) V mn

x = d d { max 1 x > 1 f = 2x 3 + 3x 2 0 < x < 1 V crt = fv max + (1 f)v mn d d max V crt V crt < V cell

(, j) L = (log 2 + 1) = (log 2 j + 1) x (,j) = x (1,1) 2 L 1 y (,j) = y (1,1) 2 L 1 x (,j y (,j) x (1,1) y (1,1) V = V (,j) (2 L 1 ) 2 = V (,j) (4 L 1 ) = x (1,1) y (1,1) (4 L 1 ) x (,j) = x (1,1) 3 2 x (1,1) + ( + 1 2 ) x (1,1) 2 L 1 y (,j) = y (1,1) 3 2 y (1,1) + (j + 1 2 ) y (1,1) 2 L 1 (, j)

V cell V cell < V mn V cell < V max

t + x x + y y = 0 x y ϱ = ϱu ϱv x = E t ϱu ϱu 2 + p ϱuv u ( E t + p ) y = ϱv ϱuv ϱv 2 + p v ( E t + p ) ϱ u v x y E t p

E t = p γ 1 + 1 ( 2 ϱ u 2 + v 2) γ W t + f j x j = 0 = 1,..., 4 j = 1, 2 x 1 = x x 2 = y R R = W t + f j x j = 0 Ω

Ω Ω R dω = Ω ( W t + f j x j ) dω = 0 Ω ( fj ) dω x j

Ω ) dω = x j ( fj S f j n j ds j = 1, 2 n j x y

( ) fj n j ds f j n j S S faces n j S f j n j ϱ ( ) f j n j #» ϱu ( ) + pn x = ϱv ( ) + pn ( y Et + p )( ) #» = ( x, y ) = ( n x, n y ) = ( u, v ) P P

P P P = P P P P = P ± P x ± 1 2 P x 2! x 2 x 2 ± 1 3 P 3! x 3 x 3 ±... P / x P Q = P + ( ) P Q = Q + ( ) Q P ( ) P = ( ) P x, P ( ) ( ) Q y Q = x, Q y

P Q P + P x Q P + P x P x ( ) P ( ) xq x P + yq y P y x + P y x + P y y y 0 E = N n=0 [ P x x n + P y y n n ] 2 N N Q 1 Q 2 Q 6 P P Q5 Q 3 Q 4 Q 5

E ( P x E ( P x ) = ) = { [ } N P 2 x x n + P y y n n ] x n = 0 { [ } N P 2 x x n + P y y n n ] y n = 0 n=1 n=1 P x P x N { } P xn x n + n=1 y N { } P yn x n + n=1 y N { } N { } yn x n = xn n n=1 n=1 N { } N { } yn y n = yn n n=1 n=1 2 2 { } xn x N { }] P [ n n=1 yn x N { }] n N { } n=1 yn x N { } x n n=1 yn y n P n=1 xn n = N { } n=1 yn n y [ N n=1 ( ) P ( ) = P + Π P P W

= P ( ) mn = mn, Q ( ) max = max, Q Φ(z) = x2 + 2x x 2 + x + 2 [2, 8] Π P 1 W j = W P j ( Wmax,j Wj P Π Pj = Φ W j Wj P ( Wmn,j Wj P Φ W j Wj P ) ) W j > W P j W j < W P j j j = 1,..., 4 W j P Q x = x ( P, Q )

P Q P Q P x Q = 2( 1 P x + Q ) 1 x P Q y = 1 2( P y + Q y 2 Ãx ( Q P ) ) 1 2 Ãy ( Q P ) f P Q j = 1 f P 2( j + f Q ) 1 j 2 Ãj ( W Q W P ) = 1,..., 4 j = 1, 2 à Ãj 4 4 Ãj j j = 1, 2 x, y P Q à j ( P, ) A j ( ) A j = j / Ãj( P, ) ( P ) = P j Q j Ãj Ãj à j = P j Λ j P 1 j Λ j P j Ãj j A x

