7 1 () Brgers 1.1 a + b y + c y + d + e + f + g =0. (1.1) y b 4ac > 0 t c =0 b 4ac =0 t = κ b 4ac < 0 + y =4πGρ
8 1 1., y y (, +1) (-1, ) (, ) (+1, ) y (, -1) 1.1: (,y ) y y ±1 = ± y ±1 = y ± y (, ), = (,y ) (,y ) +1, = ( +, y )=, + 1, = (, y )=, (1.) (1.3) ( ) +! ) ( +1, 1, = +! ( ) ( ) + 3 3! 3 3! ( 3 ) +... (1.) 3 ( 3 3 ) +... (1.3) ( ) + O( 3 ) (1.4) ( ) = +1, 1, + O( ) (1.5) (, ) ( / ) ( ) ( ) = +1, 1, (1.6)
1.3. 9 1 ( ) ( ) (1.) (1.3) = +1,, =, 1, () (1.7) () (1.8) ( +1, + 1, =, + ) + O( 4 ) (1.9) ( ) = +1,, + 1, + O( ) (1.10) ( / ) ( ) = +1,, + 1, (1.11) ( ) y =,+1, +, 1 y. (1.1) 1.3 1.3.1 t + c = 0 (1.13) c c>0 c (1.13) (, t) =( ct, 0) (1.14)
10 1 t = 0 t > 0 c 1 c 1 1.: 1 t>0 t =0 ct 1. 0 = 1 <0 = =0 t>0 1.3. FTCS 1 (1.13) t n t t n+1 = t n + t n = (,t n ) n + c n +1 n 1 = 0 (1.15) t FTCS (Forward n Tme and Centered Dfference n Space) = n 1 ν(n +1 n 1 ) (1.16) ν ν c t (1.17) 1.16 t n t n+1 = t n + t t n 1 (t n+1 ) FTCS 1.3 t n+1 t n 1 1. ( )
1.3. 11 t bondary n+1 n -1 +1 1.3: FTCS. (t =0) () 3. t t end (a) t ( ) FTCS (1.16) (b) ( ) (c) t 3 3 1 1 0 0-1 40 45 50 55 60-1 0 0 40 60 80 100 1.4: FTCS =1,..., 50 =1 =51,..., 100 =0 ν = c t/ =0.5 1 3 1 50 100 FTCS 1.4
1 1 1.3.3 FTCS Von Nemann FTCS 1 1.4 n = cos(θ) (1.18) (1.16) θ k θ = k θ = π n 1.5 1 θ = π/3 6 1 θ = π θ = π/3 =0 1 3 4 =0 1 3 4 5 6 1.5: n θ = π/3 = cos(θ) : θ = π = cos(θ)+ νsnθsn(θ) (1.19) n+ = (1 ν sn θ)cos(θ)+νsnθsn(θ) (1.0) = Re [ (1 νsnθ) e θ] (1.1) Re n+k = Re [ (1 νsnθ) k e θ] (1.) ( ) (1.18)
1.3. 13 n = gn e θ (1.3) g 1 g 1 Von Nemann n = gn ep(θ) FTCS g = 1 1 ν(eθ e θ ) (1.4) = 1 νsnθ (1.5) g =1+ν sn θ 1 (1.6) θ =0 FTCS FTCS t n+1 = t n + t ±1 = ± = n + t t + 1 t t +... (1.7) n +1 = n + + 1 + 1 3 6 3 3 +... (1.8) n 1 = n + 1 1 3 6 3 3 +... (1.9) FTCS (1.7) (1.8) (1.9) n = 1 ν(n +1 n 1 ) (1.30) t t + 1 ( t t = ν + 1 3 ) 6 3 3 +... (1.31) t + c = 1 t t c 3 6 3 +... (1.3)
14 1 t = c t (1.33) = c (1.34) t + c = c t c 6 3 3 +... (1.35) 1 FTCS 1.3.4 La-Fredrch FTCS n (n +1 + n 1 )/ n = g n ep(θ) g = 1 (n +1 + n 1) ν (n +1 n 1) (1.36) g = 1 (eθ + e θ ) 1 (eθ e θ ) (1.37) = cosθ νsnθ (1.38) g = cos θ + ν sn θ (1.39) 1.6 g θ (g, θ) La-Fredrch ν = c t/ ν 1 Corant, Fredrch, Lewy (CFL ) (1.36) = 1 ν n +1 + 1+ν n 1 (1.40) ν 1 t = t n+1 n 1 +1 n +1 t = t n 1
