es MNIBIO 21gen25 part II III correzione

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12 points Exercise 1. - PART II - MATLAB: Take Ω = (−1,3), µ = 0.01, β = 10, boundary conditions u(−1) = 2, u(3) = 1, consider the following problem −µ u′′ + β u′ = x2 and the function esame provided by the form. (a) (2 points) Describe in the box below the Centered scheme applied to the problem introduced above. Use the notation U_j to represent Uj (explaining their meaning). Report also the matrix/vector representation. Solution: Subdivide the space domain [ −1,3] in nodes xj = −1 +jh, j = 0, . . . ,M, with h = 3−(−1) M , and let Uj ≃ u(xj) be the approximation of the continuous solution. Then, the Centered scheme applied to the problem above reads: −µ Uj+1 − 2Uj + Uj−1 h2 + β Uj+1 − Uj−1 2h = x2 j , j = 1, . . . ,M − 1. In matrix/vector representation, we have:  µ h2 A + β 2h B  U = F , where (U)j = Uj, F = h x2 1 + 2µ h 2 + 2β 2h , . . . , x2 j , . . . , x2 M −1 + 1µ h 2 − 1β 2h iT , Aii = 2, Ai,i−1 = Ai,i+1 = −1 and zero elsewhere, Bii = 0, Bi,i−1 = −1, Bi,i+1 = 1 and zero elsewhere. (b) (1 point) Run the proposed code with the Centered scheme, h = 0.2 and the data reported in the text, writing in the box below all the instructions used to call the function. Solution: Function esame wants as input the expression of variable ϕ which regulates the stabiliza- tion. In this case, we do not have any stabilization, thus ϕ = 0. The Matlab instructions to call and run the function are: a = -1; b = 3; mu = 0.01; beta = 10; ua = 2; ub = 1; f = @(x) x.^2; h = 0.2; phizero = @(x) 0*x; % no stab [x, u] = esame(a, b, h, ua, ub, f, mu, beta, phizero); (c) (1 point) Run the proposed code with the Upwind method, h = 0.2 and with the data reported in the text, writing in the box below all the instructions used to call the function. Solution: In this case we want to add a…