First partial exam Sample text

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EXAMPLE 2 OF EXAM - NUMERICAL METHODS Prof. C. Vergara TIME : 1h 45' Rispondere direttamente su Forms o su un foglio di carta in base alle istruzioni indicate punto per punto. Per le risposte scritte su un foglio di carta, eseguire poi la scansione e il caricamento su Forms solo alla fine della prova, nel momento in cui vi verrà esplicitamente richiesto. IMPORTANTE: il file della scansione va nominato con il proprio codice persona + il numero della domanda, si veda sotto Pontos:7/32 Part I Quiz Answer to the following questions: Correto 1/1 Pontos Which of the following is the Lax_Wendroff scheme for a conservation law with convection given by β > 0? 1 = (1− )( − );𝑈 𝑛+1𝑗 Δ𝑡𝛽 ℎ 𝑈 𝑛𝑗+1 𝑈 𝑛𝑗 = − ( − )+ ( −2 + );𝑈 𝑛+1𝑗 𝑈 𝑛𝑗 Δ𝑡𝛽 ℎ 𝑈 𝑛𝑗 𝑈 𝑛𝑗−1 Δ𝛽2 𝑡2 2ℎ2 𝑈 𝑛𝑗+1 𝑈 𝑛𝑗 𝑈 𝑛𝑗−1 = − ( − )+ ( −2 + );𝑈 𝑛+1𝑗 𝑈 𝑛𝑗 Δ𝑡𝛽 2ℎ 𝑈 𝑛𝑗+1 𝑈 𝑛𝑗−1 Δ𝛽2 𝑡2 2ℎ2 𝑈 𝑛𝑗+1 𝑈 𝑛𝑗 𝑈 𝑛𝑗−1 = (1− )( − );𝑈 𝑛+1𝑗 Δ𝑡𝛽 ℎ 𝑈 𝑛𝑗+1 𝑈 𝑛𝑗−1 Correto 1/1 Pontos The method is unconditionally stable; It is stable only if h is small enough; Regarding the Crank-Nicolson method for the heat equation, which of the following is false? 2 29/12/2024, 17:22 EXAMPLE 2 OF EXAM - NUMERICAL METHODS https://forms.office.com/Pages/ResponsePage.aspx?id=K3EXCvNtXUKAjjCd8ope66irfrBtR-9Mn7CDwbBlTypUQkw4VjdZOUJPV002VFhSWENL… 1/8 It requires at each time step the solution of a linear system. The method is of order 2; Correto 1/1 Pontos The Forward Euler approximation of the heat equation in algebraic form is given by (A is the matrix related to the approximation of Poisson problem with +2 on the main diagonal and μ the diffusion parameter): 3 (𝐼− 𝐴) = + Δ𝑡 𝜇Δ𝑡 ℎ2 𝑈 𝑛+1 𝑈 𝑛 𝐹𝑛+1 (𝐼+ 𝐴) = + Δ𝑡 𝜇Δ𝑡 ℎ2 𝑈 𝑛+1 𝑈 𝑛 𝐹𝑛+1 = (𝐼− 𝐴) + Δ𝑡 𝑈 𝑛+1 𝜇Δ𝑡 ℎ2 𝑈 𝑛 𝐹𝑛 𝐴 = + Δ𝑡 𝜇Δ𝑡 ℎ2 𝑈 𝑛+1 𝑈 𝑛 𝐹𝑛+1 Correto 1/1 Pontos It is stable under the condition |c| ≤ (∆x /…