1.3. 15 ν = 1 ν=0.5 g -1 θ 1 1.6: La-Fredrch t bondary n+1 n c t -1 +1 1.7: La-Fredrch ν<1 ν >1 1.7 La-Fredrch t = t n+1 t = t n ν = c t/ <1 ν>1 c t < t n 1 n +1 t > / c 1 +1 1.8 La-Fredrch La-Fredrch 1.3.5 1 1.9 1
16 1 3 1 50 100 0-1 0 0 40 60 80 100 1.8: La-Fredrch ν =0.5 50 100 t bondary c > 0 n+1 n c t -1 +1-1 +1 1.9: 1 1 c >0 n t + c n n 1 = 0 (1.41) = n c t (n n 1 ) (1.4) 1.9 1 t n+1 t n t n+1 g = 1 ν(1 e θ ) (1.43)
1.3. 17 = (1 ν + νcosθ) νsnθ (1.44) g = (1 ν + νcosθ) + ν sn θ (1.45) = 1 ν(1 ν)(1 cosθ) (1.46) 0 ν 1 θ g 1 1.4 =(1 ν) n + νn 1. (1.47) 0 ν 1 t = t n+1 t = t n 1.9 ( c t, t n ) n 1 n 3 3 1 1 50 100 16 3 0 0-1 0 0 40 60 80 100-1 0 0 40 60 80 100 1.10: 1 ν =0.5 50 100 ν =0.80 16 3 1.10 ν 1,0.75,0.5 1 g θ ( g,θ) 1 (1.4) 1 t + c = 1 c (1 ν) 1 6 c( ) (ν 3ν +1) 3 +... (1.48) 3
18 1 1.3.6 La-Wendroff La-Wendroff = n + t t + 1 t t + O( t3 ) (1.49) 3 / t = c / / t = c / = n c t + 1 c t + O( t3 ) (1.50) / / = n 1 c tn +1 n 1 + 1 ( ) t c ( n +1 n + n 1) (1.51) La-Wendroff La-Wendroff La-Wendroff von Nemann g = 1 ν (eθ e θ )+ ν (eθ +e θ ) (1.5) = 1 νsnθ + ν cosθ ν (1.53) g = [1 ν (1 cosθ)] + ν sn θ (1.54) = 1 ν (1 ν )(1 cosθ) (1.55) ν 1 θ g 1 1 La-Wendroff La-Wendroff / +1/ = n +1 + n = n c t 1 c t (n +1 n ) (1.56) (n+1/ +1/ n+1/ 1/ ) (1.57) 1.11 1.1 La-Wendroff 1 La-Wendroff Godnov 1 / t + c / =0
1.3. 19 t bondary n+1 n+1/ n -1 +1 1.11: La-Wendroff t n 1 +1 t n+1/ 1/ +1/ t n+1 = k a k n +k (1.58) t n (, t n ) t n+1 (, t n+1 ) 0 ν 1 n 1 n n 1 n n +1 +1 Godnov (1994) κ ũ n+1 = + κ t (n +1 n + n 1) (1.59) κ Q v κ +1/ = Q v n +1 n (1.60)
0 1 3 3 1 1 50 100 16 3 0 0-1 0 0 40 60 80 100-1 0 0 40 60 80 100 1.1: La-Wendroff ν =0.5 50 100 ν =0.80 16 3 1.4 1 t + c = 0 (1.61) t + f = 0 (1.6) f = c (1.63) (1.6) 1.13 ( 1/ << +1/ ) (1.6) = 1/ = +1/ +1/ d + f( +1/ ) f( 1/ )=0. (1.64) t 1/ n = +1/ 1/ (, t n )d (1.65) ±1/ f ±1/ = n t (f n +1/ f n 1/) (1.66)
1.5. Brgers 1 f -1/ f +1/ -1 +1 1.13: (1.66) f n ±1/ f n ±1/ FTCS La-Fredrch f n +1/ = 1 (f n +1 + f n ) (1.67) f n +1/ = 1 [ (1 1 ν )f +1 n +(1+ 1 ] ν )f n (1.68) Upwnd () f +1/ n = 1 [ (f n +1 + f n ) c (n +1 n )] (1.69) c >0 f n +1/ = f n c<0 f +1/ = f n +1 La-Wendroff f +1/ n = 1 [ ] (1 ν)f n +1 +(1+ν)f n (1.70) 1.5 Brgers c Brgers t + = 0 (1.71)
