es MNIBIO 6Feb25 part II III correzione

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12 points Exercise 1. Take Ω = ( −L,L), L = 0 .5, µ = 0 .1, the initial time t0 = 0, the final time T = 0 .5, boundary conditions u = 0 at x = ±L, and the following initial condition u0 = 1 − cos(2πx/L). (a) (2 points) Describe in the box below the Forward Euler/Centered method applied to the problem introduced above, specifying also the continuous formulation of the latter. Report also the ma- trix/vector representation. Use the notation U^n_j to represent U n j , U^n+1_j to represent U n+1 j , etc, (explaining the meaning of U n j , .... ). Solution: The continuous problem is the following heat equation: ∂u ∂t − 0.1 ∂2u ∂x2 = 0, t ∈ (0,0,5], x ∈ (−0.5,0.5), u(0,x) = 1 − cos(2πx/0.5), u(t, − 0.5) = u(t,0.5) = 0. The Forward Euler/Centered method reads as follows: U n+1 j − U n j ∆t − 0.1 U n j+1 − 2U n j + U n j−1 h2 = 0, n = 0,1, . . . ,N − 1, j = 1, . . . ,M − 1, where U n j ≃ u(tn,xj) is the numerical approximation at instant tn and node xj, with tn = n∆t, n = 0,1, . . . ,N, xj = −0.5 +j h, j = 0,1, . . . ,M, ∆t and h being the time and space discretization param- eters. Introducing matrix A such that Aii = 2, Ai,i−1 = Ai,i+1 = −1, zero elsewhere, we have the following matrix/vector representation of the FE/C method: U n+1 =  I − 0.1 ∆t h2 A  U n, where vectors U n collect componets U n j . (b) (2 points) Run the proposed code for the Forward Euler/Centered method with ∆ t = 0.01 over 50 space intervals, writing in the box below all the instructions used to call the function. Solution: The Matlab commands are as follows. Note the value of the theta variable set equal to zero in the last argument, in accordance with the FE/C method: L = 0.5; t0 = 0.0; T = 0.5; mu = 0.1; u0 = @(x) 1 - cos(2*pi*x/L); M = 50; h = 2*L/M; dt = 0.01; [x,t,u] =…