A y Ãx Ãy λ 1x = ũ + c λ 2x = ũ c λ 3x = ũ λ 4x = ũ λ 1y = ṽ + c λ 2y = ṽ c λ 3y = ṽ λ 4y = ṽ x y ũ c ϱ = ϱ P ϱ Q ϱp u P + ϱ Q u Q ũ = ϱp + ϱ Q ϱp v P + ϱ Q v Q ṽ = ϱp + ϱ Q h ϱp h P + c = (γ 1)[ h 1 ] (4.2.16) ϱ Q h Q = ϱp + 2ũ2 ϱ Q Λ x P x Λ y P y Ãj 4 4 Λ x = Λ y = ũ + c 0 0 0 0 ũ c 0 0 0 0 ũ 0 0 0 0 ũ ṽ + c 0 0 0 0 ṽ c 0 0 0 0 ṽ 0 0 0 0 ṽ P x = P y = 1 1 1 0 ũ + c ũ c ũ 0 ṽ ṽ ṽ 1 h + ũ c h ũ c 1 2 q2 ṽ 1 1 1 0 ũ ũ ũ 1 ṽ + c ṽ c ṽ 0 h + ṽ c h ṽ c 1 2 q2 ũ q q 2 = ũ 2 + ṽ 2

f j n j S = faces k=faces { [1 ( f P 2 j + f Q,k ) 1 j 2 Ãk j ( W Q,k W P,k) ] } n k j S k k ( P ) à k j = P j Λ j P 1 k j f P j n k j = f P,n f Q,k j n k j = f Q,k,n à k j n k j = Ãk n f j n j S = faces k=faces { [1 ( f P 2,n + f Q,k ) 1,n 2 Ãk n ( W Q,k W P,k) ] } S k P Q Λ P P Ãj

Ω P ( W ) dω t Ω P ( W ) dω = W dω t t Ω P l l = 1, 2 x, y V grd l Ω P n l W d( Ω) Ω P ( W ) dω = W dω t t Ω P W W dω = W t Ω P t ΩP W t Ω P ( 3W P,m+1 m+1 4W P,m + W P,m 1 ) Ω P 2 t (m 1) m (m + 1) Ω P P t (m+1)

τ τ + t + x x + y y = 0 W τ + W t + f j x j = 0 W τ m +1 ΩP ( W P,m +1 ) Ω P W P,m τ P m (m + 1) (m + 1) τ P

τ P = CF L ΩP C CF L ( V ) P C = k + c P Sk P Sk P = 1 { n P Q } 2 k S k faces k = 1, 2 x, y Vk P P c P n P Q k S k P R P = k=faces { [1 ( f P 2,n + f Q,k ) 1,n 2 Ãk n ( W Q,k + W P,k) ] } S k + ( 3W P,m+1 4W P,m + ( W P,m +1 + W P,m 1 ) Ω P 2 t W P,m ) Ω P τ P

( ) 0 = + ( ) = ( + ) = 0 R ( W j ) = R ( Wj + W j ) = 0 = 1,..., 4 j = 1,..., 4 R ( W j ) = R ( Wj + W j ) = R ( Wj ) + R W j W j + = 0 W j R W j W j = R j 4 4

[ ] [ ] NUMERICS W j = P HY SICS [ ] NUMERICS [ ] P HY SICS M 4 4 M M

4 4 4M 4M M M M M

RP W P j P, j = 1,..., 4 j = 1,..., 4 ( dag ) P Roe = W P j ( k=faces k=faces { [1 ( f P 2,n + f Q,k ) 1,n 2 Ãk n ( W Q,k W P,k) ] }) S k = { 1 2 ( f,n P + f Q,k ),n Wj P S k 1 2 Ãk n ( W Q,k W P,k) Wj P S k 1 ( Ãk n ) } ( W Q,k W P,k) S k 2 W P j ( f Q,k ),n W P j Q P ( W Q W P j,k ) Q P ( Ãk n ) W P j ( W Q,k W P,k) ( Ãk n ) W P j R W j

( dag ) P Roe = k=faces { 1 2 ( f,n P ) W P j S k 1 2 Ãk n ( W P W P j,k ) } S k ( f,n P ) W P j P A P n ( W P,k ) W P j ( dag ) P Roe = 1 2 k=faces { } ( ) A P n Ãk n S k ( dag ) P tme = W P,m+1 j ( [3W P,m+1 4W P,m + W P,m 1 ] Ω P ) 2 t m + 1 m 1 m m + 1 ( dag ) P tme = 3 ΩP 2 t I I 4 4 ( dag ) P pseudo = W P,m +1 j ( [W P,m +1 ]Ω P ) W P,m τ

(m + 1) m ( dag ) P pseudo = ΩP τ P I ( dag ) P = 1 2 k=faces { } ( ) A P n Ãk n S k + 3 ΩP 2 t I + ΩP τ P I P Q P ( off dag ) P,k Roe = W Q,k j ( = k=faces k=faces { { [1 ( f P 2,n + f Q,k ) 1,n 2 Ãk n ( W Q,k W P,k) ] }) S k 1 2 ( f,n P + f Q,k ),n W Q,k j S k 1 2 Ãk n ( W Q,k W P,k) 1 ( Ãk n ) 2 W Q,k j W Q,k j S k ( W Q,k W P,k) S k } P ( f,n P ) W Q,k j ( f Q,k ),n W Q,k j

