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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 coefficient a = 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 1-(10x - 1)2 for 0 < x <0.2, T(x,0) = 0 for 0.2 <x< 1. Let u = 0.08. The exact solution is T(x, t) = S 1- [10(x – ut) – 1]? for 0 < (x – ut) <0.2, 0 otherwise (a) Solve the problem for 0 <t<8 using (i) Explicit Euler time advancement and the second-order central differ- ence for the spatial derivative. (ii) Leapfrog time advancement and the second-order central difference for the spatial derivative. Plot the numerical and exact solutions for t = 0, 4, 8. You probably need at least 51 points in the x direction to resolve the disturbance. Discuss your solutions and the computational parameters that you have chosen in terms of what you know about the stability and accuracy of these schemes. Try several appropriate values for u At/Ax.