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Ξα 9. Consider the convection–diffusion equation ar ar a2T + u 0<x<1, at ax ax2 with the boundary conditions T(0,t)= 0 T(1,t) = 0. This equation describes propagation and diffusion of a scalar such as temper- ature or a contaminant in, say, a pipe. Assume that the fluid is moving with a
constant velocity u in the x direction. For the diffusion coefficienta = 0, the so- lution consists of pure convection and the initial disturbance simply propagates downstream. With non-zero a, propagation is accompanied by broadening and damping. Part 1. Pure convection (a = 0) Consider the following initial profile T(x,0) = 1-(10x – 1)2 for 0 < x <0.2, for 0.2 <x<1. {: Let u=0.08. The exact solution is T(x, t) = :{. – [1068 – ut) – ip eher wise – ut) 0.2 otherwise (a) Solve the problem for ( <t< 8 using (1) Explicit Euler time advancement and the second-order central differ- ence for the spatial derivative.
(c) With the results in part (a)(i) as the motivation, the following scheme, which is known as the Lax-Wendroff scheme, has been suggested for the solution of the pure convection problem Ty9+) = Ty" – (7,9. – 1) + (739. – 27+ 1) 2*), T ) + " = (Y (n1 72 2 2 where y = uAt/Ax. What are the accuracy and stability characteristics of this scheme? Repeat part (a)(i) with the Lax-Wendroff scheme using y = 0.8, 1, and 1.1. Discuss your results using the modified equation analysis.