( W P,k W Q,k j ) P ( Ãk n ) W Q,k j ( W Q,k W P,k) S k [ ] ( ) P,k off dag Roe = 1 ( ) A Q,k n 2 Ãk n S k A Q,k n Q ) ( W Q ) ( f P,k,n W Q,k j W Q j ( off dag ) P,k tme = W Q,m+1 j ( [3W P,m+1 k 4W P,m + W P,m 1 ] Ω P ) 2 t P Q ( off dag ) P,k tme = 0 ( off dag ) P,k pseudo = 0 P [ ] ( ) P,k 1 ( ) off dag = A Q,k n 2 Ãk n S k

P [ ] 1 ( ) ( ) P,k A Q,k n off dag = 2 Ãk n S k, 0, (dag) 1 m.................. + m 1... (off) P k m... (dag)p m... (off) P k m... + = m P.................. (dag) M m + m M m P (dag) P m P,m +1 = P,m {(off dag) P m P,m } cells new P = [ [ (dag) P ] 1 old P old { (off dag) P old old} ] P cells P

= 0 0 #» pn x = wall pn y 0

+ P M

0 m = 1 m m = 0 W m,m R P, (dag) P, (off dag) P j = 0 P new m,m m,m +1 m = m + 1 j = j + 1 P old = P new m = m + 1 j < j jacob m,m +1 = m + new m,m +1 R P RMS(R P ) < ε t m < t total

N H 2N H + 1 2N H + 1

d(t) dt + (t) = 0 = () N H (t). = 0 + (t). = 0 + N H n=1 N H n=1 { an cos(ωnt) + bn sn(ωnt)} { an cos(ωnt) + bn sn(ωnt)} ω ω = 2π T T 2N H + 1 d { } (t) = d { 0 + dt dt d(t) dt 0 d 0 = + dt N H = d(t) dt n=1 N H n=1 { an cos(ωnt) + } } bn sn(ωnt) d{ NH n=1 { an cos(ωnt) + bn sn(ωnt) dt { ωn an sn(ωnt) + ωn bn cos(ωnt) } } }

0 + N H n=1 { ancos(ωnt) + bn sn(ωnt)} + 0 + N H n=1 N H n=1 { } ωn ansn(ωnt) + ωn bn cos(ωnt) = 0 { ( an + ωn bn ) cos(ωnt) + ( bn ωn an ) sn(ωnt) } 2N H + 1 0 = 0 an + ωn bn = 0 n = 1,..., N H bn ωn an = 0 n = 1,..., N H 4N H + 1 2N H +1 an bn an bn (t) (t) an bn an bn T N T = 2N H + 1 #» HB = (t 0 ) (t 0 + t) (t 0 + T t) #» HB = (t 0 ) (t 0 + t) (t 0 + T t) t T t = 2N H + 1 = 2π 2N H + 1 1 ω = α 2π α = ω 2N H + 1 1 ω 2N H + 1

2N H + 1 2N H + 1 #» HB = t T t = 2 2 + 1 = T 5 = 2π 5 1 ω = α ω α = 2π 5 N H = 2 (t) = 2 0 + { an cos(ωnt) + bn sn(ωnt)} n=1 (t) = 0 + a1 cos(ωt) + b1 sn(ωt) + a2 cos(2ωt) + b2 sn(2ωt) 0 a1 b1 a2 b2 #» HB #» HB = (t 0 ) (t 0 + t) (t 0 + 2 t) (t 0 + 3 t) (t 0 + T t) (0) #» (1) HB = (2) (3) (4) cos(ωt) = ejωt + e jωt 2 cos(2ωt) = ej2ωt + e j2ωt 2

