By S L Sobolev

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62), let Γ2 (η) = Ω2 (η) so that it integrates and gives Ω2 (η) β(x, t) = B(t)Ω2 (η). 67) Γ2 (η) ≡ 0. 67), and integrate with respect to x to get B (t) α(x, t) = A(t) − xt + B(t)Ω3 (η). 69) and have α(x, t) = A(t) − xt B (t) . 70) Since Γ3 (η) = Ω3 (η), we also have Γ3 (η) ≡ 0. 61), we get t + A(t) − xt B (t) − B(t)Γ1 (η) ηx + tηt = 0. 72) are dt dη dx = = . 73) is clearly η = constant; this is the similarity variable. 73), we have dx B (t) A(t) − lB(t) + x=1+ . 75) gives the similarity variable η = xB(t) − ©2000 CRC Press LLC 1+ A(t) − lB(t) B(t)dt .

Again, the analytic results of Murray (1970) were confirmed numerically. 1) with g(u) = uα , h(u) = uβ . With this choice it becomes possible to find explicit solutions for many cases either by the method of characteristics or by reduction to an ODE via similarity analysis. Apart from finding explicit solutions, the concern here is to demonstrate the limiting nature of the similarity solution. We follow the work of Bukiet, Pelesko, Li, and Sachdev (1996). An important contribution of this paper is the development of a characteristicbased numerical scheme for nonlinear scalar hyperbolic equations, which involves the solving of ODEs.

11), we have 1 α α+1 t hX xs = (α + 1) . 15) x > xs , t ≥ t0 . ii) λ = 0, β = 1. 16) so that du dx + λu = 0 along the characteristic curves = uα . 17) gives u = u0 e−λt where u = u0 at t = 0. Since u varies from 0 to h in the rarefaction wave, decay of the height of the top hat is given by max u = he−λt . 18) where we have inserted the 1C xF = 0 at t = 0. 19) which, on integration and use of 1C xs = X at t = 0, gives xs = X + hα (1 − e−λαt ). 20) gives the shock trajectory. 21) yielding t0 = − 1 (α + 1)λX ln 1 − .