u(x, t) = u 0 (x ct). This Riemann invariant u is constant along characteristics λ with x = x 0 +ct (u(x, t) = u 0 (x 0 )):

Transcription

1 x, t ), h x The Frst-Order Wave Eqato The frst-order wave advecto) eqato s c > 0) t + c x = 0, x, t = 0) = 0x). The solto propagates the tal data 0 to the rght wth speed c: x, t) = 0 x ct). Ths Rema varat s costat alog characterstcs λ wth x = x 0 +ct x, t) = 0 x 0 )): d = dt λ t + dx dt x = t + c x = 0. Note that gog backwards tme t t smply propagates the tal data to the left wth speed c. method type order stablty effect pwd explct frst t x/c dffsve Lax-Fredrchs explct frst t x/c dffsve Lax-Wedroff explct secod t x/c dspersve Table 1: Nmercal methods for the wave eqato t + c x = 0. The pwd mercal method +1 t = c 1 x +1 = c t x 1) s frst-order accrate ad stable for t x/c CFL codto).

2 Cosstecy: The LTE = t τ s gve by x, t + t) = x, t) c t x, t) x h, t)) + t τ. h Taylor expadg, we get + t t + t2 2 tt + = c t x h ) 2 xx + τ = t 2 tt ch 2 xx + + t τ Stablty: To aalyze stablty, we wll derve the modfed eqato for the pwd method. I the Taylor expaso above wth the LTE o the RHS, we delete the LTE ad the do ot se the wave eqato: + t t + t2 2 tt + = c t x h 2 xx + or t + c x = ch 2 xx t 2 tt + c 2 h c t) xx D m xx makg se of the LHS t c x to leadg order. Ths modfed eqato shows the leadg order effects of the mercal method o the orgal PDE. The modfed eqato s stable for t h/c, sce that case the mercal dffso coeffcet D m 0. Ths dervato s ot strctly speakg a proof of stablty, bt t does strogly sggest that pwd s stable for the wave eqato.) Note that pwd s dffsve sce the leadg order effect of the method o the wave eqato s to trodce the dffsve xx term. The codto that t h/c for stablty s called the CFL codto. It says that the doma of depedece of the PDE mst be clded the doma of depedece of the dfferece method. The CFL codto s ecessary bt ot sffcet for stablty see the FTCS scheme below). The dowwd method whch volates the CFL codto, ad s oly sed by mstake) +1 = c t ) x +1 s codtoally stable, sce ow for the modfed eqato we have t + c x = ch 2 xx t 2 tt + c 2 h + c t) xx D m xx 2 )

3 ad D m < 0, makg the modfed eqato eqvalet to the always stable) backward heat eqato. Smlarly the cetral dfferece FTCS forward tme cetral space) scheme +1 = c t 2 x +1 1) s codtoally stable. For the modfed eqato we have or to leadg order ad D m < 0. ) + t t + t2 2 tt + = c t x + h2 6 xxx + The Lax-Fredrchs LF) method t + c x = c2 t 2 xx D m xx +1 = ) t c 2 x +1 1) s frst-order accrate, stable for t x/c, ad coservatve. The modfed eqato for LF t + c x = ch 2r 1 r2 ) xx D m xx, r = c t h shows that t s dffsve. r s called the Corat mber. Note that D m 0 whe r 1. The Lax-Wedroff LW) method +1 = c t 2 x +1 1) 1 t2 ) + c2 2 x s secod-order accrate, stable for t x/c, ad coservatve. derved by Taylor expadg It s t + t) + t t + t2 2 tt = c t x + c 2 t2 2 xx 3

4 sg t = c x, ad the replacg the x dervatves wth the three-pot cetral dfferece approxmatos. Sce LW agrees wth the Taylor seres expaso throgh secod order ad we sed secod-order accrate cetral dervatves x), t s secod-order accrate. The modfed eqato for LW s t + c x = ch2 6 r2 1) xxx ɛh 3 xxxx wth ɛ = cr1 r 2 )/8 0 whe r 1, sggestg that LW s CFL stable. Note that LW s dspersve sce the leadg order effect of the method o the wave eqato s to trodce the dspersve xxx term. For olear hyperbolc coservato laws w t +fw) x = 0, the two-step Lax- Wedroff method ca be sed. Frst, termedate soltos are compted at + 1, ± 1 sg Lax-Fredrchs: 2 2 w = 1 2 w + w+1) t ) f 2 x +1 f w = 1 ) w w t ) f 2 x f 1. The the ew solto at tme level + 1 s compted from these two termedate soltos sg leapfrog: w +1 = w t f x f Two-step LW s secod-order accrate, stable for t x/c, ad coservatve. It ca be derved as a fte volme method sg the 2D Gass Theorem to dscretze w t + fw) x = 0. ). The Secod-Order Wave Eqato The secod-order) wave eqato s c > 0) 2 w t 2 c2 2 w x 2 = 0, wx, t = 0) = w 0x), w t x, t = 0) = v 0 x). The solto propagates the tal data to the left ad rght wth speed c: wx, t) = fx ct) + gx + ct). 4

5 The Rema varat f s costat alog characterstcs dx/dt = c ad the Rema varat g s costat alog characterstcs dx/dt = c. Aga ote that gog backwards tme t t smply propagates the tal data oppostely to the left ad rght wth speed c. The D Alembert solto to the wave eqato expresses the Rema varats f ad g terms of the tal codtos: wx, t) = 1 2 w 0x ct) w 0x + ct) + 1 2c x+ct x ct v 0 s)ds. To solve the wave eqato mercally, we covert t to a frst-order system = cw x, v = w t ): t = cv x v t = c x wth x, t = 0) = cw 0x), vx, t = 0) = v 0 x). Vo Nema Stablty Upwd Method for t + c x = 0: Set j eqal to a sgle Forer mode j = e kx j, ad derve the growth factor +1 j = Gk) j for ths mode. ff +1 j = j r j j 1), r = c t/h +1 j = 1 r 1 e kh)) e kx j = Gk) j Gk) = 1 r + re kh Gk) 2 = 1 r) 2 + r 2 + r1 r) e kh + e kh) = 2r 2 2r r1 r) coskh) 1 Ths pwd s stable ff t x/c. r 1)1 coskh)) 0 ff r 1. FTCS for t + c x = 0: Satsfes the CFL codto, bt s codtoally stable. +1 j = j r 2 j+1 j 1) 5

6 wheever skh) j = 1 r e kh e kh)) e kx j = Gk) j 2 Gk) = 1 r skh) Gk) 2 = 1 + r 2 s 2 kh) > 1 Coservatve Hyperbolc Methods For hyperbolc coservato laws, the Lax-Fredrchs ad Lax-Wedroff methods are coservatve, whle the pwd method s ot pwd s coservatve for t + c) x = 0 oly f c does ot chage sg). Two-step Lax-Wedroff s mafestly coservato form. For Lax-Fredrchs, the mercal flx for w t + fw) x = 0 s F = 1 2 fw ) + fw +1 )) x 2 t w +1 w ). 6