sn(ωt) = ejωt e jωt 2j sn(2ωt) = ej2ωt e j2ωt 2j = j e jωt e jωt 2 = j e j2ωt e j2ωt 2 (t) = 0 + e jωt + e jωt a1 + j 2 e jωt e jωt b1 2 + a2 ej2ωt + e j2ωt + j 2 e j2ωt e j2ωt b2 2 = 0 + e j2ωt[ a2 + j ] b2 + e jωt[ a1 + j ] b1 2 2 +e jωt[ a1 j ] b1 + e 2jωt][ a2 j ] b2 2 2 2,, 2 2 = a2 + j b2 2 1 = a1 + j b1 2 1 = a1 j b1 2 2 = a2 j b2 2 0 = 0 (5.2.5) (t) = 2 e j2ωt + 1 e jωt + 0 + 1 e jωt + 2 e j2ωt T #» HB = [ 0 1 2 3 4 ] T t 0 = 0 t = 0 t 0 = 2 e 0jω t + 1 e 0jω t + 0 + 1 e 0jω t + 2 e 0jω t t = 1 t 1 = 2 e j2ω t + 1 e jω t + 0 + 1 e jω t + 2 e j2ω t t = 2 t 2 = 2 e j4ω t + 1 e j2ω t + 0 + 1 e j2ω t + 2 e j4ω t t = 3 t 3 = 2 e j6ω t + 1 e j3ω t + 0 + 1 e j3ω t + 2 e j6ω t t = 4 t 4 = 2 e j8ω t + 1 e j4ω t + 0 + 1 e j4ω t + 2 e j8ω t

e ±jkω t = ϕ ±k k = 8,..., 8 t = 0 t 0 = 2 ϕ 0 + 1 ϕ 0 + 0 ϕ 0 + 1 ϕ 0 + 2 ϕ 0 t = 1 t 1 = 2 ϕ 2 + 1 ϕ 1 + 0 ϕ 0 + 1 ϕ 1 + 2 ϕ 2 t = 2 t 2 = 2 ϕ 4 + 1 ϕ 2 + 0 ϕ 0 + 1 ϕ 2 + 2 ϕ 4 t = 3 t 3 = 2 ϕ 6 + 1 ϕ 3 + 0 ϕ 0 + 1 ϕ 3 + 2 ϕ 6 t = 4 t 4 = 2 ϕ 8 + 1 ϕ 4 + 0 ϕ 0 + 1 ϕ 4 + 2 ϕ 8 ϕ ϕ ±k = e ±jkω t e ±jkω t = cos(±kω t) + jsn(±kω t) 2N H + 1 ϕ p = ϕ p±q(2n H+1), p, q Z t = 0 t 0 = 0 ϕ 0 + 1 ϕ 0 + 2 ϕ 0 + 2 ϕ 0 + 1 ϕ 0 t = 1 t 1 = 0 ϕ 0 + 1 ϕ 1 + 2 ϕ 2 + 2 ϕ 3 + 1 ϕ 4 t = 2 t 2 = 0 ϕ 0 + 1 ϕ 2 + 2 ϕ 4 + 2 ϕ 6 + 1 ϕ 8 t = 3 t 3 = 0 ϕ 0 + 1 ϕ 3 + 2 ϕ 6 + 2 ϕ 9 + 1 ϕ 12 t = 4 t 4 = 0 ϕ 0 + 1 ϕ 4 + 2 ϕ 8 + 2 ϕ 12 + 1 ϕ 16 ϕ 2 3 1 4 1 1 1 1 1 0 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 1 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 2 = 3 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 4 0 1 2 3 4

M 1 1 1 1 1 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 M = 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 M ϕ = e jω t t j2π = e T 1 + ϕ + ϕ 2 + ϕ 2N H = 0 1 + ϕ + ϕ 2 + + ϕ 2N H = 1 ϕ2n H+1, ϕ R 1 ϕ ϕ = e jω t 1 + ϕ + ϕ 2 + + ϕ 2N H = 1 e(2n H+1)j2π t T t j2π 1 e T (2N H +1) t=t ========= 1 + ϕ + ϕ 2 + + ϕ 2N H = 1 ej2π e j2π =1 t ==== j2π 1 e T 1 + ϕ + ϕ 2 + ϕ 2N H = 0, ϕ C k r q ϕ { 2N H ϕ k(r q) 0, r q = 2N H + 1, r = q k=0 r q r q = 1 2N H k=0 ϕ k 1 = ϕ 0 + ϕ 1 + ϕ 2 ϕ 2N H 2N H ====== ϕ k = 0 k=0

r q = 2 2N H k=0 ϕ k 2 = ϕ 0 + ϕ 2 + ϕ 4 ϕ 4N H N H = 2 4 ϕ k 2 = ϕ 0 + ϕ 2 + ϕ 4 + ϕ 6 + ϕ 8 k=0 ϕ 6 = ϕ 1 ϕ 8 = ϕ 3 4 ϕ k 2 = ϕ 0 + ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 k=0 r q r = q 2N H k=0 2N H ϕ 0 = {1} = 2N H + 1 k=0 M 1 1 1 1 1 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 M = 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 ϕ k k Z ϕ k t jk2π = e T = cos ( k 2π t ) ( 2π t) + jsn k T T ϕ k = cos ( k 2π t ) ( 2π t) jsn k = e jk2π t T = ϕ k T T M 1 1 1 1 1 1 1 1 1 1 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 M = 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 = 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 M 1

