05 02 19

Pagina 1

16 punti Exercise 1. Take h = 0 .1, ∆t = 0 .05, Ω = ( − 4,4], the initial time t0 = 0, the final time Tf = 1 .0, boundary condition u = 0 at x = − 4, and the following initial condition u0 = cos 2(π x), |x| ≤ 0.5 and zero elsewhere. Consider the following function function [u]=esame(h,dt,Tf,x0,xf,Uinflow,funU0) a = 2.0; lambda = a*dt/h; Nh = floor((xf-x0)/h)+1; x = linspace(x0,xf,Nh); u0 = eval(funU0); uold = u0; u = zeros(size(u0)); Nt=floor(Tf/dt)+1; for n = 1:Nt; t = n*dt; u(2:Nh-1) = ... ; u(1) = Uinflow; u(Nh) = (1-lambda)*uold(Nh) + lambda*uold(Nh-1); uold = u; plot(x, u, ’linewidth’,2); disp(sprintf(’t = %f \n’, t)); pause; end (a) (4 punti) Introducing the suitable notation, describe the Forward Eu ler method (2p) and the continuous problem that is discretized (2p). Version n. 1 – Page 2 Scaricato da Chiara Mk ([email protected]) lOMoARcPSD|3743663 (b) (6 punti) Complete and run the proposed code, writing all the instruct ions used in Matlab to run it (2p). Report the values of the numerical solution at ti me T=1 for x = 2 and x = 3 (2p). Comment the result in view of the theory (2p). (c) (6 punti) Introduce the Upwind method (1p) and modify the code to implem ent it (1p). Report the plot of the solution at time t = 1 .0 on the paper with suitable scales and legend (2p). Com- ment the results in view of the theory (2p). Version n. 1 – Page 3 Scaricato da Chiara Mk ([email protected]) lOMoARcPSD|3743663 16 punti Exercise 2. Consider the following advection-diffusion problem: − µ △ u + β · ∇ u = f x ∈ Ω, with u = 0 for x ∈ ΓD, µ∇ u ·n = ψ for x ∈ ΓN , ΓD ∪ ΓN = ∂Ω, ΓD ∩ ΓN = ∅. (a) (4 punti) Derive the weak formulation of the proposed problem (1p), sp ecifying the functional space (1p), the integration by parts (1p) and the treatment o f the boundary…