1 d/dt = / t + / d dt = 0 (1.7) = 1.14 + 1.14: Brgers 1.01 0 16 3 48-1.01 48 3 16 0 1.00-1.00 0.99-0.99-40 -0 0 0 40 60 80 100-40 -0 0 0 40 60 80 100 1.15: Brgers >0 <0 t/ =0.8 Brgers
1.5. Brgers 3 t + ( ) = 0 (1.73) f() f() = / c c>0 f n +1/ = f n +1/ ( (t)+ +1 (t))/ +1 (t)+ (t) > 0 f n +1/ = f n = (t) / +1 (t)+ (t) 0 f n +1/ = f n +1 = +1 (t) / 1.15 () =1+ɛsn(k) (ɛ =0.01, 0 k π) Brgers 1.15 1 c =1 1.15 1 Brgers 1.16 () =1+0.1sn(k) Brgers / 1.15 1.10 0 3 64 96 18 160 1.05 1.00 0.95 0.90 0.85 0 50 100 150 00 1.16: () =1+0.1sn(k) Brgers t/ =0.8
4 1 1.6 Brgers Godnov La-Wendroff La-Wendroff f n +1/ = c[n + 1 (1 ν)(n +1 n )] (1.74) 1 c>0 f n +1/ = c La- Wendroff 1 La-Wendroff f n +1/ = c[ n + 1 (1 ν)b +1/( n +1 n )] (1.75) B +1/ (1.75) (1.66) n n 1 n = ν[1 1 (1 ν)b 1/]+ 1 ν(1 ν)b +1/ r (1.76) r n n 1 n +1 n n -1 (1.77) n+1 n 1.17: 1.17 n n 1 (1.76)
1.6. 5 0 n+1 n n 1 n (1.76) 1 (1.78) ν B 1/ B +1/ r 1 ν (1.79) CFL 0 ν 1 B 1/ B +1/ r (1.80) 0 B +1/ (1.81) 0 B +1/ r (1.8) 1.18 r <0 B +1/ =0 B +1/ B = r 1 LW mnmod O 1 r 1.18: B +1/ (r) LW La-Wendroff mnmod mnmod La-Wendroff B +1/ =1( LW) r <1/
6 1 mnmod ( mnmod) 0 (r<0) mnmod(r) = r (0 r 1) 1 (r>1) (1.83) 1.7 TVD 1 U = d d. (1.84) d du/dt =0 Total Varaton (TV) TV( n ) n +1 n (1.85) Total Varaton TV( ) TV( n ) (1.86) Total Varaton Dmnshng (TVD) TVD 1.8 T t = κ T B t = η B
1.8. 7 1 t =0 (, t) (, 0) t (> 0) (, t) t = κ (1.87) κ t = 0 t > 0 1.19: (eplct ) 1 (1.87) t n t t n+1 = t n + t (FTCS n = (,t n ) n t (1.88) = κ n +1 n + n 1 (1.88) = n + κ t (n +1 n + n 1 ) (1.89) t n t n+1 = t n + t t n 1 (t n+1 ) (
8 1 Von Nemann FTCS n = g n ep(θ) FTCS g n+1 e θ = g n e θ + κ t gn [e (+1)θ e θ + e ( 1)θ ]. (1.90) g =1 κ t (1 cosθ). (1.91) g 1 1 1 κ t (1 cosθ) 1. (1.9) 0 κ t (1 cosθ) =κ t θ sn 1. (1.93) θ ( ) 0 κ t 1. (1.94) FTCS 1 t t 1/4 (mplct ) t n+1 (mplct) eplct Crank-Ncolson λ n t = κ [λ n+1 +1 + 1 +(1 λ) n +1 n + ] n 1 (1.95) A = b( n ). (1.96) 1. A b
1.8. 9. Von Nemann λ >1/ κ t/ > 0 t Crank-Ncolson (1) (1994) () (000) (3) Nmercal Comptaton of Internal and Eternal Flows, C. Hrsch, John Wley & Sons, 1990 (1)