M M 1 1 1 1 1 1 1 1 1 1 1 5 0 0 0 0 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 0 5 0 0 0 M M = 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 = 0 0 5 0 0 = 5 I 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 0 0 0 5 0 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 0 0 0 0 5 M M 5 k=1 {ϕ ( 1)(k 1) ϕ ( 1)(k 1)} ϕ m =ϕ m 5 k=1 { ϕ ( 1)(k 1) ϕ ( 1)(k 1)} = 5 k=1 { ϕ 0} = 5 M M 5 {ϕ ( 1)(k 1) ϕ (j 1)(k 1)} ϕ m =ϕ 5 { m ϕ ( 1)(k 1) ϕ (j 1)(k 1)} = 5 { k=1 k=1 ϕ ( j)(k 1)} 4 { = ϕ k( j)} j 2N H +1 5 { ϕ ( 1)(k 1) ϕ (j 1)(k 1)} = 0 k=1 k=0 k=1 N H M M 1 = 1 5 M M 1 = M 1 2N H + 1 = M 1 N T 0,, 4 0 1 1 1 1 1 1 2 = 1 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 5 3 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 4 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 0 1 2 3 4 (t)

(t) (t) = 2 { 0 + an cos(ωnt) + bn sn(ωnt)} n=1 (t) = 0 + a1 cos(ωt) + b1 sn(ωt) + a2 cos(2ωt) + b2 sn(2ωt) 0 1 1 1 1 1 0 1 2 = 1 1 ϕ 1 ϕ 2 ϕ 3 ϕ 4 1 1 ϕ 2 ϕ 4 ϕ 6 ϕ 8 2 5 3 1 ϕ 3 ϕ 6 ϕ 9 ϕ 12 3 4 1 ϕ 4 ϕ 8 ϕ 12 ϕ 16 4 2 = 3 = a2 + j b2 2 1 = 4 = a1 + j b1 2 1 = a1 j b1 2 2 = a2 j b2 2 0 = 0 (5.2.14) 2 = 3 1 = 4 #» #» #» = [ 0 ] T #» 1 2 3 4 = [ ] T 0 1 2 3 4 = 0,..., 4 #» = [ 0 1 2 2 1 ] T #» = [ 0 1 2 2 1 ] T a2 b2 2 3 a2 b2 2 3

2 3 2 = 0 + ϕ 2 1 + ϕ 4 2 + ϕ 6 3 + ϕ 8 4 3 = 0 + ϕ 3 1 + ϕ 6 2 + ϕ 9 3 + ϕ 12 4 t j2π ϕ ϕ = e T 2 = 0 + 1 cos( 4π t T [ +j 1 sn( 4π t T 3 = 0 + 1 cos( 6π t T [ +j 1 sn( 6π t T ) + 2 cos( 8π t T ) + 2 sn( 8π t T ) + 2 cos( 12π t T ) + 2 sn( 12π t T t t ) + 3 cos( 12π ) + 4 cos( 16π T T ) t t ] ) + 3 sn( 12π ) + 4 sn( 16π T T ) t t ) + 3 cos( 18π ) + 4 cos( 24π T T ) t t ] ) + 3 sn( 18π ) + 4 sn( 24π T T ) ) ) Re ( 2 = Re ( 3 ) ) Im ( 2 = Im ( 3 ) ) Re ( 2 = Re ( 3 0 + 1 cos( 4π t T ) + 2 cos( 8π t T ) + 3 cos( 12π t T ) + 4 cos( 16π t T ) = 0 + 1 cos( 6π t T ) + 2 cos( 12π t T ) + 3 cos( 18π t T ) + 4 cos( 24π t T ) 1 cos( 4π t T ) + 2 cos( 8π t T ) + 4 cos( 6π t t 2 5π T T ) = 1 cos( 6π t T ) + 3 cos( 8π t T t 2 5π T ) + 4 cos( 4π t t 4 5π T T ) 5 t/t=1 ========== 1 cos( 4π t cos(a±2kπ)=cos(a) T ) + 2 cos( 8π t T ) + 4 cos( 6π t T ) = 1 cos( 4π t T ) + 3 cos( 6π t T ) + 4 cos( 8π t T ) 0 = 0 ) ) Im ( 2 = Im ( 3 1 sn( 4π t T ) + 2 sn( 8π t T ) + 3 sn( 12π t T ) + 4 sn( 16π t T ) = 1 sn( 6π t T ) + 2 sn( 12π t T ) + 3 sn( 18π t T ) + 4 sn( 24π t T ) 1 sn( 4π t T ) + 2 sn( 8π t T ) + 4 sn( 6π t t 2 5π T T ) = 1 sn( 6π t T ) + 3 sn( 8π t t 2 5π T T ) + 4 sn( 4π t t 4 5π T T )

