120 q q q 10121a let denote the jump in a quantity

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Unformatted text preview: uation (10.1.21b) is called the Rankine-Hugoniot jump condition and the discontinuity is called a shock wave. We can use the Rankine-Hugoniot condition to nd a discontinuous solution of Example 10.1.5. Example 10.1.6. For t < 1, the discontinuous solution of (10.1.16, 10.1.18) is as given in Example 10.1.5. For t 1, we hypothesize the existence of a single shock wave, passing through (1 1) in the (x t)-plane. As shown in Figure 10.1.9, the solution of Example 10.1.5 can be used to infer that u; = 1 and u+ = 0. Thus, f (u;) = (u;)2 =2 = 1=2 and f (u+) = (u+)2=2 = 0. Using (10.1.21b), the velocity of the shock wave is _ = 1: 2 Integrating, we nd the shock location as = 1 t + c: 2 14 Hyperbolic Problems ξ = (t + 1)/2 t 1 0 x 1 Figure 10.1.9: Characteristics and shock discontinuity for Example 10.1.6. u(x,0) 0 u(x,1/2) 1 2 x 0 u(x,1) 0 1 2 x u(x,3/2) 1 2 x 0 1 2 x Figure 10.1.10: Solution u(x t) of Example 10.1.6 as a function of x at t = 0, 1/2, 1, and 3/2. The solution is discontinuous for t > 1. Since the shock passes through (1 1), the constant of integration c = 1=2, and = 1 (t + 1): 2 (10.1.22) 10.1. Conservation Laws 15 The characteristics and shock wave are shown in Figure 10.1.9 and the solution u(x t) is shown as a function of x for several times in Figure 10.1.10. Let us consider another problem for Burgers' equation with di erent initial conditions that will illustrate another structure that arises in the solution of nonlinear hyperbolic systems. Example 10.1.7. Consider Burgers' equation (10.1.16) subject to the initial conditions 8 < 0 if x < 0 0 u (x) = : x if 0 x < 1 : (10.1.23) 1 if 1 x Using (10.1.17) and (10.1.23), we see that the characteristic passing through (x0 0) satis es 8 if x < 0 < x0 x = : x0(1 + t) if 0 x < 1 : (10.1.24) x +t if 1 x 0 These characteristics, shown in Figure 10.1.11, may be used to verify that the solution, shown in Figure 10.1.12, is continuous. Additional considerations and di culties with nonlinear hyperbolic systems are discussed in Lax 20]. Examp...
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