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 )):

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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 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 = ch3 8 r1 r2 > 0 whe r < 1, sggestg that t s 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 sds. 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. t for Lear vs. Nolear Hyperbolc PDEs For lear hyperbolc PDEs, we typcally get the exact solto projected oto the grd f the CF L factor s set to 1 t = CF L x c max, 0 < CF L 1 where c max s the maxmm characterstc speed magtde. For olear hyperbolc PDEs, CF L = 1 s ofte comptatoally stable thogh theoretcally stable the x 0 lmt, ad sally we take 0.1 CF L 0.9. Whe dfferet CF L factors gve stable reslts for the olear problem, the soltos geerally are the same graphcally to wth a le wdth. For lear hyperbolc PDEs, CF L < 1 leads to excessve dffso or dsperso of waves. 5