5 t/t=1 ========== 1 sn( 4π t sn(a±2kπ)=sn(a) T ) + 2 sn( 8π t T ) + 4 sn( 6π t T ) = 1 sn( 4π t T ) + 3 sn( 6π t T ) + 4 sn( 8π t T ) 0 = 0 #» #» #» #» 0 = 0 = 1 [ ] 0 + 1 + 2 + 3 + 4 5 a1 = 2Re ( 1 ) 2 = 5 Re[ 0 + ϕ 1 1 + ϕ 2 2 + ϕ 3 3 + ϕ 4 ] 4 b1 = 2Im ( 1 ) 2 = 5 Im[ 0 + ϕ 1 1 + ϕ 2 2 + ϕ 3 3 + ϕ 4 ] 4 a2 = 2Re ( 2 ) 2 = 5 Re[ 0 + ϕ 2 1 + ϕ 4 2 + ϕ 6 3 + ϕ 8 ] 4 b2 = 2Im ( 2 ) 2 = 5 Im[ 0 + ϕ 2 1 + ϕ 4 2 + ϕ 6 3 + ϕ 8 ] 4 ϕ k, k Z a = ω t 0 = 1 [ ] 0 + 1 + 2 + 3 + 4 5 a1 = 2 [ ] 0 + cos(a) 1 + cos(2a) 2 + cos(3a) 3 + cos(4a) 4 5 b1 = 2 [ ] 0 0 + sn(a) 1 + sn(2a) 2 + sn(3a) 3 + sn(4a) 4 5 a2 = 2 [ ] 0 + cos(2a) 1 + cos(3a) 2 + cos(6a) 3 + cos(8a) 4 5 b2 = 2 [ ] 0 0 + sn(2a) 1 + sn(4a) 2 + sn(6a) 3 + sn(8a) 4 5 0 1 1 1 1 1 a1 b1 = 1 2 2 cos(a) 2 cos(2a) 2 cos(3a) 2 cos(4a) 0 2 sn(a) 2 sn(2a) 2 sn(3a) 2 sn(4a) 5 a2 2 2 cos(2a) 2 cos(4a) 2 cos(6a) 2 cos(8a) 0 2 sn(2a) 2 sn(4a) 2 sn(6a) 2 sn(8a) b2 0 1 2 3 4

E 1 1 1 1 1 E = 1 2 2 cos(a) 2 cos(2a) 2 cos(3a) 2 cos(4a) 5 0 2 sn(a) 2 sn(2a) 2 sn(3a) 2 sn(4a) 2 2 cos(2a) 2 cos(4a) 2 cos(6a) 2 cos(8a) 0 2 sn(2a) 2 sn(4a) 2 sn(6a) 2 sn(8a) #» #» 0 = 0 ω a1 = b1 ω b1 = a1 2ω a2 = b2 2ω b2 = a2 0 0 0 0 0 0 0 0 0 1 0 0 a1 a1 ω 0 1 0 0 0 b1 = b1 0 0 0 0 2 a2 a2 0 0 0 2 0 b2 b2 C C = 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 2 0 ωc #» = #» #» =E #» #» =E #» ==== ωce #» = E #» ωe 1 CE #» = #» ωd #» + #» = 0 D D = E 1 CE D E C E 1 (t) E 1 = 1 1 0 1 0 1 cos(a) sn(a) cos(2a) sn(2a) 1 cos(2a) sn(2a) cos(4a) sn(4a) 1 cos(3a) sn(3a) cos(6a) sn(6a) 1 cos(4a) sn(4a) cos(8a) sn(8a)

D E 1 C 0 0 1 0 2 E 1 0 sn(a) cos(a) 2 sn(2a) 2 cos(2a) C = 0 sn(2a) cos(2a) 2 sn(4a) 2 cos(4a) 0 sn(3a) cos(3a) 2 sn(6a) 2 cos(6a) 0 sn(4a) cos(4a) 2 sn(8a) 2 cos(8a) D D,j = 2 [ sn [ a(j ) ] + 2sn [ 2a(j ) ]] 5 = 1,..., 5 j = 1,..., 5 t 0 = 0 1 1 1 1 2 2 cos(a) 2 cos(2a) 2 cos [ (N T 1)a ] E = 1 N T 0 2 sn(a) 2 sn(2a) 2 sn [ (N T 1)a ] 2 2 cos(n H a) 2 cos(2n H a) 2 cos [ (N T 1)N H a ] 0 2 sn(n H a) 2 sn(2n H a) 2 sn [ (N T 1)N H a ] E D 1 1 0 1 1 1 cos(a) sn(a) cos(n Ha) sn(n Ha) E 1 = 1 cos(2a) sn(2a) cos(n H2a) sn(n H2a) 1 cos [ (N T 1)a ] sn [ (N T 1)a ] cos [ (N T 1)N ] Ha sn [ (N T 1)N ] Ha C D n = 2n j = 2n + 1 C,j = n = 2n + 1 j = 2n 0

1 n N H n N D D,j = 2 N T N H k=1 { ksn [ ak(j ) ]} d #» dτ + ωd #» + #» = 0 d dt d dt = ωd N T = 2N H +1

0 0 0 0 1 1 1 1 2 2 3 2 2NH 2NH 2NH 2NH N T #» #» #» = E #»

N T D d #» = 0 dt

t + x x + y y = 0 = 1,..., 4 j = 1, 2 W t + f j x j = 0 ωd #» + #» = 0 #»

#» #» = #» x x + #» y y ωd #» ωd #» + #» x x + #» y y = 0 ωd µν W ν + f µ j x j = 0 [1, 4] j µ ν [1, 2N H + 1] µ 2N H + 1 ν 2N H + 1 2N H + 1

Ω ( ωd µν W ν + f µ ) j dω = 0 x j µ µ µ f j n j S = { [1 ( f P µ 2,n + f Q,k ) 1,n 2 Ãk n ( W Q,k W P,k) ] } S k faces k=faces µ µ ωd µν W ν dω ωd µν W ν dω = ωd µν W ν Ω P Ω P Ω Ω ( ωd µν W ν + f µ ) j dω x j k=faces { [1 2 ( f P,n + f Q,k ) 1,n 2 Ãk n ( W Q,k W P,k) ] } S k µ +ωd µν W ν Ω P

W µ τ + ωd µν W ν + f µ j x j = 0 dw µ dτ µ Ω W µ τ µ W dω τ ΩP W µ τ ΩP ( W P,m +1 ) W P,m µ Ω P τ P Ω ( W µ τ + ωd µν W ν + f µ j x j + { [1 k=faces ) dω ( W P,m +1 2 ( f P,n + f Q,k ) 1,n ) W P,m µ Ω P τ P + ωd µνw ν Ω P 2 Ãk n ( W Q,k W P,k) ] } S k µ #» P R P µ #» P Res P µ

( Wl P ωd µν W ν Ω) P l [1, 4] Wl P µ ( Wl P ωd µ1 W 1 Ω + ωd µ2 W 2 Ω + + ωd µ2nh +1W 2NH +1Ω) P µ (ωd µνw ν ) Wl P ν Wl P µ D µ ( dag ) P µ = [ 1 2 k=faces { } ] ( ) A P n Ãk n S k + ΩP τ P I µ

W Q,o l ( ωd µν W ν Ω) o P µ P W Q,o l ( ωd µ1 W 1 Ω + ωd µ2 W 2 Ω + + ωd µ2nh +1W 2NH +1Ω) µ P P [ ] ( ) P,k µ 1 ( ) A Q,k n off dag = 2 Ãk n S k, µ 0, P µ #» #» P new = [ [ (d ag) #» P ] 1 old #» P old { (of f #» dag) P old #» old} ] P cells #» = E #»

Ŵ µ = E µν W ν #» m = E #» m Ŵ µ,m = E µν Wν m #» = #» new #» old #» = #» m +1 #» m.. #» #».. = 1 N.O.C N.O.C P =1 { } #» P N.O.C A.E.H µ = 1 N.O.C N.O.C P =1 { Dff P,µ } #» = 2N H + 1 error = error error error

A.E.H µ < Err µ, µ #» error #» 2N H + 1

0,µ m µ m = 0 µ = 1 #» P (dag) P (off dag) P j = 0 #» P new j = j + 1 #» P old = #» P new µ = µ + 1 j < j jacob #» m +1 = #» m + #» new #» old = #» m = m + 1 new µ < 2N H + 1 #» #».. C.C

a

V ds p d d d = p ds S w = d = p ds S w S w ds p ( ) = = p ds = p ds S w S w

S w = p n k r k ds S w k = 1, 2 F obj = 1 T T 0 dt = 1 T T 0 S w p n k r k dsdt T 1 T F obj = T 0 S w p n k r k dsdt F aug = T 0 S w p n k r k dsdt + T 0 Ω Ψ R dωdt b q δf aug = δ { T } p n k r k dsdt + δ { T } Ψ R dωdt δb q δb q 0 S }{{ w δb q 0 Ω }}{{} T 0 T 0 S w δp δb q n k r k dsdt S w p δ δb q { nk r k ds } dt

2 T Ψ W T Ψ ( W ) l dωdt A lj dωdt 0 Ω t b q 0 Ω x j b q T T 0 0 S T 0 S Ψ f k b q n k dsdt + x k Ψ R n k dsdt b q δ n k Ψ f k dsdt + δb q S T T 0 0 T 0 S w Ψ k+1 δp δb q n k dsdt f k δx σ Ψ n k dsdt S w x σ δb q Ψ k+1 p δ n k dsdt S w δb q, l = 1,..., 4 k, j, σ = 1, 2 q T Ψ W T Ψ W l dωdt A lj dωdt 0 Ω t b q 0 Ω x j b q = T 0 Ω { ( Ψ l t A Ψ ) Wl lj x j b q ( Wl } dωdt ) b q Ψ l t A Ψ lj = 0 x j T δp T f T k δp n k r k dsdt + Ψ n k dsdt + Ψ k+1 n k dsdt 0 S w δb q 0 S b q 0 S w δb q T f T } k δp = Ψ n k dsdt + { n k r k + Ψ k+1 n k dsdt 0 S b q 0 S w δb q ( ) ( ) fk δp b q δb q

Ψ = 0 S S Ψ k + n k r k = 0 Sw k+1 n δf T x T k f k δx σ = Ψ R n k dsdt Ψ n k dsdt δb q 0 S b q 0 S w x σ δb q T + p δ { nk r k ds } T } δ n k dt + {Ψ k+1 p Ψ f k dsdt 0 S w δb q 0 S w δb q ( ) xk b q S w δf T x T k f k δx σ = Ψ R n k dsdt Ψ n k dsdt δb q 0 S w b q 0 S w x σ δb q T + p δ { nk r k ds } T } δ n k dt + {Ψ k+1 p Ψ f k dsdt 0 S w δb q 0 S w δb q Ψ µ l τ ωd µνψ νl ( A lj Ψ x j ) µ = 0 µ 2N H + 1

HB Ψ µ S = 0 S HB Ψ µ k+1 n k + n k r µ k = 0 S w

M = V γ p ϱ ϱ = 1.2 kg m 3 V = 100 m s p = 10 5 P a M 0.293

γ a 3 o + 2 o sn(2πt)

t = 1.5sec c L c D

c L c D c L c D c L c D 0.04 c D η = coef HB coef T M coef T M δ = 100 η coef c L c D HB T M

1.5 t = 1.5sec t = 1.5sec

t = 2.25sec t = 2.25sec

( 1 5) ( 1 7) ϱ = 1.2 kg m 3 V = 280 m s p = 10 5 P a M = V γ p ϱ M 0.82

a 2.5 o + 1.5 o sn(1.333πt)

c L c D

c L c D

t = 1.5sec t = 1.5sec t = 1.725sec t = 1.725sec t = 1.95sec t = 1.95sec

t = 2.175sec t = 2.175sec t = 2.4sec t = 2.4sec

t = 1.95sec t = 1.95sec t = 1.95sec t = 1.95sec ( 1 8)

a = 2 o + 1.5 o sn(5t) + 2 o sn(5t)cos(15t)cos(10t) T = 1.256sec

ϱ = 1.2 kg m 3 V = 80 m s p = 10 5 P a M = V γ p ϱ M 0.234

c L c D

10 20 100 t = 0 + kt k N t = 0

( 1 3) (0, 0) (1, 0)

ϱ = 1.2 kg m 3 V = 150 m s p = 10 5 P a M = V γp/ϱ M 0.44 (α ) 4.5 o + 1.5 o sn(2.222πt)

b new q = b old q + η δf δb q η η η = 0.005 max( SD ) max( SD ) η

η HB = 165 10 10 η T M = 166 10 10

1

t = 0sec t = 0sec t = 0.18sec t = 0.18sec t = 0.36sec t = 0.36sec

t = 0.54sec t = 0.54sec t = 0.72sec t = 0.72sec (0, 0.9)

( 1 5)

N H 2N H + 1 2N H + 1

( 1 ( 1 8) 3) 3.8 17